# All important topics and formula's for JEE 2020-With Explanation

## ALL TOPICS WITH FORMULA'S

Here I am covering all the important formulas for JEE-2020/jee main/jee adavance preparation,expected jee 2020 questions.
JEE advance 2020 date June 20 to June 22

JEE 2020 MATHS

### 1. Complex Number 2. Theory of Equation (Quadratic Equation) 3. Sequence & Progression(AP, GP, HP, AGP, Spl. Series) 4. Permutation & Combination 5. Determinant 6. Matrices 7. Logarithm and their properties 8. Probability 9. Function 10 Inverse Trigonometric Functions 11. Limit and Continuity & Differentiability of Function 12. Differentiation & L' hospital Rule 13. Application of Derivative (AOD) 14. Integration (Definite & Indefinite) 15. Area under curve (AUC) 16. Differential Equation 17. Straight Lines & Pair of Straight Lines 18. Circle 19. Conic Section (Parabola 30, Ellipse 32,  Hyperbola 33)          20. Binomial Theorem and Logarithmic Series 21. Vector & 3-D 22. Trigonometry-1 (Compound Angle) 23. Trigonometry-2 (Trigonometric Equations & Inequations)24. Trigonometry-3 (Solutions of Triangle) 25. Syllbus IITJEE Physics, Chemistry, Maths & B.Arch 26. Suggested Books for IITJEE

1.COMPLEX NUMBERS

1. DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b ∈ R & i = −1.  It is  denoted by  z  i.e.  z = a + ib. ‘a’  is called as real part of  z (Re z) and ‘b’ is called as imaginary part of  z (Im z).

EVERY  COMPLEX  NUMBER  CAN  BE  REGARDED  AS

Purely real Purely imaginary      Imaginary    if b = 0       if a = 0         if b ≠ 0

Note :

(a) The set  R  of real numbers is a proper subset of the Complex Numbers. Hence the  Complete Number system  is  N ⊂  W ⊂  I  ⊂  Q  ⊂  R  ⊂  C.

(b) Zero is both purely real as well as purely imaginary but not imaginary.

(c) i = −1 is called the imaginary unit. Also  i² = − l  ;  i3 = −i  ;   i4 = 1  etc.

(d) a b = ab only if atleast one of either a or b is non-negative.

2. CONJUGATE  COMPLEX :

If  z = a + ib  then  its  conjugate  complex  is  obtained  by  changing  the  sign  of  its  imaginary part  & is denoted by z .  i.e.  z = a − ib.

Note  that  :

(i) z + z =  2 Re(z) (ii) z − z =  2i Im(z) (iii) z z = a² + b²   which  is  real

(iv) If  z  lies  in  the 1st quadrant then z lies in the 4th quadrant and − z lies in the 2nd  quadrant.

3. ALGEBRAIC  OPERATIONS :

The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if  we say that complex number is positive or negative.

e.g.   z > 0,  4 + 2i < 2 + 4 i   are  meaningless .

However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers,

z12 + z22 = 0 does not imply z1 = z2 = 0.www.bloggermayank.online

4. EQUALITY IN COMPLEX NUMBER :

Two complex  numbers  z1 = a1 + ib1  &  z2 = a2 + ib2  are  equal  if  and  only  if  their  real  & imaginary parts coincide.

5. REPRESENTATION  OF  A  COMPLEX  NUMBER  IN  VARIOUS  FORMS: (a) Cartesian Form (Geometric Representation) :

Every complex number   z = x + i y  can be represented by a point on the cartesian plane  known as  complex plane (Argand diagram) by the  ordered pair (x, y). length OP is called modulus of the  complex number denoted  by z & ΞΈ is called the argument or amplitude eg          . z = x2 + y2

ΞΈ = tan−1  y (angle made by OP with positive x−axis)

x

NOTE  :(i) z is always non negative . Unlike real numbers z = −zz if zif z <> 00  is not correct

(ii) Argument of a complex number is a many valued function . If  ΞΈ  is the argument of a complex number then  2 nΟ + ΞΈ  ;  n ∈ I will also be the argument of that complex number. Any two  arguments of a complex number differ  by 2nΟ.

(iii) The unique value of ΞΈ  such that  – Ο < ΞΈ ≤ Ο is called the principal value of the argument.

(iv) Unless otherwise stated, amp z  implies principal value of  the argument.

(v) By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i  the argument is not defined and this is the only complex number which is given by its modulus.www.bloggermayank.online

(vi) There exists a one-one correspondence between the points of the plane and the members  of the set of complex numbers.

(b) Trignometric / Polar  Representation :

z = r (cos ΞΈ + i sin ΞΈ)   where | z | = r  ;   arg  z  =  ΞΈ  ;   z =  r (cos ΞΈ − i sin ΞΈ) Note: cos ΞΈ + i sin ΞΈ  is also written as  CiS ΞΈ.

eix +e−ix eix −e−ix

Also cos x =    &  sin x =   are known as Euler's identities.

2 2

(c) Exponential  Representation :

z = reiΞΈ  ; | z | = r   ;   arg z  =  ΞΈ   ;   z = re− iΞΈ

6. IMPORTANT  PROPERTIES  OF CONJUGATE / MODULI / AMPLITUDE :

If  z ,  z1 ,  z2 ∈ C  then   ;

(a) z + z = 2 Re (z)    ;    z − z = 2 i Im (z)    ;    (z) = z     ;    z1+z2  = z1 + z2  ;

 z 

z1 −z2 =  z1 − z2    ;   z z1 2 = z1 . z2     z12  = zz12   ;   z2 ≠ 0

(b) | z | ≥ 0  ;  | z | ≥  Re (z)  ;    | z | ≥ Im (z) ;    | z | = | z | = | – z | ;    zz = | z |2 ;

z1 |z |1 ≠ 0 ,   | zn | = | z |n   ;

| z1 z2 | = | z1 | . | z2 |       ;     z2 = | z |2 ,  z2

| z1 + z2 |2 + | z1 – z2 |2 = 2  [| z |1 2 + | z |2 2]www.bloggermayank.online

z1− z2  ≤  z1 + z2  ≤  z1 + z2[ TRIANGLE  INEQUALITY ] (c) (i) amp (z1 . z2) =  amp  z1 + amp z2 + 2 kΟ. k ∈ I

 z1 

(ii) amp   z2  = amp z1 − amp z2 + 2 kΟ   ;     k ∈ I

(iii) amp(zn) = n amp(z)  +  2kΟ .

where proper value of  k  must be chosen  so  that  RHS  lies  in  (− Ο , Ο ].

(7) VECTORIAL  REPRESENTATION OF A COMPLEX :

Every complex number can be  considered as  if  it is  the  position vector of that point.  If the point  P

represents the complex number  z  then,  OP = z   &   OP  =  z.

NOTE  :(i) If OP = z = r ei ΞΈ  then OQ = z1 = r ei (ΞΈ + Ο)  = z . e iΟ. If  OP and OQ are  of  unequal magnitude

Ξ Ξ

then OQ = OPeiΟ

(ii) If  A, B, C & D  are four  points  representing  the complex numbers  z1, z2

, z3  &  z4  then

AB  CD   if   z4 −z3  is purely  real ;

z2 −z1

z4 −z3

AB ⊥ CD   if  z2 −z1 is purely imaginary ]

(iii) If  z1, z2, z3  are the vertices of an equilateral triangle  where  z0  is its circumcentre

then  (a)   z12 + z22 + z23 − z1 z2 − z2 z3 − z3 z1 = 0   (b)   z12 + z22 + z23 = 3 z20

8. DEMOIVRE’S  THEOREM :

Statement : cos n ΞΈ + i sin n ΞΈ  is  the  value  or  one  of  the  values  of  (cos ΞΈ + i sin ΞΈ)n ¥ n ∈ Q.  The theorem is  very  useful  in  determining  the  roots  of  any  complex  quantity

Note : Continued  product  of  the  roots  of  a  complex  quantity  should  be  determined using theory of  equations.

− +1 i 3 −1−i 3

9. CUBE  ROOT  OF  UNITY :(i)The cube roots of unity are  1 , , .

2 2

(ii) If  w  is  one  of  the  imaginary  cube  roots  of  unity  then  1 + w + w² = 0. In general

1 + wr + w2r = 0  ;   where  r ∈ I  but  is  not  the  multiple  of  3. (iii) In polar form the cube roots of unity are :

cos 0 + i sin 0 ;  cos  + i sin  ,   cos  + i sin

(iv) The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle.www.bloggermayank.online (v) The  following  factorisation  should  be  remembered :

(a, b, c  ∈  R  &  Ο  is the cube root of unity) a3 − b3 = (a − b) (a − Οb) (a − Ο²b) ; x2 + x + 1 = (x − Ο) (x − Ο2) ;

a3 + b3 = (a + b) (a + Οb) (a + Ο2b) ; a3 + b3 + c3 − 3abc = (a + b + c) (a + Οb + Ο²c) (a + Ο²b + Οc)

10. nth  ROOTS  OF  UNITY :www.bloggermayank.online

If  1 , Ξ±1 ,  Ξ±2 ,  Ξ±3 ..... Ξ±n − 1  are  the  n ,  nth  root  of  unity  then :

(i) They  are  in  G.P.  with  common  ratio  ei(2Ο/n) &

p

n−1  =  0   if  p  is  not  an  integral  multiple  of  n

=  n   if  p  is  an  integral  multiple  of  n

(iii) (1 − Ξ±1) (1 − Ξ±2) ...... (1 − Ξ±n − 1)  =  n &

(1 + Ξ±1) (1 + Ξ±2) ....... (1 + Ξ±n − 1) = 0   if  n  is  even  and  1 if  n is odd. (iv) 1 . Ξ±1 . Ξ±2 . Ξ±3 ......... Ξ±n − 1  =  1 or  −1 according as n is odd or even.

11. THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED :

(i) cos ΞΈ + cos 2 ΞΈ + cos 3 ΞΈ + ..... + cos n ΞΈ = sin nsin((ΞΈΞΈ22)) cos  n2+1ΞΈ.

(ii) sin ΞΈ + sin 2 ΞΈ + sin 3 ΞΈ + ..... + sin n ΞΈ =  sin nsin((ΞΈΞΈ22)) sinn 12+ ΞΈ.

Note : If  ΞΈ = (2Ο/n)  then  the  sum  of  the  above  series  vanishes.

12. STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS :

nz1+ mz2

(A) If  z1 & z2 are two complex numbers then the complex number   z = m + n divides  the  joins  of  z1

& z2  in the ratio m : n.

Note:(i) If a , b , c  are  three  real  numbers  such  that   az1 + bz2 + cz3 = 0    ; where  a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers  z1 , z2 & z3 are collinear. (ii) If the vertices  A, B, C  of a  ∆  represent the complex nos.  z1, z2, z3  respectively, then :

z1 + z2 + z3

(a) Centroid  of the  ∆ ABC =   :   (b) Orthocentre  of the  ∆ ABC =

3

(a secA z) 1+(bsecB z) 2 +(csecC z) 3 z tan A1 + z2 tan B + z3 tan C

a secA + bsecB + csecC   OR  tan A + tan B + tan C

(c) Incentre  of the  ∆ ABC = (az1 + bz2 + cz3) ÷ (a + b + c) .

(d) Circumcentre  of the  ∆ ABC =  :

(Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ÷ (sin 2A + sin 2B + sin 2C) .

(B) amp(z) = ΞΈ  is a ray emanating from the origin inclined at an angle  ΞΈ  to the x− axis. (C) z − a = z − b  is  the  perpendicular  bisector  of  the  line  joining  a  to  b.

(D) The equation of a line joining  z1 & z2 is given by ; z = z1 + t (z1 − z2)  where  t  is  a  perameter.www.bloggermayank.online (E)z = z1 (1 + it) where  t  is a real parameter is a line through the point  z1 & perpendicular to oz1.

(F) The equation of a line passing through  z1 &  z2  can be expressed in the determinant form as   z

= 0. This is also the condition for three complex numbers to be collinear.

(G) Complex equation of a straight line through two given points z1 & z2 can be written as z z( 1 − z2 )− z z( 1 −z2 )+(z z1 2 − z z1 2 )= 0, which on manipulating takes the form as

Ξ± z +Ξ± z + r = 0  where r is real and Ξ± is a non zero complex constant. (H) The  equation  of  circle  having  centre  z0 &  radius  Ο  is : z − z0 =  Ο   or

zz − z0 z − z0 z + z0 z0 − Ο² = 0   which is of the form  zz+Ξ±z+Ξ±z+r = 0 ,  r is real  centre  − Ξ± & radius Ξ±Ξ±−r . Circle will be real if Ξ±Ξ±−r ≥ 0.

(I) The equation of the circle described on the line  segment joining  z1 & z2 as diameter is :

(i)  arg z −−zz12 =  ± Ο2    or   (z − z1) (z − z 2) + (z − z2) (z − z 1) = 0

z

(J) Condition for four given points  z1 , z2 , z3 & z4  to  be concyclic is, the number

z3 −−zz12 .zz44 −−zz12 is real. Hence the equation of a circle through 3 non collinear points z1, z2 & z3 can be

z3

(z −z )(z −z ) (z −z )(z −z ) (z − z )(z − z ) taken as ( 2)( 3 1) is  real  ⇒ ( 2)( 3 1) = (2)( 31) z −z1 z3 −z2 z −z1 z3 −z2 z − z1 z3 − z2

13.(a) Reflection  points  for  a  straight  line :Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers  z1 & z2 will be the reflection points for the straight line  Ξ± z +Ξ±z + r = 0 if and only if ; Ξ±z1+Ξ±z2 +r =0, where r is real and Ξ± is non zero complex constant.

(b) Inverse  points  w.r.t. a circle :www.bloggermayank.online

Two points P & Q are said to be inverse w.r.t. a circle with centre 'O' and radius Ο,  if  :

(i)   the point O, P, Q are collinear and on the same side of O.     (ii)  OP . OQ = Ο2. Note that the two points  z1 & z2  will be the inverse points  w.r.t. the circle zz+Ξ±z+Ξ±z+r=0  if and only if  z1z2+Ξ±z1+Ξ±z2+r=0 .

14. PTOLEMY’S  THEOREM :www.bloggermayank.online

It  states  that  the  product  of  the  lengths  of  the  diagonals  of  a  convex  quadrilateral  inscribed  in  a circle  is  equal  to  the  sum  of  the  lengths  of  the  two  pairs of  its  opposite sides. i.e. z1 − z3 z2 − z4 = z1 − z2 z3 − z4 + z1 − z4 z2 − z3.

15. LOGARITHM  OF  A  COMPLEX  QUANTITY :

(i) Loge (Ξ± + i Ξ²) =   12 Loge (Ξ±² + Ξ²²) + i2nΟ+ tan−1 Ξ±Ξ²   where  n ∈ I.

− Ο+2n Ο

(ii) ii  represents  a  set  of  positive  real  numbers  given  by  e  2 ,  n ∈ I.

2.THEORY OF EQUATIONS  (QUADRATIC EQUATIONS)

The general form of a quadratic equation in  x  is , ax2 + bx + c = 0 , where  a , b , c ∈ R & a ≠ 0.

− ±b b2−4ac

RESULTS  :1. The  solution  of  the  quadratic  equation ,  ax² + bx + c = 0  is  given  by  x =

2a The  expression  b2 – 4ac = D  is  called  the  discriminant  of  the  quadratic equation.

2. If  Ξ± & Ξ²  are  the  roots  of  the  quadratic  equation  ax² + bx + c = 0,  then; (i)  Ξ± + Ξ² = – b/a (ii)  Ξ± Ξ² = c/a (iii)  Ξ± – Ξ² = D /a .

3.NATURE  OF  ROOTS:(A)Consider the quadratic equation ax² + bx + c = 0  where a, b, c ∈ R & a≠ 0 then (i) D > 0  ⇔  roots  are  real & distinct  (unequal). (ii)D = 0  ⇔  roots  are  real & coincident

(equal). (iii) D < 0  ⇔  roots  are  imaginary  . (iv) If  p + i q  is one root of a  quadratic  equation, then  the other must be the  conjugate  p − i q  &  vice versa.  (p , q ∈ R  &  i = −1 ).

(B) Consider the quadratic equation ax2 + bx + c = 0 where a, b, c ∈ Q & a ≠ 0 then; (i)      If  D > 0  &  is a perfect  square ,  then  roots  are  rational & unequal.

(ii) If  Ξ± = p + q is  one root in this case, (where p is rational & q is a surd)  then the other

root must be the conjugate of it  i.e. Ξ² = p − q & vice versa.

4. A quadratic  equation  whose  roots  are  Ξ± & Ξ²  is  (x − Ξ±)(x − Ξ²) = 0  i.e.

x2 − (Ξ± + Ξ²) x + Ξ± Ξ² = 0 i.e.  x2 − (sum of  roots) x +  product  of  roots = 0.

5.Remember that a quadratic equation cannot have three different roots & if  it  has, it becomes an  identity.

6. Consider  the  quadratic  expression ,  y = ax² + bx + c  ,  a ≠ 0  &  a , b , c ∈ R  then

(i) The  graph  between  x , y  is  always  a  parabola .  If  a > 0  then  the  shape  of  the parabola is concave upwards &  if a < 0  then the shape of the parabola is concave  downwards.

(ii) ∀  x ∈ R ,  y > 0  only  if  a > 0  &  b² − 4ac < 0  (figure 3) . (iii) ∀  x ∈ R ,  y < 0  only  if  a < 0  &  b² − 4ac < 0  (figure 6) .

Carefully go through the 6 different shapes of the parabola given below.

Fig. 1    Fig. 2

x x x

Roots  are  real  & Roots  are Roots  are  complex

Fig. 4 Fig. 5

Roots  are  real  & Roots  are Roots  are  complex

7. SOLUTION  OF  QUADRATIC  INEQUALITIES: ax2 + bx + c > 0 (a ≠ 0).

(i) If  D > 0, then  the  equation  ax2 + bx + c = 0  has  two different roots x1 < x2.

Then a > 0   ⇒   x ∈ (−∞, x1) ∪ (x2, ∞) a < 0   ⇒   x ∈ (x1, x2)www.bloggermayank.online

(ii) If  D = 0, then roots are equal,  i.e.  x1 = x2. In that case a > 0   ⇒   x ∈ (−∞, x1) ∪ (x1, ∞) a < 0   ⇒   x ∈ Ο

P(x)

(iii)Inequalities of  the form     0 can  be  quickly  solved using  the  method of intervals. Q x( )

8.MAXIMUM & MINIMUM  VALUE   of  y = ax² + bx + c occurs at  x = − (b/2a) according as ;  a < 0  or a > 0 .  y ∈  4ac − b2 , ∞ if a > 0  &  y ∈ −∞ ,  4ac − b2 if a < 0 .

4a 4a

9.COMMON  ROOTS  OF  2  QUADRATIC  EQUATIONS  [ONLY  ONE  COMMON  ROOT]  :

Let  Ξ±  be  the  common  root  of  ax² + bx + c = 0  &  a′x2 + b′x + c′ = 0 Thereforea Ξ±² + bΞ± + c = 0  ;

a′Ξ±² + b′Ξ± + c′ = 0. By  Cramer’s  Rule Ξ±2 = Ξ± = 1     Therefore, Ξ± = bc′−b′c a′c−ac′ ab′−a′b

ca′−c′a = bc′−b′c .So  the  condition  for  a  common  root  is  (ca′ − c′a)² = (ab′ − a′b)(bc′ − b′c). ab′−a′b a′c−ac′

10. The condition that a quadratic function  f (x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c  may be  resolved into  two  linear  factors  is  that  ;

a h g

abc + 2 fgh − af2 − bg2 − ch2 = 0   OR  h b f = 0.

g f c

11. THEORY  OF  EQUATIONS  : If  Ξ±1, Ξ±2, Ξ±3, ......Ξ±n  are the roots of  the equation;

f(x) = a0xn + a1xn-1 + a2xn-2 + .... + an-1x + an = 0 where  a0, a1, .... an are all  real & a0 ≠ 0 then, ∑ Ξ±1 = −  aa01 ,

∑ Ξ±1 Ξ±2 = + aa2 ,  ∑ Ξ± Ξ± Ξ± = −  aa03 , ....., Ξ±1 Ξ±2 Ξ±3 ........Ξ±n = (−1)n  aa 0n

Note : (i) If Ξ±  is a  root  of the equation f(x) = 0, then the polynomial f(x) is exactly divisible by (x − Ξ±) or (x − Ξ±)  is a factor of  f(x) and conversely .

(ii) Every equation of nth degree (n ≥ 1) has exactly n roots & if the equation has more than  n  roots, it is an identity.

(iii) If the coefficients of the equation f(x) = 0 are all real and  Ξ± + iΞ²  is its root, then  Ξ± − iΞ²  is also a root. i.e. imaginary roots occur in conjugate pairs.

(iv) If the coefficients in the equation are all rational & Ξ± + Ξ² is one of its roots, then  Ξ± − Ξ² is also a root where  Ξ±, Ξ² ∈ Q & Ξ²  is not a perfect square.

(v) If  there  be  any  two  real  numbers 'a' & 'b'  such  that  f(a) & f(b) are of opposite signs,  then  f(x) = 0  must  have  atleast  one  real  root  between 'a' and 'b' .

(vi)Every  eqtion f(x) = 0  of degree  odd  has atleast one real root of a sign opposite to that of its last term.

12. LOCATION  OF  ROOTS  : www.bloggermayank.online

Let  f (x) = ax2 + bx + c,  where  a > 0  &  a, b, c ∈ R.

(i) Conditions for both the roots of  f (x) = 0  to be greater than a specified number ‘d’ are b2 − 4ac ≥ 0;  f (d) > 0  &  (− b/2a) > d.

(ii) Conditions  for  both  roots  of  f (x) = 0  to  lie  on  either  side  of  the  number ‘d’ (in other words the number ‘d’ lies between the roots  of  f (x) = 0)  is  f (d) < 0.

(iii) Conditions  for  exactly  one  root  of  f (x) = 0  to  lie  in  the  interval  (d , e)  i.e. d < x < e are  b2 − 4ac > 0  &  f (d) . f (e) < 0.

(iv) Conditions that both roots of  f (x) = 0 to be confined between  the numbers p & q are (p < q).  b2 − 4ac ≥ 0;   f (p) > 0;   f (q) > 0  &  p < (− b/2a) < q.

13. LOGARITHMIC INEQUALITIES

(i) For  a > 1  the  inequality  0 < x < y  &  loga x <  loga y  are  equivalent.

(ii) For  0 < a < 1  the  inequality  0 < x < y  &  loga x > loga y  are  equivalent.

(iii) If  a > 1   then  loga x < p 0 < x < ap

(iv) If  a > 1   then  logax > p x > ap

(v) If  0 < a < 1 then  loga x < p x > ap

(vi) If  0 < a < 1 then  logax > p ⇒ 0 < x < ap www.bloggermayank.online

3.Sequence & Progression(AP, GP, HP, AGP, Spl. Series)

DEFINITION : A sequence is a set of terms in a definite order with a rule for obtaining the terms.

e.g.  1 , 1/2 , 1/3 , ....... , 1/n , ........ is a sequence.

AN ARITHMETIC  PROGRESSION (AP) :AP is a sequence whose terms increase or decrease by a fixed number. This fixed number  is called the common difference. If a is the first term & d the common difference, then AP can be written as a, a + d, a + 2 d, ....... a + (n – 1)d, ........ nth term of this AP tn = a +

n a + (n – 1)d] = n [a

(n – 1)d,  where  d = an – an-1. The sum of the first n terms of the AP is given by  ;   Sn = 2 [2 2

+ l]. where l  is the last term.

NOTES :(i) If each term of an  A.P. is increased, decreased, multiplied or divided by the same non zero  number, then the resulting sequence is also an AP.

(ii) Three numbers in AP can be taken as a – d ,  a ,  a + d ;  four numbers in  AP  can be  taken  as  a – 3d,  a – d,  a + d,  a + 3d ;   five numbers in  AP are  a – 2d , a – d ,  a, a + d,  a + 2d  &  six terms in  AP are  a – 5d,

a – 3d, a – d, a + d, a + 3d, a + 5d  etc.

(iii) The common difference can be zero, positive or negative.

(iv) The sum of the two terms of an AP equidistant from the beginning  &  end  is constant and equal to the sum of first  &  last terms.

(v) Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it.

(vi) tr = Sr − Sr−1 (vii) If  a , b , c  are  in  AP ⇒ 2 b = a + c.

GEOMETRIC  PROGRESSION  (GP) :

GP is a sequence of numbers whose first term is non zero & each of the succeeding terms  is equal to the proceeding terms multiplied by a constant . Thus in a GP the ratio of  successive terms is constant. This constant factor is called the COMMON RATIO of  the series & is obtained by dividing any term by that which immediately proceeds it.  Therefore  a, ar,  ar2, ar3, ar4, ...... is a GP with a as the first term & r as common ratio.www.bloggermayank.online

a r( n −1)

(i) nth  term = a rn –1 (ii) Sum of the Ist n terms  i.e. Sn = r−1 , if  r  ≠ 1 .

(iii) Sum of an infinite GP when r < 1 when  n → ∞  rn → 0  if  r < 1  therefore,

a

S∞ = 1 −r (| r |<1) .

(iv) If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP.

(v) Any 3 consecutive terms of a GP can be taken as  a/r, a, ar  ;  any 4 consecutive terms of a GP can be taken as  a/r3, a/r, ar, ar3  &  so on.

(vi) If a, b, c  are in  GP ⇒ b2 = ac.www.bloggermayank.online

HARMONIC  PROGRESSION  (HP) :A sequence is said to HP if the reciprocals of its terms are in AP.If the sequence a1, a2, a3, .... , an  is an HP then  1/a1, 1/a2, .... , 1/an  is an AP & converse. Here we do not have the formula for the sum of the  n  terms of an  HP. For HP whose first term is a & second term is b,  the nth term is  tn =  b + −(nab1)(a b− ) .

are  in  HP ⇒ b = 2+ac  or   ac = a bb c−− .

If  a, b, c

a c

MEANS ARITHMETIC  MEAN :

If three terms are in AP then the middle term is called the  AM between the other two, so  if a, b, c  are in

AP,  b is AM of a & c . AM for any n positive number a1, a2, ... , an is  ; a1+ + + +a2 a3 ..... an

A = .www.bloggermayank.online

n n - ARITHMETIC  MEANS  BETWEEN  TWO  NUMBERS  :

If a, b  are any two given numbers &  a, A1, A2, .... , An,  b  are in AP then A1, A2, ... An are  the n  AM’s between  a & b .

A1 = a + bn −+ a1 ,  2 = a +  2 (nb+−1a) , ...... , n = a +  n b(n +−1a)     = a + d ,    = a + 2 d, ...... , An = a + nd ,

b−a

where  d = n+1

NOTE : Sum of n  AM’s inserted between a & b  is equal to n times the single AM between a & b     i.e.

n

∑ Ar = nA  where  A is the single AM between  a & b. r = 1

GEOMETRIC  MEANS :

If  a, b, c  are in GP,  b  is the GM between  a & c. b² = ac,   therefore  b = a c ;  a > 0,  c > 0. n-GEOMETRIC  MEANS  BETWEEN a, b :

If  a, b  are  two  given  numbers  &  a, G1, G2, ..... , Gn,  b  are  in GP. Then G1, G2, G3 , ...., Gn are n  GMs between  a & b .

G1 = a(b/a)1/n+1,  G2 = a(b/a)2/n+1, ...... ,  Gn = a(b/a)n/n+1

= ar ,                  = ar² ,       ......         = arn,  where  r = (b/a)1/n+1

NOTE : The product of  n GMs  between  a & b  is equal to the nth power of the single GM between a & b

n

i.e. Ο Gr = (G)n  where G is the single GM between a & b. r =1

HARMONIC  MEAN :

If a, b, c  are in HP,  b  is the HM between  a & c,  then  b = 2ac/[a + c].

THEOREM :

If A, G, H are respectively AM, GM, HM between a & b both being unequal & positive then,

(i) G² = AH

(ii) A > G > H (G > 0). Note that A, G, H constitute a GP.

ARITHMETICO-GEOMETRIC  SERIES :

A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series.  e.g.  1 + 3x + 5x2 + 7x3 + .....

Here  1, 3, 5, ....  are  in  AP &  1, x, x2, x3 ..... are in GP. www.bloggermayank.online

Standart appearance of an Arithmetico-Geometric Series is

Let  Sn = a + (a + d) r + (a + 2 d) r² + ..... + [a + (n − 1)d]  rn−1

SUM  TO  INFINITY  :

If r < 1  &  n → ∞  then  Limitn →∞ rn = 0 .  S∞ =   .

SIGMA  NOTATIONS THEOREMS  :

n n n n n n

(i) ∑ (ar ± br) = ∑ ar ± ∑ br.(ii) ∑ k ar =  k ∑ ar.(iii) ∑ k = nk  ;  where k is a constant.

r=1 r=1 r=1 r=1 r=1 r=1

n n (n+1)

RESULTS (i) ∑ r =    (sum of the first n natural nos.)

r=1 2

n n (n+1) (2n+1)

(ii) ∑ r² =    (sum of the squares of the first n natural numbers) r=1 6

2

(iii) ∑n r3 =  n2 (n+1)2 ∑n r  (sum of the cubes of the first n natural numbers) r=1 4 r = 1 

(iv) ∑n r4  =  n (n + 1) (2n + 1) (3n² + 3n − 1)www.bloggermayank.online r=1 30

METHOD  OF  DIFFERENCE  : If  T1, T2, T3, ...... , Tn  are  the  terms  of a sequence then some times the terms  T2 − T1, T3 − T2 , ....... constitute  an  AP/GP.  nth term of the series is determined & the sum to n terms of the sequence can easily be obtained.

Remember that  to find  the sum of n terms of a series each term of which is composed of r  factors in AP, the first factors of several terms being in the same AP, we “write down the nth term, affix  the  next factor at the end, divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial conditions”.     4.PERMUTATION AND COMBINATION

DEFINITIONS :1. PERMUTATION : Each of the arrangements in a definite order which can be made by taking some or all of a number of things is called a PERMUTATION.

2.COMBINATION : Each of the groups or selections which can be made by taking some or all of a number of things without reference to the order of the things in each group is called a COMBINATION.

FUNDAMENTAL PRINCIPLE OF COUNTING :

If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order   is m × n. This can be extended to any number of events.

RESULTS :(i) A Useful Notation : n! = n (n − 1) (n − 2)......... 3. 2. 1  ;  n ! = n. (n − 1) !0! = 1! = 1 ; (2n)! = 2n. n ! [1. 3. 5. 7...(2n − 1)] Note that factorials of negative integers are not defined.

(ii) If nPr denotes the number of permutations of n different things, taking r at a time, then

n!

nPr = n (n − 1) (n − 2)..... (n − r + 1) =  (n −r)! Note that , nPn = n !.

(iii) If nCr  denotes the number of combinations of n different things taken r at a time, then n! n

nCr = −r)! = rP!r  where  r ≤ n  ;  n ∈ N  and  r ∈ r!(n

(iv) The number of ways in which (m + n) different things can be divided into

(m+n)!

two groups containing m & n things respectively is :   If m = n, the groups are equal & in this case the m!n!

(2n)! number of subdivision is   ; for in any one way it is possible to interchange the two groups n!n!2!

without obtaining a new distribution. However, if 2n things are to be divided

(2n)!

equally between two persons then the number of ways =   .

n!n!

(v) Number of ways in which (m + n + p) different things can be divided into three groups containing m , n &

(m+ n + p)! ≠ n ≠ p. p things respectively is , m m!n!p!

(3n)!

If  m = n = p  then the number of groups =   .However, if 3n things are to be divided equally among n!n!n!3!

three people then the number of ways = .

(vi)The number of permutations of n things taken all at a time when p of them are similar & of one type, q of them are similar & of another type, r of them are similar & of a third type & the remaining n! n – (p + q + r) are all different is :   .www.bloggermayank.online p!q!r!

(vii) The number of circular permutations of n different things taken all at a time is ; (n − 1)!. If clockwise & anti−clockwise circular permutations are considered to be same, then it is  ! .

Note : Number of circular permutations of n things when p alike and the rest different taken all at a time

(n −1)!

distinguishing clockwise and anticlockwise arrangement is   . p!

(viii) Given n different objects, the number of ways of selecting atleast one of them is , nC1 + nC2 + nC3 +.....+ nCn = 2n − 1. This can also be stated as the total number of combinations of n distinct things.

(ix) Total number of ways in which it is possible to make a selection by taking some or all out of p + q + r +...... things , where p are alike of one kind, q alike of a second kind , r alike of third kind & so on is given by : (p + 1) (q + 1) (r + 1)........ –1.www.bloggermayank.online

(x)Number of ways in which it is possible to make a selection of m + n + p = N things , where p are alike of one kind , m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr in the expansion of (1 + x + x2 +...... + xp) (1 + x + x2 +...... + xm) (1 + x + x2 +...... + xn).

Note : Remember that coefficient of xr in (1 − x)−n = n+r−1Cr (n ∈ N). For example the number of ways in which a selection of four letters can be made from the letters of the word PROPORTION is given by coefficient of x4 in (1 + x + x2 + x3) (1 + x + x2) (1 + x + x2) (1 + x) (1 + x) (1 + x).

(xi) Number of ways in which n distinct things can be distributed to p persons if there is no restriction to the number of things received by men = pn.www.bloggermayank.online

(xii) Number of ways in which n identical things may be distributed among p persons if each person may receive none , one or more things is ; n+p−1Cn.

(xiii) a.nCr = nCn−r ; nC0 = nCn = 1;b.nCx = nCy ⇒ x = y or x + y = n c.nCr + nCr−1 = n+1Cr

nCr is maximum if : (a) r = n2  if n is even. (b) r = n 12−  or n 12+ if n is odd. (xiv)

(xv) Let N = pa. qb. rc...... where p , q , r...... are distinct primes & a , b , c..... are natural numbers then: (a) The total numbers of divisors of N including 1 & N is = (a + 1)(b + 1)(c + 1).....

(b) The sum of these divisors is

= (p0 + p1 + p2 +.... + pa) (q0 + q1 + q2 +.... + qb) (r0 + r1 + r2 +.... + rc)....

(c) Number of ways in which N can be resolved as a product of two

1 (a +1)(b +1)(c +1).... if N is not a perfect square factors is =  2

[(a +1)(b +1)(c +1)....+1] if N is a perfect square

(d) Number of ways in which a composite number N can be resolved into two factors which are relatively prime (or coprime) to each other is equal to 2n−1 where n is the number of

different prime factors in N. [ Refer Q.No.28 of Ex−I ] (xvi) Grid Problems and tree diagrams.

DEARRANGEMENT : Number of ways in which n letters can be placed in n directed letters so that no

 1

letter goes into its own envelope is = n!2! − 31! + 41!+........... (+ −1)n  n1! .

(xvii) Some times students find it difficult to decide whether a problem is on permutation or combination or both. Based on certain words / phrases occuring in the problem we can fairly decide its nature as per the following table :www.bloggermayank.online

PROBLEMS OF COMBINATIONS PROBLEMS OF PERMUTATIONS  Selections , choose Arrangements

Distributed group is formed Standing in a line seated in a row

Committee problems on digits

Geometrical problems Problems on letters from a word  5.DETERMINANT

a1 b1

1 The symbol  is called the determinant of order two .

a2 b2

Its  value  is  given  by  : D = a1 b2 − a2 b1

a1 b1 c1

2. The  symbol  a2 b2 c2 is  called  the  determinant  of  order  three .

a3 b3 c3

b1 c1b1 c1

Its  value  can  be  found  as  : D = a   +  a     OR

1b3 c33b2 c2

b ca c

D = a1 b2 c2  −  b1 a23 c23  +  c ....... and  so  on .In this manner we can expand a

3 3

determinant in 6 ways using elements of  ; R1 , R2 , R3  or  C1 , C2 , C3  .

3. Following  examples  of  short hand  writing  large  expressions  are :

(i) The  lines   : a1x + b1y + c1 = 0........ (1) a2x + b2y + c2 = 0........ (2) a3x + b3y + c3 = 0........ (3)

a1 b1 c1

are  concurrent  if  ,a2 b2 c2 = 0 .

a3 b3 c3

Condition for the consistency of  three  simultaneous linear equations  in 2 variables. (ii)    ax² + 2 hxy + by² + 2 gx + 2 fy + c = 0   represents  a  pair  of  straight  lines  if

a h g

abc + 2 fgh − af² − bg² − ch² = 0 =  h b f

g f c

(iii) Area  of  a  triangle  whose  vertices  are  (xr , yr) ;  r = 1 , 2 , 3  is  :

x1 y11

D =   x2 y2 1 If  D = 0  then  the  three  points  are  collinear .

x3 y31

(iv) x

Equation of a straight line  passsing  through  (x1 , y1) & (x2 , y2)  is   x1

x2 y

y1 y2 1

1 = 0

1

4. MINORS  :The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands For  example, the

b2 c2a1 c1

minor of a1 in (Key Concept 2)  is  b3 c3 &  the minor of b2 is  a3 c3  . Hence  a  determinant  of  order two  will  have  “4 minors”  &  a  determinant  of  order  three will  have  “9 minors”  .

5. COFACTOR  :If Mij represents the minor of some typical element then the cofactor is defined as  : Cij = (−1)i+j . Mij  ;  Where i & j denotes the row & column in which the particular element  lies. Note that the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as :  D = a11M11

− a12M12 + a13M13 OR  D = a11C11 + a12C12 + a13C13  & so on .......

6. PROPERTIES  OF  DETERMINANTS  :

P− 1 : The  value  of  a  determinant  remains  unaltered , if  the  rows & columns  are inter changed .  e.g.

a1 b1 c1a1 a2 a3

if   D = a2 b2 c2 = b1 b2 b3 = D′ D & D′  are   transpose  of  each  other .  If  D′ = − D  then  it

a3 b3 c3c1 c2 c3

is  SKEW  SYMMETRIC determinant  but  D′ = D ⇒ 2 D = 0 ⇒ D = 0 ⇒ Skew  symmetric  determinant of  third order  has  the  value  zero .www.bloggermayank.online

P− 2 : If  any  two  rows (or  columns)  of  a  determinant  be  interchanged ,  the  value of determinant  is changed  in  sign  only .    e.g.

a1 b1 c1a2 b2 c2

Let   D  = a2 b2 c2   &   D′ = a1 b1 c1Then   D′ = − D  .

a3 b3 c3a3 b3 c3

P− 3 : If  a  determinant  has  any  two  rows  (or columns)  identical , then  its  value  is zero . e.g.

a1 b1 c1

Let  D =  a1 b1 c1   then  it  can  be  verified  that  D = 0.

a3 b3 c3

P− 4 : If all the elements of any row (or column) be  multiplied  by  the  same number , then  the  determinant is  multiplied  by  that  number.

a1 b1 c1Ka1 Kb1 Kc1

e.g.     If  D  =  a2 b2 c2and  D′ =  a2 b2 c2Then   D′= KD

a3 b3 c3a3 b3 c3

P−5 : If  each  element  of any row (or column) can be expressed as  a sum of  two terms then  the determinant  can  be  expressed  as  the  sum  of  two  determinants .  e.g.

a1+x b1+y c1+za1 b1 c1x y z a2 b2 c22 b2 c2 + a2 b2 c2 a3 b3 c3a3 b3 c3a3 b3 c3

P− 6 : The  value  of  a  determinant  is  not  altered  by  adding  to  the  elements  of  any  row (or column) the  same  multiples  of  the  corresponding  elements  of  any other  row (or column) .e.g. Let   D

a1 b1 c1a1+ma2 b1+mb2 c1+mc2

= a2 b2 c2 and D′ = a2 b2 c2  . Then  D′ = D  .

a3 b3 c3a3+na1 b3+nb1 c3+nc1

Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must  remain

unchanged  .P− 7 : If  by  putting  x = a  the  value  of  a  determinant  vanishes  then  (x − a)  is  a  factor of  the  determinant  .

a1 b1l1 m1a l1 1+b l1 2 a m1 1+b m1 2

7.MULTIPLICATION  OF  TWO  DETERMINANTS   :(i)

a2 b2l2 m2a l2 1+b l2 2 a m2 1+b m2 2

Similarly  two  determinants  of  order  three  are  multiplied.

a1 b1A1 B1 C1

(ii)If D = a2 b2A2 B2 Ci,Bi,Ci  are  cofactors

a3 b3A3 B3 C3

a1 b1 A2 A3D 0

PROOF  :Consider   a2 b2 B2 B3 = 0 D 0        Note : a1A2 + b1B2 + c1C2 = 0   etc. therefore

a3 b3 C2 C30 0

A1 A2 A3 A1 A2 A3A1 B1 C1

,   D x B1 B2 B3 = D     ⇒ B1 B2 B3 = D²  OR  A2 B2 C2 = D²

C1 C2 C3 C1 C2 C3CA3 B3 C3

8. SYSTEM  OF  LINEAR  EQUATION  (IN  TWO  VARIABLES)  :

(i) Consistent  Equations   : Definite  &  unique  solution .  [ intersecting lines ]

(ii) Inconsistent  Equation  : No  solution  .  [ Parallel line ] (iii) Dependent  equation   : Infinite  solutions  .  [ Identical  lines ] Let   a1x + b1y + c1 = 0     &     a2x + b2y + c2 = 0     then  :

a1 = b1 ≠ c1 ⇒Given  equations  are  inconsistent& a1 = b1 = c1 ⇒Given  equations  are  dependent a2 b2 c2 a2 b2 c2

9. CRAMER'S  RULE  :[ SIMULTANEOUS  EQUATIONS  INVOLVING  THREE  UNKNOWNS ]

Let  ,a1x + b1y + c1z = d1 ...(I) ;  a2x + b2y + c2z = d2 ...(II) ; a3x + b3y + c3z = d3 ...(III)

D1 D2 3

Then ,  x =   ,  Y =   ,  Z =   . D

a1 b1 c1b1 c1a1 d1 c1a1 b1 d1

Where  D = a2 b2 c2  ;  D1 = d b2 c2  ;   D2 =  a2 d2 c2  &  D3 =  a2 b2 d2

a3 b3 c3b3 c3a3 d3 c3a3 b3 d3

NOTE :(a) If  D ≠ 0 and  alteast  one of  D1 , D2 , D3 ≠ 0 ,  then  the  given  system  of  equations  are consistent  and  have  unique  non  trivial  solution .

(b) If  D ≠ 0  &  D1 = D2 = D3 = 0 ,  then  the  given  system  of equations  are   consistent  and  have  trivial solution only .www.bloggermayank.online

(c) If  D = D1 = D2 = D3 = 0 ,  then  the  given  system  of equations  are

a1x + b1y +c1z=d1 

consistentand  have  infinite  solutions  . In case      a x b y c z d2 + 2 + 2 = 2    represents these parallel a x b y c z d3 + 3 + 3 = 3 

planes then also     D = D1 = D2 = D3 = 0      but the system is inconsistent.

(d) If  D = 0   but  atleast  one  of  D1 , D2 , D3 is not  zero  then the equations are inconsistent and have no solution .

10. If  x , y , z  are  not  all  zero ,  the  condition  for  a1x + b1y + c1z = 0  ;   a2x + b2y + c2z = 0  & a3x + b3y

a1 b1 c1

+ c3z = 0  to  be  consistent  in  x , y , z  is  that  a2 b2 c2 = 0.Remember  that  if

a3 b3 c3

a  given  system  of  linear  equations  have  Only  Zero Solution  for  all  its  variables  then  the  given equations  are  said  to  have  TRIVIAL SOLUTION 6.MATRICES

USEFUL  IN  STUDY  OF  SCIENCE, ECONOMICS AND ENGINEERING

1. Definition  : Rectangular array of m n  numbers . Unlike determinants it has no value.

a11 a12 ...... a1n   a11 a12 ...... a1n 

a21 a 22 ...... a2n   a21 a22 ...... a2n  A  =   : : : :           or            : : : : 

 a m1 am2 ...... a mn a m1 a m2 ...... a mn 

Abbreviated  as  : A  =  [a i j ]1  ≤  i  ≤  m ; 1  ≤  j  ≤  n, i denotes the row andj denotes the column is called a matrix of order m × n.

2. Special  Type  Of  Matrices  :

(a) Row Matrix  : A  =  [ a11 , a12 , ...... a1n ] having one row . (1 × n) matrix.  (or row vectors)

 a11 

(b) Column Matrix  : A =   a21  having one column. (m × 1) matrix   (or column vectors)

 : 

 a m1

(c) Zero or Null Matrix :    (A =  Om × n)An  m  ×  n  matrix all whose entries are zero .

 0 0   0 0 0

  ×  2    null matrix &   B  =   0 0 0 is   3  ×  3  null matrix A  =   0 0  is  a  3

 0 0   0 0 0

1 2 3 4

(d) Horizontal Matrix : A matrix of order  m × n  is a horizontal matrix if n > m. 

2 5 1 1

2 5

(e) Verical Matrix : A matrix of order m × n is a vertical matrix if m > n.  1 1

3 6

2 4

(f) Square  Matrix  :   (Order  n)If number of row  =  number of column ⇒     a square matrix. Note

 a a 

11 12

(i)In a square matrix the pair of elements  aij & aj i are called Conjugate Elements .e.g. a21 a22

(ii)The elements  a11 ,  a22 ,  a33 , ...... ann are called Diagonal Elements . The line along which  the diagonal elements lie is called " Principal  or  Leading "  diagonal. The qty     Ξ£  ai i  =  trace of the matrice written as ,    i.e.  tr A Triangular Matrix Diagonal Matrix denote as ddia (d1 , d2 , ....., dn)  all elements except the leading diagonal are zero diagonal Matrix Unit or Identity Matrix

Note: Min. number of zeros in a diagonal matrix of order n = n(n – 1)  "It is to be noted that with square matrix there is a corresponding determinant formed by the elements of A in the same order."

3. Equality Of  Matrices : Let A  =  [a i j ]   &    B  =  [b i j ]  are equal if , (i) both have the same order . (ii) ai j  =  b i j  for each pair of  i & j.

4.Algebra Of  Matrices :Addition  : A  +  B  =  [ ai j + bi j ]   where  A & B are of the same type. (order)

(a)     Addition of matrices is commutative. i.e. A  +  B  =  B  +  A,   A = m × n; B = m × n

(b)Matrix  addition is associative .(A + B) + C  =  A + (B + C)  Note :  A ,  B & C  are of the same type.

(c)  Additive inverse.       If    A + B  =  O  =  B  +  A A  =  m × n

a b c ka kb kc

5. Multiplication  Of  A  Matrix  By  A  Scalar  :   IfA = b c a ;k A = kb kc ka

c a b kc ka kb

6.Multiplication  Of  Matrices : (Row by Column)AB  exists  if ,   A =  m  ×  n    &  B =  n × p   2 × 3      3 × 3 AB  exists ,  but  BA  does not ⇒   AB  ≠  BA

b1 

 A = prefactor

AB ,    , a , ...... a ) &B =  b2

Note :  In the product  B = postfactor A =  (a1 2 n  :  1 × n       n × 1

bn 

B  =  [a1 b1 + a2 b2 + ...... + an bn] If  A  = [aij ]  m × n   &   B  = [ bi j ]    n  ×  p

n

matrix , then        (A B)i j  =  ∑  ai r . br j Properties  Of  Matrix  Multiplication  :

r = 1

1. Matrix multiplication is not commutative .

A   =  10 10 ; B  =  10 00 ; AB  =  10 00 ;BA  = 10 10 ⇒ ≠  BA  (in general)

2. AB  =  12 12    −11 −11   =   00 00 ⇒    =  O   ⇒/    A =  O   or   B  =  O

Note: If A and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero. Also if  [AB] = O ⇒ | AB |  ⇒ | A | | B | = 0 ⇒ | A | = 0  or | B | = 0 but not the converse.  If A and B are two matrices such that (i) AB = BA    ⇒   A and B commute each other (ii) AB = – BA  ⇒  A and B anti commute each other

3. Matrix  Multiplication  Is  Associative :

If  A , B & C  are conformable for the product  AB  &  BC, then (A . B) . C  =  A . (B . C)

4. Distributivity  :

A B( + C) = AB + AC

+ B C) = AC + BC Provided A, B & C are conformable for respective products (A

5. POSITIVE  INTEGRAL  POWERS  OF  A  SQUARE  MATRIX  :

For a square matrix A , A2 A = (A A) A  = A (A A)  =  A3 .

Note that for a unit matrix I of any order ,  Im  =  I  for  all  m  ∈  N.

6. MATRIX POLYNOMIAL :

If    f (x) = a0xn + a1xn – 1 + a2xn – 2 + ......... + anx0  then we define a matrix polynomial f (A) = a0An + a1An– 1 + a2An–2 + ..... + anIn where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial f (x).

DEFINITIONS :(a) Idempotent Matrix : A square matrix is idempotent provided  A2 = A.

Note that An = A ∀ n > 2 , n ∈ N.

(b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m ∈ N, if

Am = O , Am–1  ≠ O.

(c) Periodic Matrix : A square matrix is which satisfies the relation AK+1 = A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true.

Note that period of an idempotent matrix is 1.

(d) Involutary Matrix : If A2 = I , the matrix is said to be an involutary matrix. Note that A = A–1 for an involutary matrix.

7. The Transpose  Of  A  Matrix  :   (Changing rows & columns)

Let  A be any matrix . Then ,  A  =  ai j       of  order    m  ×  n

⇒ AT  or  A′   =  [ aj i ]    for  1  ≤ i  ≤  n   &   1  ≤ j  ≤  m     of  order    n  ×  m

Properties of Transpose :  If  AT  &  BT  denote the transpose of  A and B ,

(a) (A ± B)T  =  AT  ±  BT ;  note that  A  &  B  have the same order.

IMP. (b) (A B)T  =  BT  AT    A  &  B  are conformable for matrix product AB.

(c) (AT)T  =  A (d) (k A)T  =  k AT k  is a scalar .

General  : (A1 , A2 , ...... An)T   =  ATn , ....... , A2T , A1T  (reversal law for transpose)

8. Symmetric  &  Skew  Symmetric  Matrix  :

A  square matrix  A  =  [a i j ]  is said to be , symmetric if , ai j  =  aj i    ∀   i  &  j (conjugate  elements are equal)

(Note A = AT)

n(n +1)

Note: Max. number of distinct entries in a symmetric matrix of order  n  is   . 2

and  skew symmetric if , ai j  =  − aj i    ∀   i  &  j (the pair of conjugate elements are additive inverse of each other) (Note A = –AT ) Hence   If  A  is  skew  symmetric,  then ai i  =  − ai i    ⇒   ai i  =  0 ∀   i Thus the digaonal elements of a skew symmetric matrix are all zero , but not the converse .

Properties  Of  Symmetric  &  Skew  Matrix  :      P − 1 A  is symmetric  if AT  =  A  A is skew symmetric if AT  =  − A

P − 2 A  +  ATis a symmetric matrix  A  −  AT is a skew symmetric matrix . Consider  (A + AT)T  =  AT  + (AT)T      =   AT  +  A   =  A  +  AT A  +  AT  is  symmetric . Similarly we can prove that   A  −  AT

is skew symmetric .

P − 3 The sum of two symmetric matrix is a symmetric matrix and the sum of two skew symmetric matrix is a skew symmetric matrix . Let AT  =  A   ; BT  =  B where  A & B  have the same order . (A +

B)T  =  A + B  Similarly we can prove the other

P − 4 If  A & B  are symmetric matrices then ,

(a) A B  +  B A  is a symmetric matrix   (b) AB − BA  is a skew symmetric matrix .

P − 5 Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.

A  =     (A +  AT)  +    (A −  AT)

P           Q

Symmetric Skew Symmetric

[ ] a11 a12 a13

9. Adjoint  Of  A  Square  Matrix  :  Let A  =  ai j  =  a21 a22 a23   be a square  matrix

a31 a32 a33

C11 C12 C13

and let the matrix formed by  the cofactors of [ai j ] in determinant  A   is =  C21 C22 C23 .   Then

C C C 

(adj A)  =  CC1211 CC2221 CC3231V. Imp. Theorem :  A (adj. A) = (adj. A).A = |A| I31 n , If  A be a square32 33

C13 C23 C33

matrix of order n. Note : If  A  and B are non singular square matrices of same order, then (i) | adj A | = | A |n – 1 (ii) adj (AB)   =   (adj B) (adj A)

(iii) adj(KA)    =   Kn–1 (adj A), K is a scalar

Inverse  Of  A  Matrix (Reciprocal Matrix) :A  square matrix  A said to be invertible (non singular) if there exists a matrix B such that, A B  =  I  =  B A

B  is called the inverse (reciprocal) of A and is denoted by  A −1 . Thus

A −1  =  B  ⇔  A B  =  I  =  B A .    We have , A . (adj A)  =  A  In

A −1  A  (adj A)  =  A −1 In  |Ξ|; In  (adj A)  =  A −1   A  In A −1  =   (adj A|A| )

Note : The necessary and sufficient condition for a square matrix A to be invertible is that A ≠  0. Imp. Theorem :  If A & B are invertible matrices ofthe same order , then  (AB) −1  =  B −1  A −1. This is reversal law for inverseNote :(i)If  A  be an invertible matrix , then  AT  is also invertible  & (AT) −1 = (A −1)T.

(ii) If  A  is invertible,   (a)     (A −1) −1  =  A  ;  (b)     (Ak) −1  =  (A− 1)k = A–k, k  ∈  N

(iii) If A is an Orthogonal Matrix.     AAT = I = ATAwww.bloggermayank.online (iv) A square matrix is said to be orthogonal if ,  A −1  =  AT .

1

(v) | A–1 | =   SYSTEM  OF  EQUATION  &  CRITERIAN  FOR CONSISTENCY | A |

GAUSS - JORDAN METHOD x + y + z  =  6, x − y + z = 2, 2 x + y − z = 1

 x y z+ +  6

or 2x y zx y z− ++ −   =  12 112 −111 −111   xyz  =  162

(adj. A).B A X  =  B   ⇒ A −1  A  X  =  A −1  B   ⇒ X  =   −1  B  =   | A | .

Note  :(1)If A ≠  0, system is consistent having unique solution

(2)If  A  ≠  0   &  (adj A) .  B  ≠  O  (Null matrix) , system is consistent having unique non − trivial solution .(3) If  A  ≠  0   &  (adj A) .  B  =  O    (Null matrix) , system is consistent having trivial solution

(4) If      A  =  0  ,   matrix method fails

Consistent (Infinite solutions) Inconsistent (no solution)

================================================================================ 7.LOGARITHM AND THEIR PROPERTIES

THINGS  TO  REMEMBER :1.LOGARITHM  OF  A  NUMBER :The  logarithm  of  the  number  N

to  the  base  'a'  is  the  exponent  indicating  the  power  to which  the  base  'a'  must  be  raised  to  obtain the  number  N.  This  number  is  designated  as  loga N. Hence :  logaN = x ⇔  ax = N   ,  a > 0, a ≠ 1 &  N >0If  a = 10then  we  write  log b  rather  than  log10 b If  a = e ,  we  write  ln b  rather  than  loge b.The existence  and  uniqueness  of  the  number  loga N  follows  from  the  properties  of  an experimental functions  . From  the  definition  of  the logarithm of the number N to the base 'a' , we have an identity log N

:a a = N  ,  a > 0  ,  a ≠ 1  &  N > 0  This  is  known  as  the  FUNDAMENTAL  LOGARITHMIC  IDENTITY

NOTE  : loga1 = 0(a > 0  ,  a ≠ 1);  loga a = 1(a > 0  ,  a ≠ 1)   and log1/a a = - 1 (a > 0  ,  a ≠ 1)

2. THE  PRINCIPAL  PROPERTIES  OF  LOGARITHMS :Let  M & N  are  arbitrary  posiitive  numbers

,  a > 0 , a ≠ 1 , b > 0 , b ≠ 1  and  Ξ±  is  any  real number  then  ;

(i)loga (M . N) = loga M + loga N (ii) loga (M/N) = loga M − loga N

log M

(iii) loga MΞ± =  Ξ± . loga M (iv) logb M = a log b a

NOTE :   logba . logab = 1  ⇔  logba = 1/logab.   logba . logcb . logac = 1

logy x . logz y . loga z = logax.   eln ax = ax

3. PROPERTIES  OF  MONOTONOCITY  OF  LOGARITHM :

(i) For  a > 1  the  inequality  0 < x < y  &  loga x <  loga y  are  equivalent.

(ii) For  0 < a < 1  the  inequality  0 < x < y  &  loga x > loga y  are  equivalent.

(iii) If  a > 1   then  loga x < p 0 < x < ap

(iv) If  a > 1   then  logax > p x > ap  (v)  If  0 < a < 1 then  loga x < p ⇒ x > ap

(vi) If  0 < a < 1 then  logax > p 0 < x < ap NOTE THAT :

If the number & the base are on one side of the unity , then the logarithm is  positive ;  If the number & the base are on different sides of unity, then the logarithm is negative.

The base  of  the logarithm  ‘a’  must not equal unity otherwise numbers not  equal to unity will not  have a  logarithm &  any  number  will  be the logarithm  of  unity.

For a non negative number  'a'  &  n ≥ 2 ,  n ∈ N      n a = a1/n.

================================================================================

8.PROBABILITYTHINGS TO REMEMBER :RESULT − 1

(i) SAMPLE–SPACE : The set of  all possible outcomes of an experiment is called the SAMPLE–SPACE(S).

(ii) EVENT  : A sub set of sample−space is called an EVENT.

(iii) COMPLEMENT OF AN EVENT A :  The  set  of  all  out  comes  which    are  in  S  but  not  in A is called  the

COMPLEMENT OF THE EVENT  A  DENOTED  BY  A OR  AC .

(iv) COMPOUND EVENT  :  If  A & B are  two  given  events  then  A∩B  is called  COMPOUND  EVENT and is denoted by  A∩B or  AB  or  A & B .

(v) MUTUALLY EXCLUSIVE EVENTS  :  Two  events  are  said  to  be  MUTUALLY  EXCLUSIVE  (or disjoint or incompatible)  if the occurence of one precludes (rules out) the simultaneous  occurence of the other . If  A & B are two mutually exclusive events then P (A & B) = 0.

(vi) EQUALLY LIKELY EVENTS  : Events are said to be EQUALLY LIKELY when each event is as likely to occur  as any other  event.

(vii) EXHAUSTIVE EVENTS  : Events  A,B,C ........ L are said to be  EXHAUSTIVE EVENTS if no  event outside this set can result as an outcome of an experiment . For example, if A & B are two

events defined on a sample space S, then A & B are exhaustive  ⇒  A ∪ B = S⇒  P (A ∪ B) = 1 .

(viii) CLASSICAL DEF. OF  PROBABILITY  :  If n represents the total  number of equally likely , mutually exclusive and exhaustive outcomes of  an experiment and m of them are  favourable to the  happening  of   the event A, then  the  probability  of happening  of  the  event A is given  by  P(A) = m/n  .

Note : (1) 0 ≤  P(A)  ≤   1 (2) P(A) + P( A ) = 1,   Where  A = Not  A

x

(3) If x cases are favourable to A &  y cases are favourable to A then P(A) =   and  P( A ) =

(x + y)

y

We  say that ODDS IN FAVOUR OF A are x: y & odds  against A are  y : x (x + y)

Comparative study of  Equally likely , Mutually Exclusive and Exhaustive events.

Experiment          Events      E/L        M/E        Exhaustive

1. Throwing of a die    A : throwing an odd face {1, 3, 5} No Yes        No

B : throwing a composite face {4,. 6}

2. A ball is drawn from E1 : getting a W ball

an urn containing 2W, E2 : getting a R ball No Yes Yes

3R and 4G balls E3 : getting a G ball

3. Throwing a pair of A : throwing a doublet

dice {11, 22, 33, 44, 55, 66}

B : throwing a total of 10 or      more {46, 64, 55, 56, 65, 66} Yes No No

4. From a well shuffled E1 : getting a heart

pack of cards a card is E2 : getting a spade Yes Yes Yes

drawn        E3 : getting a diamond

E4 : getting a club

5. From a well shuffled A = getting a heart

pack of cards a card is drawn B = getting a face card No No No

RESULT − 2www.bloggermayank.online

AUB = A+ B = A or  B  denotes  occurence  of  at least A or B. For 2  events  A & B : (See fig.1)

(i) P(A∪B) = P(A) + P(B) − P(A∩B) =

P(A. B) + P( A .B) + P(A.B) = 1 − P( A . B)

(ii) Opposite of′  " atleast A or  B " is   NIETHER

A NOR  B  .e.  A + B = 1-(A or B) = A∩ B Note that     P(A+B) + P( A ∩ B ) = 1.

(iii) If A & B are mutually exclusive then  P(A∪B) = P(A) + P(B).

(iv) For any two events A & B,  P(exactly one of  A , B occurs)

(v) If  A & B are  any  two  events  P(A∩B) = P(A).P(B/A) = P(B).P(A/B), Where P(B/A) means  conditional probability  of  B given A &  P(A/B) means conditional probability  of A given B. (This can be easily seen from the figure)

(vi) DE MORGAN'S LAW  : − If A & B are two subsets of a universal set  U , then

(a) (A∪B)c = Ac∩Bc     & (b) (A∩B)c = Ac∪Bc

(vii) A ∪ (B∩C) = (A∪B) ∩ (A∪C)    & A ∩ (B∪C) = (A∩B) ∪ (A∩C)

RESULT − 3

For any three events A,B and C we have  (See Fig. 2)

(i) P(A or B or C) = P(A) + P(B) + P(C) − P(A∩B) − P(B∩C)−

P(C∩A) + P(A∩B∩C)

(ii) P (at least two of A,B,C  occur)  =

P(B∩C) + P(C∩A) +

P(A∩B) − 2P(A∩B∩C)

(iii) P(exactly two of A,B,C  occur) = P(B∩C) + P(C∩A) +

P(A∩B) − 3P(A∩B∩C) www.bloggermayank.online

(iv) P(exactly one of A,B,C  occurs) =

P(A) + P(B) + P(C) − 2P(B∩C) − 2P(C∩A) −  2P(A∩B)+3P(A∩B∩C)Fig. 2

NOTE : If three  events A, B and C are pair wise mutually exclusive then they must be mutually exclusive. i.e  P(A∩B) = P(B∩C) = P(C∩A) = 0 ⇒ P(A∩B∩C) = 0. However  the converse of  this  is not  true.

RESULT − 4 INDEPENDENT EVENTS   :  Two events A & B are said to be independent if occurence  or non  occurence of one does not effect the probability of the occurence or non occurence of  other.

(i) If the occurence of one event  affects the  probability  of the occurence  of the other event  then the events are said to be DEPENDENT or CONTINGENT . For two independent events

A and B :  P(A∩B) = P(A). P(B). Often this is taken as the definition of independent  events.

(ii) Three events A , B & C are independent if & only if all the following conditions hold ;

P(A∩B) = P(A) . P(B) ; P(B∩C) = P(B) . P(C) P(C∩A) = P(C) . P(A) & P(A∩B∩C) = P(A) . P(B) . P(C)

i.e. they must be pairwise as well  as mutually independent  . Similarly  for  n  events  A1 , A2 , A3 , ...... An  to  be independent , the number of  these  conditions is equal to nc2 + nc3 + ..... + ncn = 2n − n − 1.

(iii)The probability  of  getting exactly r success in n  independent trials is given  by P(r) =  nCr pr qn−r where:  p = probability of success in a single trial  q = probability of failure in a single trial. note : p + q = 1 Note : Independent events are not in general mutually exclusive & vice versa.

Mutually exclusiveness can be used when the events are taken from the same experiment & independence can be used when the events are taken from different experiments .

RESULT − 5  :   BAYE'S THEOREM   OR   TOTAL PROBABILITY THEOREM  :

If  an event A can occur only with one of  the n mutually exclusive and exhaustive events B1, B2, .... Bn & the probabilities P(A/B1) , P(A/B2) ....... P(A/Bn) are known then,   (B1/A) = ∑n P B( i ).P A( (/ Bi ) )

P

P B( i ).P A / Bi

i=1

PROOF :The  events A occurs with one of the n mutually exclusive & exhaustive events B1,B2,B3,........Bn; A = AB1 + AB2 + AB3 + ....... + ABn

n

P(A) = P(AB1) + P(AB2) +.......+ P(ABn) =    P(ABi )

i=1

NOTE : A  ≡ event what we have  ;

B1   ≡ event what we want  ; B2, B3, ....Bn  are alternative  event .

Now, P(ABi ) = P(A) . P(Bi/A)

= P(Bi ) . P(A/Bi)

( i ) P B( i ) . P A B( / i ) P B( i ) . P A B( / )

P B / A = i

P A( ) n

∑ P AB( i )

i=1

P B( i / A) = ∑P B( i ) .i P A B( ( / i ) ) P B( ) . P A B/ i

RESULT − 6 If p1 and  p2 are the probabilities of speaking the truth of two indenpendent witnesses A

p1 p2 and B thenP (their combined statement is true) =     . In this case it has been p p1 2 + −(1 p1)(1− p2)

assumed that we have no knowledge of the event except the statement made by A and B. However if p is the probability of the happening of the event before their statement then

pp1 p2

P (their combined statement is true) =     .

pp p1 2 + −(1 p)(1− p1)(1− p2 )

Here it has been assumed that the statement given by all the independent witnesses can be given  in two ways only, so that if all the witnesses tell falsehoods they agree in telling the same falsehood. If this is not the case and c is the chance of their coincidence testimony then the

Pr. that the statement is true = P p1 p2 Pr. that the statement is false = (1−p).c (1−p1)(1−p2) However chance of coincidence testimony is taken only if the joint statement is not contradicted by any witness.

RESULT − 7  (i)   A PROBABILITY DISTRIBUTION  spells out how a total probability of 1 is distributed over several values of a random variable .www.bloggermayank.online

∑ p xi i ∑ i

(ii)Mean of any probability distribution of a random variable is  given by  :Β΅= = p xi

∑pi

( Since Ξ£ pi = 1 ) (iii) Variance of a random variable is given by,  Ο² =  ∑ ( xi − Β΅)² . pi Ο² =  ∑ pi x²i − Β΅²    ( Note  that  SD = + Ο2 )(iv) The probability distribution for a binomial variate

‘ X ’ i s g i v e n b y   ; P ( X = r ) = nCr pr qn−r  where all symbols have the same meaning as given in result 4. The

recurrence formula  P r( +1) = n − r .  p   ,  is very helpful for quickly computing

P r( ) r +1 q

P(1) , P(2). P(3) etc.  if  P(0) is known .   (v) Mean of  BPD = np ;   variance of BPD = npq  . (vi) If p represents a persons chance of success in any  venture  and ‘M’ the  sum  of money  which he will receive in case of success, then his expectations or probable value = pM expectations = pM

RESULT − 8  :  GEOMETRICAL APPLICATIONS   : The following statements are axiomatic  :

(i) If a  point  is  taken  at  random  on  a  given  staright  line  AB, the chance that  it falls  on  a  particular segment  PQ  of  the  line  is  PQ/AB .   (ii) If  a  point  is  taken  at  random  on  the  area  S  which  includes   an area  Ο ,  the chance  that  the  point  falls  on  Ο  is  Ο/S .

9.FUNCTIONS

THINGS TO REMEMBER    : 1. GENERAL DEFINITION   :

If to every value (Considered as real unless other−wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and  only one  finite value of the quantity y, then  y is said to be a function (Single valued) of x or a dependent variable defined on the set E ;  x is the argument or independent variable .

If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as the  meaning  of a  single  valued function, if not otherwise stated.    Pictorially : →x

input

→f x( )=y , y  is called the image of x & x is the pre-image of y under f.  Every function from  A → B output

satisfies the following conditions .

(i) f ⊂  A x B  (ii) ∀  a ∈ A ⇒ (a, f(a)) ∈ f  and (iii)(a, b) ∈ f   &   (a, c) ∈ f  ⇒  b = c

2. DOMAIN,  CO−DOMAIN  &  RANGE  OF  A  FUNCTION :

Let  f :  A → B, then  the  set  A is  known  as the domain of  f & the set B is known as co-domain of f . The set of all  f images of elements of A is known as the range of  f . Thus Domain of  f = {a  a ∈ A, (a, f(a)) ∈ f} Range of  f = {f(a)  a ∈ A, f(a) ∈ B} It should be noted that range is a subset of co−domain . If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.

3. IMPORTANT  TYPES  OF  FUNCTIONS  : (i) POLYNOMIAL  FUNCTION :

If a function f is defined by f (x) = a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an where n  is a non negative integer and a0, a1, a2, ..., an are real numbers and a0 ≠ 0, then f  is called a polynomial function of degree n

NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function .  i.e.  f(x) = ax ,  a ≠ 0

(b) There  are  two polynomial  functions ,  satisfying  the  relation ; f(x).f(1/x) = f(x) + f(1/x).  They are  :

(i) f(x) = xn + 1  & (ii)  f(x) = 1 − xn   ,  where n is a positive integer .

(ii) ALGEBRAIC  FUNCTION   :y is an algebraic function of x, if it is a function that satisfies an algebraicequation of the formP0 (x) yn + P1 (x) yn−1 + ....... + Pn−1 (x) y + Pn (x) = 0  Where n is a positive integer and P0 (x), P1 (x) ........... are Polynomials in x.

e.g.  y = x is an algebraic function, since it satisfies the equation y² − x² = 0.

Note that all polynomial functions are Algebraic but not the converse.  A function that is not algebraic is called TRANSCEDENTAL FUNCTION .www.bloggermayank.online

g(x)

(iii) FRACTIONAL  RATIONAL  FUNCTION   : A rational function is a function of the form.   y =  f (x)  =   , h(x)

where   g (x) &  h (x) are polynomials  & h (x) ≠ 0.

(iv) ABSOLUTE  VALUE  FUNCTION   : A function y = f (x) = x is called  the  absolute  value  function  or

x if x ≥ 0

Modulus function.  It is defined as  :  y = x= −x if x < 0

(V) EXPONENTIAL  FUNCTION  :A  function  f(x) = ax = ex ln a (a > 0 ,  a ≠ 1, x ∈ R) is called anexponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = loga x . Note that  f(x) & g(x) are inverse of each other & their graphs are as shown .

+ ∞

(vi) SIGNUM  FUNCTION   :

A function y= f (x) = Sgn (x) is defined as follows :

1 for x > 0 y = f (x) = 0 for x = 0 −1 for x < 0

It is also written as Sgn x = |x|/ x  ;  x ≠ 0 ;  f (0) = 0

(vii) GREATEST  INTEGER  OR  STEP UP  FUNCTION  : The function  y = f (x) = [x]  is called the greatest integer  function where [x]  denotes the greatest integer less than or equal to x . Note that for :

− 1 ≤ x <  0 ; [x] = − 1 0 ≤ x <  1

1 ≤ x <  2 ; [x] =  1 2 ≤ x  <  3

and  so  on .

Properties of  greatest  integer function  :

(a) [x] ≤ x < [x] + 1   and

x − 1 < [x] ≤ x ,  0 ≤ x − [x] < 1

(b) [x + m] = [x] + m   if  m  is an integer . y

(c) [x] + [y] ≤ [x + y] ≤  [x] + [y] + 1

(d) [x] + [− x]  = 0   if  x  is an integer 1

= − 1  otherwise .

(viii) FRACTIONAL  PART  FUNCTION  :

x

It  is  defined  as : −1 g (x) = {x} = x − [x] .

e.g. the fractional part of the no. 2.1 is

2.1− 2 = 0.1 and the fractional part  of − 3.7 is 0.3.  The period of this function is 1 and graph of thi s function is as shown .www.bloggermayank.online

4. DOMAINS  AND RANGES OF COMMON FUNCTION :

Function           Domain             Range

(y = f (x) ) (i.e. values taken by x) (i.e. values taken by f (x) )

A. Algebraic Functions

(i) xn   ,  (n ∈ N) R = (set of real numbers) R ,        if n is odd

R+ ∪ {0} ,   if n is even

1

(ii) n , (n ∈ N) R – {0} R – {0} ,    if n is odd x

R+  ,        if n is even

(iii) x1/n ,  (n ∈ N) R ,        if n is odd R ,        if n is odd

R+ ∪ {0} ,   if n is even R+ ∪ {0} ,   if n is even

Function Domain            Range

(y = f (x) ) (i.e. values taken by x) (i.e. values taken by f (x) )

1

(iv) 1/n , (n ∈ N) R – {0} ,   if n is odd R – {0} ,    if n is odd x

R+  ,        if n is even R+  ,        if n is even

B. Trigonometric Functions

(i) sin x R [–1, + 1]

(ii) cos x R [–1, + 1]

(iii) tan x R – (2k + 1) ,k I R

(iv) sec x R – (2k + 1) ,k I (– ∞ , – 1 ] ∪ [ 1 , ∞ )

(v) cosec x R – kΟ , k ∈ I (– ∞ , – 1 ] ∪ [ 1 , ∞ )

(vi) cot x R – kΟ , k ∈ I R

C. Inverse Circular  Functions (Refer after Inverse is taught )

(i) sin–1 x [–1, + 1] − Ο2 , Ο2 

(ii) cos–1 x [–1, + 1] [ 0, Ο]

(iii) tan–1 x R − Ο, Ο

 2 2 

 Ο Ο

(iv) cosec –1x (– ∞ , – 1 ] ∪ [ 1 , ∞ ) − 2 , 2  – { 0 } Ο 

(v) sec–1 x (– ∞ , – 1 ] ∪ [ 1 , ∞ ) [ 0, Ο] –   2 

(vi) cot –1 x R ( 0, Ο)

D. Exponential Functions

(i) ex R R+

(ii) e1/x R – { 0 } R+ – { 1 }

(iii) ax , a > 0 R R+

(iv) a1/x , a > 0 R  – { 0 } R+ – { 1 }

E. Logarithmic Functions

(i) logax , (a > 0 ) (a ≠ 1)      R+ R

1

(ii) logxa =  log xa (a > 0 ) (a ≠ 1)    R+ – { 1 }   R – { 0 }

F. Integral Part Functions Functions

(i) [ x ] R I

(ii) R – [0, 1 )  1 ,n∈ −I {0}

n

G. Fractional Part Functions

(i) { x } R [0, 1)

(ii) R – I (1, ∞)

H. Modulus Functions

Function Domain     Range

(y = f (x) )              (i.e. values taken by x)          (i.e. values taken by f (x) )

(i) | x | R R+ ∪ { 0 }

1

(ii) R – { 0 } R+

|x|

I. Signum Function

|x|

= ,x ≠0 R {–1, 0 , 1}

sgn (x) x

= 0 , x = 0

J. Constant Functionwww.bloggermayank.online say f (x) = c R { c }

5. EQUAL  OR  IDENTICAL  FUNCTION :

Two functions  f  &  g  are said to be equal  if :

(i) The domain of  f  =  the domain of  g.

(ii) The range of  f  =  the range of  g   and

(iii) f(x) = g(x)  ,  for every x belonging to their common domain.  eg.

1 x

f(x) =    &  g(x) =    are identical functions  . x x2

6. CLASSIFICATION  OF  FUNCTIONS : One − One Function (Injective mapping) :

A function  f :  A → B is said to be  a one−one  function  or  injective mapping if different  elements of  A have different f  images in B .  Thus for   x1, x2 ∈ A &  f(x1) , f(x2) ∈ B ,  f(x1) = f(x2)  ⇔  x1 = x2   or  x1 ≠  x2  ⇔  f(x1) ≠  f(x2) .

Diagramatically an injective mapping can be shown as

Note : (i) Any function which is entirely increasing or decreasing in whole domain, then f(x) is one−one .

(ii) If any line parallel to x−axis cuts the graph of the function atmost at one point, then the function is one−one .

Many–one function :

A function  f :  A → B  is  said  to  be  a  many  one  function  if two or more elements of A have  the  same

f image in  B . Thus  f :  A → B is  many  one  if for  ;  x1, x2 ∈ A ,  f(x1) = f(x2)   but  x1 ≠ x2 Diagramatically a many one mapping can be shown as

Note : (i) Any continuous function which has atleast one local maximum or local minimum,then  f(x) is

many−one . In  other  words,  if  a line parallel to x−axis  cuts  the  graph  of the  function  atleast at  two  points, then f is many−one .www.bloggermayank.online (ii) If a function is one−one, it cannot be many−one and vice versa .

Onto function (Surjective mapping) : If the function  f :  A → B  is such that each element in B (co−domain) is the f image of atleast one element in A, then we say that  f is a function of A 'onto' B . Thus f : A → B is surjective iff   ∀  b ∈ B,  ∃  some  a ∈ A  such that  f (a) = b .

Diagramatically surjective mapping can be shown as

Note that : if range = co−domain, then  f(x) is onto. Into function :

If  f :  A → B  is such that there exists atleast one element in co−domain which is not the image of any element in domain, then  f(x) is  into .

Diagramatically into function can be shown as

Note that : If a function is onto, it cannot be into and vice versa . A polynomial of degree even will always be into. Thus a function can be one of these four types :

(a) one−one onto (injective & surjective)

(b) one−one into (injective but not surjective)

(c) many−one onto (surjective but not injective)

(d) many−one into (neither surjective nor injective)

Note : (i) If  f is both injective & surjective, then it is called a Bijective mapping.

The bijective functions  are  also  named  as  invertible,  non singular  or biuniform functions.

(ii) If a  set  A contains n distinct  elements  then  the  number  of  different  functions defined from A→A is nn & out of it n ! are one one.

Identity function : The function  f :  A → A defined by  f(x) = x  ∀  x ∈ A is called the identity of A and is denoted by IA. It is easy to observe that identity function is a bijection .

Constant function : A function  f :  A → B is said to be a constant function if every element of A has the same f image in B . Thus  f :  A → B ;  f(x) = c ,  ∀  x ∈ A ,  c ∈ B  is a constant function. Note that the range of a constant function is a singleton and a constant function may be  one-one or many-one, onto or into .

7. ALGEBRAIC  OPERATIONS  ON  FUNCTIONS  :

If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A ∩ B. Now we define  f + g ,  f − g ,  (f .g) & (f/g) as follows :

(i) (f ± g) (x) = f(x) ± g(x)

(ii) (f . g) (x) = f(x) . g(x)

 f  f x( )  x ∈ A ∩ B  s . t  g(x) ≠ 0} .

(iii)   (x) = domain is  {x

 g g x( )

8. COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS :

Let  f :  A → B  &  g :  B → C  be two functions . Then the function gof :  A → C  defined by  (gof) (x) = g (f(x))  ∀  x ∈ A  is called the composite of the two functions  f & g .

Diagramatically  →x   →f(x)   →  g (f(x)) .Thus  the  image  of  every   x ∈ A under the function  gof is the g−image of the f−image of x .

Note that gof is defined only if  ∀  x ∈ A,  f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions  f & g, the range of f must be a subset of the domain of g. PROPERTIES  OF  COMPOSITE  FUNCTIONS  :

(i) The composite of functions is not commutative  i.e.  gof ≠ fog .

(ii) The composite of functions is associative  i.e.  if  f, g, h are three functions such that fo (goh) & (fog) oh  are defined, then  fo (goh) = (fog) oh .

(iii) The composite  of  two bijections is a bijection  i.e.  if  f & g are two bijections such that  gof is defined, then gof is also a bijection.www.bloggermayank.online

9. HOMOGENEOUS  FUNCTIONS :

A function  is  said  to  be  homogeneous  with  respect  to  any  set  of  variables when  each  of  its  termsis  of  the  same  degree  with  respect  to  those  variables  .

For  example  5 x2 + 3 y2 − xy  is  homogeneous  in  x & y . Symbolically if , f (tx , ty) = tn . f (x , y)  then  f (x , y) is homogeneous function of degree  n .

10. BOUNDED FUNCTION :

A function is said to be bounded if f(x) ≤ M , where M is a finite quantity .

11. IMPLICIT  &  EXPLICIT  FUNCTION :

A function  defined  by an  equation  not  solved for the dependent variable is called anIMPLICIT FUNCTION . For eg. the equation  x3 + y3 = 1 defines  y  as an implicit  function. If y has been expressed in terms of x alone then it is called an EXPLICIT FUNCTION.

12. INVERSE  OF  A  FUNCTION : Let  f :  A → B  be a  one−one  &  onto function,  then  their  exists a  unique  function g :  B → A  such that  f(x) = y  ⇔  g(y) = x,  ∀  x ∈ A  &  y ∈ B .  Then g is said to be inverse of f .  Thus  g = f−1 :  B → A =  {(f(x), x)  (x,  f(x)) ∈ f} .

PROPERTIES  OF  INVERSE  FUNCTION  : (i) The inverse of a bijection is unique .

(ii) If  f :  A → B  is a bijection & g :  B → A is the inverse of f, then  fog = IB and gof = IA ,  where  IA &  IB  are identity functions on the sets A & B respectively.

Note  that  the  graphs  of  f & g  are  the  mirror  images  of  each  other  in  the line  y = x . As shown in the figure given below a point (x ',y ' ) corresponding to y = x2 (x >0) changes to (y ',x ' ) corresponding

(iv) If  f & g  two bijections  f :  A → B ,  g :  B → C  then the inverse of gof exists and  (gof)−1 = f−1o g−1

13. ODD  &  EVEN  FUNCTIONS : If  f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x  ;  g (x) = x² + 3 . If f (−x) = −f (x) for all x in the domain of ‘f’ then f is said to be an odd function. e.g. f (x) = sin x  ;   g (x) = x3 + x .

NOTE : (a) f (x) − f (−x) = 0 =>  f (x) is even  &  f (x) + f (−x) = 0 => f (x) is odd .

(b) A function may neither be odd nor even .(c)Inverse  of  an  even  function  is  not  defined (d) Every even function is symmetric about the y−axis  &  every odd  function is symmetric about the origin .

(e) Every function can be expressed as the sum of an even & an odd function.

e.g. f x( ) = f x( )+f(−x) + f x( )−f(−x)

(f) only function which is defined on the entire number line & is even and odd at the same time is f(x)= 0.

(g) If f and g both  are even or both are odd then the function  f.g  will  be even but if any one of them is odd then f.g  will  be odd .

14. PERIODIC  FUNCTION : A function  f(x) is  called  periodic  if  there exists a positive number T (T > 0) called the period  of the  function  such  that  f (x + T) = f(x),  for  all  values  of  x within the domain of x e.g. The function sin x & cos x both are periodic over 2Ο & tan x is periodic over Ο NOTE : (a) f (T) = f (0) = f (−T) ,   where ‘T’ is the period .

(b) Inverse of a periodic function does not exist .www.bloggermayank.online

(c) Every constant function is always periodic, with no fundamental period .

(d) If  f (x)  has  a period  T  &  g (x)  also  has  a  period T  then it does not  mean that   f (x) + g (x)  must have a period T .   e.g.  f (x) = sinx + cosx.

1

(e) If  f(x) has a period p, then   and  f(x) also has a period p . f(x)

(f) if  f(x) has a period T then f(ax + b) has a period  T/a  (a > 0) .

15. GENERAL : If  x, y are independent variables, then :

(i) f(xy) = f(x) + f(y)   ⇒  f(x) = k ln x  or   f(x) = 0 .

(ii) f(xy) = f(x) . f(y)   ⇒  f(x) = xn ,   n ∈ R(iii) f(x + y) = f(x) . f(y)    ⇒  f(x) = akx . (iv) f(x + y) = f(x) + f(y)  ⇒  f(x) = kx,  where k is a constant . 10.INVERSE TRIGONOMETRY FUNCTION

GENERAL  DEFINITION(S):1. sin−1 x , cos−1 x , tan−1 x  etc.  denote angles  or  real  numbers  whose  sine is  x ,  whose  cosine is  x  and  whose  tangent  is  x,  provided that  the  answers  given  are   numerically smallest available .  These   are   also   written  as  arc sinx ,  arc cosx  etc .

If  there  are  two  angles  one  positive  &  the  other  negative  having  same  numerical  value, then positive  angle  should  be  taken  .

2. PRINCIPAL  VALUES  AND  DOMAINS  OF INVERSE CIRCULAR FUNCTIONS :(i)y = sin−1 x

where   −1 ≤ x ≤ 1   ;       and   sin y = x  .

(ii) y = cos−1 x   where   −1 ≤ x ≤ 1   ;   0 ≤ y ≤ Ο   and   cos y = x  .

(iii) y = tan−1 x    where   x ∈ R  ;     and  tan y = x  . (iv) y = cosec−1 x  where  x ≤ − 1  or  x ≥ 1  ;    0 and cosec y = x

(v) y = sec−1 x  where  x ≤ −1  or  x ≥ 1  ;  0    and  sec y = x  .

(vi) y = cot−1 x  where  x ∈ R ,  0 < y < Ο  and  cot y = x  .

NOTE  THAT : (a) 1st  quadrant  is  common  to  all  the  inverse  functions  . (b) 3rd  quadrant  is  not  used  in  inverse  functions  .

(c) 4th  quadrant  is  used  in  the  CLOCKWISE DIRECTION  i.e.   .

3. PROPERTIES  OF  INVERSE  CIRCULAR  FUNCTIONS  :

P−1 (i)  sin (sin−1 x) =  x  ,  −1 ≤ x ≤ 1 (ii)  cos (cos−1 x) = x  ,  −1 ≤ x ≤ 1

(iii)  tan (tan−1 x) = x  ,  x ∈ R (iv)  sin−1 (sin x) = x  ,

(v)  cos−1 (cos x) = x  ;  0 ≤ x ≤ Ο (vi)  tan−1 (tan x) = x  ;

P−2 (i)  cosec−1 x = sin−1       ;     x ≤ −1 ,  x ≥ 1 (ii)  sec−1 x = cos−1          ;     x ≤ −1  ,  x ≥ 1 x x

(iii)  cot−1 x = tan−1   1       ;    x > 0        = Ο + tan−1   1   ;    x < 0 x x

P−3 (i) sin−1 (−x) = − sin−1 x      ,    −1 ≤ x ≤ 1 (ii) tan−1 (−x) = − tan−1 x      ,    x ∈ R

(iii) cos−1 (−x) = Ο − cos−1 x    ,    −1 ≤ x ≤ 1(iv) cot−1 (−x) = Ο − cot−1 x    ,    x ∈ R

1 1

P−4 (i)  sin−1 x + cos−1 x =    1     (ii)  tan−1 x + cot−1 x =        x ∈ R

(iii)  cosec−1 x + sec−1 x =       x ≥ 1www.bloggermayank.online

P−5 tan−1 x + tan−1 y = tan−1  x + y   where   x > 0  ,   y > 0   &   xy < 1

1 − xy

= Ο + tan−1  x + y   where  x > 0  ,  y > 0  &  xy > 1

1 − xy

tan−1 x − tan−1y = tan−1 x − y   where  x > 0  ,  y > 0

1 + xy

P−6 (i)    sin−1 x + sin−1 y = sin−1 x 1 − y2 + y 1 − x2   where   x ≥ 0 ,y≥0 & (x2 + y2) ≤ 1

Note that :  x2 + y2 ≤ 1     ⇒ 0 ≤ sin−1 x + sin

(ii) sin−1 x + sin−1 y = Ο − sin−1 x  1 − y2 + y

1 − x2  where  x≥0,y ≥ 0  &  x2 + y2 > 1

Note that :  x2 + y2 >1     ⇒   < sin−1 x + sin−1 y  < Ο

(iii) sin–1x – sin–1y = sin−1[x 1 y− 2 −y 1 x− 2 ] where x > 0 , y > 0

(iv) cos−1 x + cos−1 y = cos−1 [xy ∓ 1−x2 1−y2] where  x ≥ 0 ,  y ≥ 0

P−7 If  tan−1 x + tan−1 y + tan−1 z =  tan−1 1x−+xyy +−zyz− −xyzzx if, x >0,y>0,z>0 & xy+yz+zx<1

Note : (i) If  tan−1 x + tan−1 y + tan−1 z =  Ο   then   x + y + z = xyz

(ii) If  tan−1 x + tan−1 y + tan−1 z =     then  xy + yz + zx = 1

P−8 2 tan−1 x =  sin−1 2x 2 = cos−1  1 − x22 = tan−1 2x 2 Note very carefully that :

1 + x 1 + x 1 − x

sin−1 2x 2 =   Ο− 2tan−1 x

1 + x − +(Ο 2tan−1 x)

if if x > 1 cos−1  1 − x22 =  2tan −x1

1 + x − 2tan x

x <− 1 if if x 0 x < 0

 2tan−1x

tan−1 2x 2 =   Ο+2tan−1x

1 − x ( −1 )

− Ο−2tan x

if if if x <1

x < −1 x >1

2tan−1 x if x ≤ 1 −1

REMEMBER  THAT : (i) sin−1 x + sin−1 y + sin−1 z =          ⇒ x = y = z = 1

(ii) cos−1 x + cos−1 y + cos−1 z = 3Ο       ⇒ x = y = z = −1

(iii) tan−1 1 + tan−1 2 + tan−1 3 = Ο        and   tan−1 1 + tan−1 21 + tan−1 13 =

INVERSE  TRIGONOMETRIC  FUNCTIONS

9. (a)

3. y = tan −1 x , x ∈ R , y ∈ − Ο2 , Ο2 4. y = cot −1 x , x ∈ R , y ∈ (0 , Ο)

y = tan (tan −1 x) ,  x ∈ R ,  y  ∈ R  ,  y  is aperiodic 9. (b)y = tan (tan x) ,

y = cosec −1 (cosec x), 11. (b)        y =  cosec (cosec −1 x) ,

7. (a) y = sin −1 (sin x) , x ∈ R , y ∈ − Ο , Ο, 7.(b) y = sin (sin −1 x) ,    =  x    =  x

 2 2 y is periodic with period 2Ο ; x ≥ 1 ; y ≥ 1], y  is  aperiodic

Periodic with period  2 Ο   =  x  x ∈ [− 1 , 1] ,  y ∈ [− 1 , 1]  ,  y  is x ∈ R –  (2n 1) Ο n I y ∈0 , Ο2 ∪ Ο2 , Ο

=  x periodic with period 2 Ο   =  x                       x ∈ [− 1 , 1] ,  y ∈ [− 1 , 1], y is aperiodic

=  x            =  x

11. Limit and Continuity & Differentiability of Function

THINGS TO REMEMBER :

1. Limit of a function f(x) is said to exist as, x→a when Limitx→a− f(x) = Limitx→a+ f(x) = finite quantity

2. FUNDAMENTAL THEOREMS ON LIMITS :

Let Limitx a→ f (x) = l  & Limitx→a g (x) = m. If l & m exists then :

(i) Limitx→a f (x) ± g (x) = l ± m     (ii)   Limitx→a f(x). g(x) = l. m

f (x)

(iii) Limitx→a g g( ) = m  , provided m ≠ 0www.bloggermayank.online

(iv) Limitx→a k f(x) = k Limitx→a f(x)  ; where k is a constant.

Limit f [g(x)] = fLimit g xx→a ( ) = f (m) ; provided f is continuous at g (x) = m.

(v) x→a

Limit l n (f(x) = l nLimit f xx→a ( ) l n l (l > 0).

For example x→a

3. STANDARD LIMITS :

(a) Limit→   = 1 = Limitx→0 tanx = Limitx→0 tan−1x = Limitx→0  sin−1x [ Where x is measured in radians ]

sinx

x 0

x x x x

(b) Limitx→0 (1 + x)1/x = e=  1+ x1xnote however the reLimitnh→→∞0 (1 - h )n = 0and Limitnh →→ ∞0 (1 + h )n → ∞

Limit f(x) = 1  and Limitx→a  Ο (x) = ∞ , then ; Limit→a [f(x)]Ο(x)=eLimitx→a Ο(x)[f(x) 1]−

(c) If x→a x

(d) If Limitx→a f(x) = A > 0 & Limitx→a Ο (x) = B (a finite quantity) then  ;

Limitx→a [f(x)] Ο(x) = ez where z = Limitx→a Ο (x). ln[f(x)] =  eBlnA = B

(e) Limit→0 a xx−1 = 1n a (a > 0). In particularLimitx→0  exx−1 = 1 (f)Limitx→a xxn −−aan =nan−1 x

4. SQUEEZE PLAY THEOREM :

If f(x) ≤ g(x) ≤ h(x) ∀ x  & Limitx→a f(x) = l = Limitx→a h(x)  then Limitx→a g(x) = l.

5. INDETERMINANT FORMS  :     , , 0x∞ , 0 ,° ∞° ∞−∞, and 1∞

Note :

(i) We cannot plot ∞ on the paper. Infinity (∞) is a symbol & not a number. It does not obey the laws of elementry algebra.

(ii) ∞ + ∞ = ∞   (iii)    ∞ × ∞ = ∞  (iv)    (a/∞) = 0 if a is finite a b = 0 , if & only if a = 0 or b = 0  and  a & b are finite. (v) is not defined , if a ≠ 0. (vi)a

0

6. The following strategies should be born in mind for evaluating the limits:

(a) Factorisation (b) Rationalisation or double rationalisation

(c) Use of trigonometric transformation  ; appropriate substitution and using standard limits

(d) Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below :

2 2 3 3

(i)  ax = +1 x na1 + x 1n a + x 1n a +.........a > 0 (ii)  ex

1! 2! 3!

2 3 4 3 5 7

(iii) ln (1+x) = x− + − +x x x .........for − <1 x ≤1(iv) sinx =x − x + x − x +.......

2 3 4 3! 5! 7!

x2 x4 x6 x3 2x5 -1x =x − x3 + x5 − x7 +.......

(v)  cosx = − + − +1 ...... (vi)  tan x = x + + +........ (vii)tan

2! 4! 6! 3 15 3 5 7

(viii)  sin-1x = x + 12 x3 + 1 32. 2 x5 + 1 3 52. 2. 2 x7 +....... (ix)  sec-1x =1+ +x2 5x4 + 61x6 +......

3! 5! 7! 2! 4! 6! (CONTINUITY)

THINGS TO REMEMBER : 1. A function f(x) is said to be continuous at x = c, if Limit f(x) = f(c). Symbolically x→c

f is continuous at x = c if Limit f(c - h) =Limit f(c+h) = f(c). h→0 h→0

i.e. LHL at x = c = RHL at x = c equals Value of ‘f’ at x = c.

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighbourhood of x = a, not necessarily at x = a.

2. Reasons of discontinuity:www.bloggermayank.online

(i) Limit f(x) does not exist x→c

i.e. Limitx c− f(x) ≠ Limit→ + f (x)

x c

(ii) f(x) is not defined at x= c

(iii) Limit f(x) ≠ f (c) x→c

Geometrically, the graph of the function  will exhibit a break at x= c.

The graph as shown is discontinuous at x = 1 , 2 and 3.

3. Types of Discontinuities :

Type - 1: ( Removable type of discontinuities)

In caseLimit f(x) exists but is not equal to f(c) then the function is said to have a removable discontinuity x→c or discontinuity of the first kind. In this case we can redefine the function such that Limit f(x) = f(c) & x→c

make it continuous at x= c. Removable type of discontinuity can be further classified as  : (a) MISSING POINT DISCONTINUITY : Where Limit f(x) exists finitely but f(a) is not defined. x→a

(1− x)(9− x )2 sinx

e.g. f(x) = has a missing point discontinuity at x = 1 , and f(x) =   has a missing point

(1− x) x

discontinuity at x = 0

(b) ISOLATED POINT DISCONTINUITY : WhereLimit f(x) exists & f(a) also exists but ; Limit ≠ f(a). e.g. f(x) x→a x→a

=  x2 −16 , x ≠ 4 & f (4) = 9 has an isolated point discontinuity at x = 4. x −4

 0 if x ∈I

Similarly f(x) = [x] + [ –x] = has an isolated point discontinuity at all x∈ I.

 −1 if x ∉ I

Type-2: ( Non - Removable type of discontinuities) In caseLimit f(x) does not exist then it is not possible to make the function continuous by redefining it.

x→c

Such discontinuities are known as non - removable discontinuity oR discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as :

(a) Finite discontinuity  e.g. f(x) = x − [x] at all integral x ; f(x) = tan−1  1 at x = 0 and  f(x) = 11  at x = 0 (

x

1+2x

note that f(0+) = 0 ; f(0–) = 1 )

1 1 tanx at x = Ο and f(x) = cosx at

(b) Infinite discontinuity e.g. f(x) = x − 4 or g(x) = (x − 4)2  at x = 4 ; f(x) = 2 2 x

x = 0.

(c) Oscillatory discontinuity e.g. f(x) = sin  1 at x = 0.

x

In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but Limit does not exist.

x a→

Note: From the adjacent graph note that

f is continuous at x = – 1

f has isolated discontinuity at x = 1

f has missing point discontinuity at x = 2–  f has non removable (finite type)      discontinuity at the origin.

4. In case of dis-continuity of the second kind the non-negative difference between the value of the RHL at x = c & LHL at x = c is called THE JUMP OF DISCONTINUITY. A function having a finite number of jumps in a given interval I is called a PIECE WISE CONTINUOUS or SECTIONALLY CONTINUOUS function in this interval.

5. All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains.

6. If f & g are two functions that are continuous at x= c then the functions defined by :

F1(x) = f(x) ± g(x)  ;  F2(x) = K f(x) , K any real number  ; F3(x) = f(x).g(x)  are also continuous at x= c.

f x( )

Further, if g (c) is not zero, thenF4(x) =  g x( ) is also continuous atx= c.

7. The intermediate value theorem:

Suppose f(x) is continuous on an interval I , and a and b are any two points of I. Then if y0 is a number between f(a) and f(b) , their exists a number c between a and b such that f(c) = y0.NOTE VERY CAREFULLY THAT :www.bloggermayank.online

(a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function Ο(x) = f(x). g(x)

x ≠ 0

is not necessarily be discontinuous at x = a. e.g. f(x) = x & g(x) = 0 x = 0

(b) If f(x) and g(x) both are discontinuous at x = a then the product function  Ο(x) = f(x). g(x) is not necessarily be discontinuous at x = a. e.g.f(x) = − g(x) = −11 xx ≥< 00

(c) Point functions are to be treated as discontinuous. eg. f(x) = 1−x + x 1− is not continuous at x = 1.

(d) A Continuous function whose domain is closed must have a range also in closed interval.

(e) If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is continuous at x = c. eg.

xsinx xsinx

f(x) =  2 + 2  & g(x) = x are continuous at x = 0 , hence the composite (gof) (x) = x2 +2 will also be x

continuous at x = 0 .www.bloggermayank.online

7. CONTINUITY IN AN INTERVAL :

(a) A function f is said to be continuous in (a , b) if f is continuous at each & every point ∈(a,b).

(b) A function f is said to be continuous in a closed interval[a,b if :] (i) f is continuous in the open interval (a , b) &

(ii) f is right continuous at ‘a’ i.e.Limitx→a+ f(x) = f(a) = a finite quantity

(iii) f is left continuous at ‘b’ i.e.Limitx→b− f(x) = f(b) = a finite quantity

Note  that a function f which is continuous in[a,b possesses the following properties :]

(i) If f(a) & f(b) possess opposite signs, then there exists at least one solution of the equation f(x) = 0 in the open interval (a , b).

(ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open inetrval (a , b).

8. SINGLE POINT CONTINUITY:

Functions which are continuous only at one point are said to exhibit single point continuity

x if x∈Qx if x∈Q

e . g . f ( x ) = and g(x) = are both continuous only at x = 0.

−x if x∉Q0 if x∉Q

DIFFERENTIABILITY

THINGS TO REMEMBER :

1.     Right hand & Left hand Derivatives ; By definition: f ′(a) =Limith→0  f(a + h) f(a)h − if it exist

(i) The right hand derivative of f ′ at x = a denoted by f ′(a+) is defined by : f ' (a+) =Limith 0→ +  f(a + h) f(a)h − ,

provided the limit exists & is finite.www.bloggermayank.online

(ii) The left hand derivative : of f at x = a  denoted by

f ′(a+) is defined by : f ' (a–) =Limith 0→ +  f(a −−h) f(a)h− , Provided the limit exists & is finite.

We also write f ′(a+) = f ′+(a)  &  f ′(a–) = f ′_(a).

* This geomtrically means that a unique tangent with finite slope can be drawn at x = a as shown in the figure.www.bloggermayank.online

(iii) Derivability & Continuity :

(a) If f ′(a) exists then f(x) is derivable at x= a ⇒ f(x) is continuous at x = a. (b) If a function f is derivable at x then f is continuous at x.

For : f ′(x) = Limith→0 f(x + h) f(x)h −   exists.

Also f(x + −h) f(x)=f(x + h)−f(x).h[h ≠ 0] h f(x + h)−f(x)

Therefore : [f(x + h) f(x)]− =Limith→0 h .h=f '(x).0 0=

Therefore Limith→0 [f(x + h) f(x)]− = 0 ⇒Limith→0  f (x+h) = f(x) ⇒ f is continuous at x.

Note : If f(x) is derivable for every point of its domain of definition, then it is continuous in that domain.

The Converse of the above result is not true :

“ IF f IS CONTINUOUS AT x , THEN f IS DERIVABLE AT x ” IS NOT TRUE.

e.g. the functions f(x) = x & g(x) = x sin  1  ; x ≠ 0 & g(0) = 0 are continuous at x

x = 0 but not derivable at x = 0.

NOTE CAREFULLY :

(a) Let f ′+(a) = p & f ′_(a) = q where p & q are finite then :

(i) p = q ⇒ f is derivable at x = a  ⇒ f is continuous at x = a. (ii) p ≠ q ⇒ f is not derivable at x = a.

It is very important to note that f may be still continuous at x = a.

In short, for a function f :

Differentiability ⇒ Continuity ; Continuity⇒/ derivability ;

Non derivibality⇒/ discontinuous ; But discontinuity ⇒ Non derivability

(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at x = a.

3. DERIVABILITY OVER AN INTERVAL :

f (x) is said to be derivable over an interval if it is derivable at each & every point of the interval f(x) is said to be derivable over the closed interval [a, b] if  :

(i) for the points a and b, f ′(a+) & f ′(b −) exist &

(ii) for any point c such that a < c < b, f ′(c+) & f′(c −) exist & are equal.

NOTE :

1. If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) − g(x) , f(x).g(x) will also be derivable at x = a & if g (a) ≠ 0 then the function f(x)/g(x) will also be derivable at x = a.

2. If f(x) is differentiable at x = a & g(x) is not differentiable at x = a , then the product function F(x) = f(x).

g(x) can still be differentiable at x = a e.g. f(x) = x & g(x) = x.

3. If f(x) & g(x) both are not differentiable at x = a then the product function ; F(x) = f(x). g(x) can still be differentiable at x = a  e.g. f(x) = x & g(x) = x.

4. If f(x) & g(x) both are non-deri. at x = a then the sum function F(x) = f(x) + g(x) may be a differentiable function. e.g. f(x) = x & g(x) = −x.www.bloggermayank.online

5. If f(x) is derivable at x = a ⇒/ f ′(x) is continuous at x = a.www.bloggermayank.online

e.g. f(x) =x2 sin x1 if x ≠ 0

0 if x = 0

6. A surprising result : Suppose that the function f (x) and g (x) defined in the interval (x1, x2) containing the point x0, and if f is differentiable at x = x0 with f (x0) = 0 together with g is continuous as x = x0 then the function F (x) = f (x) · g (x) is differentiable at x = x0

e.g. F (x) = sinx · x2/3 is differentiable at x = 0.

12. DIFFERENTIATION & L' HOSPITAL RULE

1. DEFINITION :

If  x and x + h  belong  to  the  domain  of  a  function  f  defined  by   y = f(x),  then

Limit→ f x( + −h) f x( )  if  it  exists ,  is  called  the  DERIVATIVE  of  f  at  x  &  is  denoted  by f ′(x)  or  dy .  h 0 h dx

have therefore ,  f ′(x) = Limith→0 f x( + −hh) f x( )

2. The derivative of a given function f at a point x = a of its domain is defined as  :

Limith→0 f a( + −hh) f a( ) , provided the limit exists & is denoted  by  f ′(a) .

Note that alternatively, we can define f ′(a) = Limitx→a f x( )x−−f aa( ) ,  provided the limit exists.

3. DERIVATIVE OF f(x) FROM THE FIRST PRINCIPLE /ab INITIO METHOD:

Limit Ξ΄y

If f(x) is a derivable function then,  Ξ΄ →x 0 Ξ΄x = LimitΞ΄ →x 0 f x( +Ξ΄Ξ΄xx)−f x( ) = f ′(x) =  dxdy

4. THEOREMS ON DERIVATIVES :

If u and v are derivable function of x, then,

(i) d (u± =v) du ± dv (ii) d (K u) = K  du , where K is any constant dx dx dx dx dx

(iii) d (u . v) = u dv ± v du  known as  “ PRODUCT  RULE ” dx dx dx

d  u = v u   where v ≠ 0   known as  “ QUOTIENT  RULE ”

(iv) dx  v v2

dy dy du

(v) If  y = f(u)  &  u = g(x)  then = .    “ CHAIN  RULE ”

dx du dx

5. DERIVATIVE  OF  STANDARDS  FUNCTIONS :

(i) D (xn) = n.xn−1 ;  x ∈ R,  n ∈ R,  x > 0 (ii)  D (ex) = ex

(iii)  D (a

x) = ax. ln a   a > 0 (iv)  D (ln x) = x1 (v)  D (logax) = x1 logae

(vi)  D (sinx) = cosx (vii)  D (cosx) = − sinx (viii)  D = tanx =  sec²x (ix)  D (secx) = secx . tanx (x)  D (cosecx) = − cosecx . cotx

(xi)  D (cotx) = − cosec²x (xii)  D (constant) = 0   where  D =  d

dx

6. INVERSE  FUNCTIONS  AND  THEIR  DERIVATIVES :

(a) Theorem : If the inverse functions f & g are defined by y = f(x)  &  x = g(y) & if f ′(x) exists &

f ′(x) ≠ 0 then  g ′(y) = 1 . This result can also be written  as, if  dy exists &    dy ≠ 0 ,  then f (x)′ dx dx

dx = 1 /  dydx or dydx . dxdy =1 or dxdy =1 /  dydx [dxdy ≠ 0] dy

(b) Results  :

(i) D(sin−1x)= 1 , − <1 x<1 (ii) D(cos−1x)= , − <1 x<1

1− x2

(iii) D(tan−1 x)=  , x∈R (iv) D (sec−1 x)= (v) D(cosec−1x)=  −1 , x >1 (vi) D(cot−1 x)=  , x∈R

Note : In general if y =  f(u) then   dy = f ′(u) .  du  .

dx dx

7. LOGARITHMIC  DIFFERENTIATION : To find the derivative of  :

(i) a function which is the product or quotient of a number of functions OR

(ii) a function of the form [f(x)]g(x)  where f & g  are both derivable, it will be found convinient to take the logarithm of the function first & then  differentiate. This is called  LOGARITHMIC

DIFFERENTIATION.

8. IMPLICIT  DIFFERENTIATION :  Ο (x , y) = 0

(i) In order to  find  dy/dx,  in  the case of  implicit  functions, we  differentiate each term w.r.t.  x regarding y as a functions of x & then collect terms in dy/dx together on one  side  to  finally  find dy/dx.www.bloggermayank.online

(ii) In answers of dy/dx in the case of implicit functions, both  x & y are present .

9. PARAMETRIC  DIFFERENTIATION :

If  y = f(ΞΈ)  &  x = g(ΞΈ)  where ΞΈ  is  a parameter ,  then  dy = dy / dΞΈ  .

dx dx / dΞΈ

10. DERIVATIVE OF A FUNCTION W.R.T. ANOTHER FUNCTION :

dy dy / dx f'( )x

Let  y = f(x)  ;   z = g(x)   then  = = . d z d z / dx g'( )x

11. DERIVATIVES OF ORDER TWO & THREE :

Let a function y = f(x) be defined on  an open interval (a, b).  It’s derivative, if it exists on (a, b)  is  a certain function f ′(x) [or (dy/dx) or y ] &  is  called  the  first  derivative of y  w.r.t. x.If it happens that the first derivative has a derivative on (a , b) then this derivative is called the second derivative of y  w. r. t.  x  &  is denoted by f ′′(x) or (d2y/dx2) or  y ′′.Similarly, the 3rd order

d y3 d 

derivative of y w. r. t.  x , if  it exists, is defined by dx3 = dx  dxd y2 2    It is also denoted by  f ′′′(x) or y ′′′.

(iii)

f x( ) g x( ) h x( )

12. If  F(x)= l x( ) m x( ) n x( ) ,where  f ,g,h,l,m,n,u,v,w are differentiable functions of  x then

u x( ) v x( ) w x( )

f'(x) g'(x) h'(x)f(x) g(x) h(x)f(x) g(x) h(x)

F ′(x) = l x( ) m x( ) n x( )l x'( ) m x'( ) n x'( )   +    l x( ) m x( ) n x( ) u x( ) v x( ) w x( )u x( ) v x( ) w x( )u'( )x v'( )x w'( )x

13. L’  HOSPITAL’S  RULE :www.bloggermayank.online

If f(x)  &  g(x) are functions of x such that  :

(i) Limitx a→ f(x) = 0 = Limitx a→ g(x)    OR   Limitx→a f(x) = ∞ = Limitx→a g(x) and

(ii) Both f(x)  &  g(x) are continuous at x = a &

(iii) Both f(x) &  g(x) are differentiable at x = a &

(iv) Both f ′(x) &  g ′(x) are continuous at x = a , Then (IV)

Limitx→a g xf(( )x) Limit  f'(x) =Limitx→a gf""( )(xx) & soon till indeterminant form  vanishes.

= x→a g x'( )

14. ANALYSIS AND GRAPHS OF SOME USEFUL FUNCTIONS :

2 tan−1 x x ≤ 1

(i) y = f(x) = sin−1   2x 2  = Ο(− 2 tan−1 x ) x > 1

1 + x − +Ο 2 tan−1 x x <−1

HIGHLIGHTS :

(a) Domain  is  x ∈ R  &

range is  −Ο2 , Ο2

(b) f  is  continuous  for all  x  but  not  diff. at  x = 1 , - 1

(v)

 2 2 for x < 1

 1 + x

(c) d y = non existent for x = 1 d x − 1 +2x2 for x > 1

(d) I  in  (- 1 , 1)  &  D  in  (- ∞ , -  1)  ∪  (1 , ∞)

ii) Consider y = f (x) = cos-1  1 − x22 = 2 tan−1−x1 if x ≥< 00

1 + x  − 2 tan x if x

HIGHLIGHTS :

(a) Domain  is  x ∈ R  &

range  is  [0, Ο)

(b) Continuous  for  all  x

but  not  diff. at  x = 0

 2 2 for x > 0 dy 1+x = 0

(c) = non existent for x

dx − for x < 0

(d) I  in  (0 , ∞)  &  D  in  (- ∞ , 0)

 2 tan−1 x x < 1

y = f

(x) = tan-1 1 −2xx2 = − −Ο(+Ο2tan2tan−1 x−1 x) xx <−> 11

HIGHLIGHTS :

(a) Domain  is   R - {1 , -1}  &

−Ο , Ο range is   

 2 2

(b) f  is  neither  continuous nor  diff.  at  x = 1 , - 1

dy  1+2x2 x ≠ 1 (c)

dx non existent x = 1

(d) I ∀ x  in its domain (e) It is bound for all x



−1 y = f (x) = sin−1 (3 x − 4 x3) =  3sin x if x



HIGHLIGHTS :www.bloggermayank.online

(a) Domain  is  x ∈ [− 1 , 1]  &

 Ο Ο

range is  − 2 , 2

(b) Not derivable at  x

dy  3 2 if x ∈ −( 12 , 12)

(c) =  1− x

dx − 3 if x ∈ − −

 1− x2 (d) Continuous everywhere in its domain



(x) = cos-1 (4 x3 - 3 x) = 

y = f

3cos−1x if x 1

HIGHLIGHTS :

(a) Domain  is  x ∈ [- 1 , 1]  & range  is  [0 , Ο]

(b) Continuous  everywhere  in  its  domain

1 1 but not derivable at x = , −

2 2

(c) I  in − 1 , 1  & D in   12 , 1 ∪ − −1 , 12

 2 2

dy  1−3 x2 if x ∈ −( 12 , 12)

(d) = dx − 3 if x ∈ − −

 1− x2

GENERAL  NOTE :

Concavity  in  each  case  is  decided  by  the  sign  of  2nd  derivative  as  :

d y2

2 > 0   ⇒ dx Concave  upwards ; d y2

dx2 < 0   ⇒ Concave  downwards

D = DECREASING ; I = INCREASING

13. APPLICATION OF DERIVATIVE (AOD). TANGENT & NORMAL

THINGS TO REMEMBER :

dx

x y1 1

II Equation of tangent at (x1, y1) is ; y − y1 =  dydxx y1 1 (x − x1).

− 1

I The value of the derivative at P (x1 , y1) gives the slope of the tangent to the curve at P. Symbolically f′ (x1) =  dy = Slope of tangent at P (x1 y1) = m (say).

III Equation of normal at (x1, y1) is ; y y1 =  dy (x  x1).

 dxx y1 1

NOTE :www.bloggermayank.online

1.The point P (x1 , y1) will satisfy the equation of the curve & the eqation of tangent & normal line.

2.If the tangent at any point P on the curve is // to the axis of x then  dy/dx = 0 at the point P.

3.If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = ∞ or dx/dy = 0.

4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ± 1.

5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = – 1.

6. Tangent to a curve at the point P (x1, y1) can be drawn even through dy/dx at P does not exist. e.g.  x = 0 is a tangent to y = x2/3 at (0, 0).

7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.

e.g. If the equation of a curve be x2 – y2 + x3 + 3 x2 y − y3 = 0, the tangents at the origin are given by  x2 – y2 = 0 i.e. x + y = 0 and x − y = 0.

IV Angle of intersection between two curves is defined as the angle between the 2 tangents drawn to the 2 curves at their point of intersection. If the angle between two curves is 90° every where then they are called ORTHOGONAL curves.

y1 1+[f (x )′ 1 ]2

V (a) Length of the tangent (PT) =

f (x )′ 1

(b) Length of Subtangent (MT) = y1

f (x )′ 1

(c) Length of Normal (PN) = y1 1+[f (x )′ 1 ]2

(d) Length of Subnormal (MN) = y1 f ' (x1)

VI DIFFERENTIALS :

The differential of a function is equal to its derivative multiplied by the differential of the independent variable. Thus if, y = tan x then dy = sec2 x dx. In general  dy = f ′ (x) d x.

Note that : d (c) = 0  where 'c' is a constant.

d (u + v − w) = du + dv − dw d (u v) = u d v + v d u Note :

1. For the independent variable 'x' , increment ∆ x and differential d x are equal but this is not the case with the dependent variable 'y' i.e. ∆ y ≠ d y.

2. The relation d y = f ′ (x) d x can be written as  dy = f ′ (x) ; thus the quotient of the differentials of 'y' and dx

'x' is equal to the derivative of 'y' w.r.t. 'x'.

MONOTONOCITY(Significance of the sign of the first order derivative)

DEFINITIONS :

1. A function f (x) is called an Increasing Function at a point x = a if in a sufficiently small neighbourhood

= a we have f a( + h) > f a( ) and  increasing;

around x

f a( − h) < f a( )

f a( + h) < f a( ) and 

Similarly decreasing if − h) > f a( ) decreasing. f a(

2. A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within the interval (but not necessarily at the end points). A function decreasing in an interval  (a, b) is similarly defined.www.bloggermayank.online

3. A function which in a given interval is increasing or decreasing is called “Monotonic” in that interval.

4. Tests for increasing and decreasing of a function at a point :

If the derivative f ′(x) is positive at a point x = a, then the function f (x) at this point is increasing. If it is negative, then the function is decreasing. Even if  f ' (a) is not defined, f can still be increasing or decreasing.

Note : If f ′(a) = 0, then for x = a the function may be still increasing or it may be decreasing as shown. It has to be identified by a seperate rule. e.g. f (x) = x3 is increasing at every point. Note that, dy/dx = 3 x².

5. Tests for Increasing & Decreasing of a function in an interval :

SUFFICIENCY TEST : If the derivative function f ′(x) in an interval (a , b) is every where positive, then the function f (x) in this interval is Increasing ;www.bloggermayank.online If f ′(x) is every where negative, then f (x) is Decreasing.

General Note :

(1) If a function is invertible it has to be either increasing or decreasing.

(2) If a function is continuous the intervals in which it rises and falls may be separated by points at which its derivative fails to exist.

(3) If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (c) is the least value of  f  in [a, b]. Similarly if f is decreasing in [a, b] then f (a) is the greatest value and f (b) is the least value.

6. (a) ROLLE'S THEOREM :

Let f(x) be a function of x subject to the following conditions :

(i) f(x) is a continuous function of x in the closed interval of a ≤ x ≤ b.

(ii) f ′ (x) exists for every point in the open interval a < x < b.

(iii) f (a) = f (b). Then there exists at least one point x = c such that a<c < b where f ′ (c) = 0. Note that if f is not continuous in closed [a, b] then it may lead to the adjacent graph where all the 3 conditions of Rolles will be valid but the assertion will not be true in (a, b).

(b) LMVT THEOREM :

Let f(x) be a function of x subject to the following conditions :

(i) f(x) is a continuous function of x in the closed interval of a ≤ x ≤ b.

(ii) f ′ (x) exists for every point in the open interval a < x < b. (iii) f(a) ≠ f(b).

Then there exists at least one point x = c such that a < c < b where f ′ (c) =  f b( ) − f a( )

b − a

Geometrically, the slope of the secant line joining the curve at x = a & x = b is equal to the slope of the tangent line drawn to the curve at x = c. Note the following :

Rolles theorem is a special case of LMVT since f (a) = f (b) ⇒ f ′ (c) =  f b( ) − f a( ) = 0.

b − a

Note : Now [f (b) – f (a)] is the change in the function f as x changes from a to b so that [f (b) – f (a)] / (b – a) is the average rate of change of the function over the interval [a, b]. Also f '(c) is the actual rate of change of the function for x = c. Thus, the theorem states that the average rate of change of a function over an interval is also the actual rate of change of the function at some point of the interval. In particular, for instance, the average velocity of a particle over an interval of time is equal to the velocity at some instant

belonging to the interval. www.bloggermayank.online

This interpretation of the theorem justifies the name "Mean Value" for the theorem.

(c) APPLICATION OF ROLLES THEOREM FOR ISOLATING THE REAL ROOTS OF AN EQUATION f (x)=0

Suppose a & b are two real numbers such that ; (i) f(x) & its first derivative f ′ (x) are continuous for a ≤ x ≤ b.

(ii) f(a) & f(b) have opposite signs.

(iii) f ′ (x) is different from zero for all values of x between a & b.

Then there is one & only one real root of the equation f(x) = 0 between a & b. MAXIMA - MINIMA

FUNCTIONS OF A SINGLE VARIABLE

HOW MAXIMA & MINIMA ARE CLASSIFIED

1. A function f(x) is said to have a maximum at x = a if f(a) is greater than every other value assumed by f(x) in the immediate neighbourhood of x = a. Symbolically

f a( ) > f a( +h)

f a( ) > f a( −h) ⇒ x = a gives maxima for a

sufficiently small positive h.

Similarly, a function f(x) is said to have a minimum value at x = b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at

f b( ) < f b( +h) ⇒ x = b gives

x = b. Symbolically if < f b( −h) f b( )

minima for a sufficiently small positive h.

Note that :

(i) the maximum & minimum values of a function are also known as local/relative maxima or local/relative minima as these are the greatest & least values of the function relative to some neighbourhood of the point in question.

(ii) the term 'extremum' or (extremal) or 'turning value' is used both for maximum or a minimum

value.

(iii) a maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.

(iv) a function can have several maximum & minimum values & a minimum value may even be greater than a maximum value.

(v) maximum & minimum values of a continuous

function occur alternately & between two consecutive maximum values there is a minimum value & vice

versa.

www.bloggermayank.online

2. A NECESSARY CONDITION FOR

MAXIMUM & MINIMUM :

If f(x) is a maximum or minimum at x = c & if f ′ (c) exists then f ′ (c) = 0.

Note :

(i) The set of values of x for which f ′ (x) = 0 are often called as stationary points or critical points. The rate of change of function is zero at a stationary point.

(ii) In case f ′ (c) does not exist f(c) may be a maximum or a minimum & in this case left hand and right hand derivatives are of opposite signs.

(iii) The greatest (global maxima) and the least (global minima) values of a function f in an interval [a, b] are f(a) or f(b) or are given by the values of x for which      f ′ (x) = 0.

dy

(iv) Critical points are those where   = 0, if it exists , dx

or it fails to exist either by virtue of a vertical tangent or by virtue of a geometrical sharp corner but not because of discontinuity of function.

3. SUFFICIENT CONDITION FOR EXTREME VALUES :  f (c-h) > 0′  ⇒ x = c is a point of local maxima, where f ′ (c) = 0.

f (c+h) < 0′

Similarly f (c-h) < 0′ ′(c) = 0.

′  ⇒ x = c is a point of local minima, where f f (c+h) > 0

Note : If f ′ (x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f(x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an extreme value of f.

4. USE OF SECOND ORDER DERIVATIVE IN ASCERTAINING THE MAXIMA OR MINIMA:

(a) f(c) is a minimum value of the function f, if f ′ (c) = 0 & f ′′ (c) > 0.

(b) f(c) is a maximum value of the function f, f ′ (c) = 0 & f ′′ (c) < 0.

Note : if f ′′ (c) = 0 then the test fails. Revert back to the first order derivative check for ascertaning the maxima or minima.

5. SUMMARY−WORKING RULE :

FIRST :

When possible , draw a figure to illustrate the problem & label those parts that are important in the problem. Constants & variables should be clearly distinguished.

SECOND :

Write an equation  for the quantity that is to be maximised or minimised. If this quantity is denoted by ‘y’, it must be expressed in terms of a single independent variable x. his may require some algebraic manipulations.

THIRD :

If y = f (x) is a quantity to be maximum or minimum, find those values of x for which dy/dx = f ′(x) = 0.

FOURTH :

Test each values of x for which f ′(x) = 0 to determine whether it provides a maximum or minimum or neither. The usual tests are :

(a) If d²y/dx² is positive when dy/dx = 0 ⇒ y is minimum.

If d²y/dx² is negative when dy/dx = 0 ⇒ y is maximum. If d²y/dx² = 0 when dy/dx = 0, the test fails.

positive for x < x0

(b) If  dydx is zero for x = x0  ⇒ a maximum occurs at x = x0.

negative for x > x0

But if dy/dx changes sign from negative to zero to positive as x advances through xo there is a minimum. If dy/dx does not change sign, neither a maximum nor a minimum. Such points are called

INFLECTION POINTS. FIFTH :

If the function y = f (x) is defined for only a limited range of values a ≤ x ≤ b then examine x = a & x = b

for possible extreme values.

SIXTH :

If the derivative fails to exist at some point, examine this point as possible maximum or minimum.

Important Note :www.bloggermayank.online

Given a fixed point A(x1, y1) and a moving point P(x, f (x)) on the curve y = f(x). Then AP will be maximum or minimum if it is normal to the curve at P.

If the sum of two positive numbers x and y is constant than their product is maximum if they are equal, i.e.  x + y = c , x > 0 , y > 0 , then xy =   [ (x + y)2 – (x – y)2 ]

If the product of two positive numbers is constant then their sum is least if they are equal. i.e. (x +

y)2 = (x – y)2 + 4xy

6. USEFUL FORMULAE OF MENSURATION TO REMEMBER :

Volume of a cuboid = lbh.

Surface area of a cuboid = 2 (lb + bh + hl).

Volume of a prism = area of the base x height.

Lateral surface of a prism = perimeter of the base x height. Total surface of a prism = lateral surface + 2 area of the base (Note that lateral surfaces of a prism are all rectangles).

Volume of a pyramid =  area of the base x height.

Curved surface of a pyramid =  (perimeter of the base) x slant height.

(Note that slant surfaces of a pyramid are triangles).

Volume of a cone =  Ο r2h.

Curved surface of a cylinder = 2 Ο rh.

Total surface of a cylinder = 2 Ο rh + 2 Ο r2.

Volume of a sphere =  Ο r3.

Surface area of a sphere = 4 Ο r2.

Area of a circular sector =  r2 ΞΈ , when ΞΈ is in radians.

7. SIGNIFICANCE OF THE SIGN OF 2ND ORDER DERIVATIVE AND POINTS OF INFLECTION : The sign of the 2nd order derivative determines the concavity of the curve. Such points such as C & E on the graph where the concavity of the curve changes are called the points of inflection. From the graph we find that if:

d y2

(i) 2 > 0  ⇒ concave upwards dx

d y2

(ii) 2 < 0  ⇒ concave downwards.

dx

d y2

At the point of inflection we find that 2 = 0 & dx

d y2

2 changes sign. dx

d y2

Inflection points can also occur if 2 fails to exist. For example, consider the graph of the function dx

defined as,www.bloggermayank.online

x3 5/ for x ∈ −∞( , )1  f (x) = [ 2 − x2 for x ∈(1, ∞)

Note that the graph exhibits two critical points one is a point of local maximum & the other a point of inflection.

14. Integration (Definite & Indefinite)

1. DEFINITION :

If  f & g  are functions of  x such that  g′(x) = f(x) then the function g  is called a PRIMITIVE  OR ANTIDERIVATIVE  OR  INTEGRAL   of  f(x)  w.r.t.  x  and  is  written  symbolically  as

∫ f(x) dx = g(x) + c ⇔  d {g(x) + c} = f(x), where  c is called the constant  of  integration. dx

2. STANDARD  RESULTS :

( )n+1

n dx = ax(+b ) + c   n ≠ −1 (ii)    dx = 1 ln (ax + b) + c

(i)   (ax + b)

a n+1 ax+b a

1 +

(iii)  ∫ eax+b dx =   eax+b + c (iv)  ∫ apx+q  dx =  1 a px q (a > 0) + c a p na

(v)  ∫ sin (ax + b) dx = −  1 cos (ax + b) + c (vi)  ∫ cos (ax + b) dx =  1 sin (ax + b) + c

a a

(vii)  ∫ tan(ax + b) dx =  1 ln sec (ax + b) + c(viii) ∫ cot(ax + b) dx =  1 ln sin(ax + b)+ c a a

(ix) ∫ sec² (ax + b) dx =  1 tan(ax + b) + c (x)  ∫ cosec²(ax + b) dx = −  1 cot(ax + b)+ c a a

sec (ax + b)  .  tan (ax + b) dx = 1 sec (ax + b) + c (xi) a

(xii) ∫ cosec (ax + b)  .  cot (ax + b) dx = −1 cosec (ax + b) + c a

(xiii) ∫ secx dx = ln (secx + tanx) + c OR ln tan  Ο4 +  2x + c

(xiv) ∫ cosec x dx = ln (cosecx − cotx) + c OR   ln tan  x + c   OR   − ln (cosecx + cotx)

2

(xv) ∫ sinh x dx = cosh x + c    (xvi) ∫ cosh x dx = sinh x + c (xvii) ∫ sech²x dx = tanh x + c

(xviii) ∫ cosech²x dx = − coth x + c (xix) ∫ sech x . tanh x dx = − sech x + c

(xx) ∫ cosech x . coth x dx =  − cosech x  +  c (xxi) ∫ = sin−1  x + c a

(xxii) ∫ 2dx+x2 = 1a tan−1  xa + c (xxiii) ∫ xa + c a

(xxiv) = ln  2 OR sinh−1  xa + c

(xxv) = ln x+ x −a  OR cosh−1 + c

(xxvi)   a2−x2 21a ln a xa x+− + c

(xxvii) ∫ x2d−x 2 = 1 ln x ax a+−  + c a 2a

2

(xxviii) dx = sin−1  x + c a

(xxix) dx = sinh−1  x + c

a

(xxx) dx = cosh−1  x

2 a

(xxxi) ∫ eax. sin bx dx =  a2e+axb2 (a sin bx − b cos bx) + c

(xxxii) ∫ eax . cos bx dx =  a2e+axb2 (a cos bx + b sin bx) + c

3. TECHNIQUES  OF  INTEGRATION :

(i) Substitution  or  change of independent variable .

Integral  I = ∫ f(x)  dx  is changed to  ∫ f(Ο (t))  f ′ (t)  dt ,  by a suitable substitution x = Ο (t)  provided the later integral is easier to integrate .

∫ u.v dx = u ∫  v dx − ∫  du.∫ vdx dx   where  u & v are differentiable (ii) Integration by part :

dx

function . Note : While using integration by  parts, choose  u  &  v  such that

∫ v dx  is  simple & (b) ∫ du.∫ vdx dx      is  simple  to integrate. (a)

dx

This is generally obtained, by keeping the order of  u & v  as per the order of the letters in  ILATE,   where

;   I − Inverse  function, L − Logarithmic function ,

A − Algebraic function,    T − Trigonometric function     &    E − Exponential function (iii) Partial fraction ,  spiliting a bigger fraction into smaller fraction by known methods .

4. INTEGRALS  OF  THE  TYPE :www.bloggermayank.online

(i) ∫ [ f(x)]n f ′(x) dx OR   ∫ [f′(x)  dx     put  f(x) = t   &   proceed . f(x)]n

dx dx 2

(ii) ∫ 2 ,  ∫ 2 ,  ∫ ax + bx + c dx ax + +bx c ax + +bx c

Express  ax2 + bx + c  in the form of perfect square & then apply the standard results . px + q px + q

(iii) 2 dx ,  ∫ 2 dx  . ax + +bx c ax + +bx c

Express   px + q = A (differential co-efficient of denominator) + B .

(iv) ∫ ex [f(x) + f ′(x)] dx =  ex . f(x) + c (v) ∫ [f(x) + xf ′(x)] dx = x f(x) + c

(vi) dx   n ∈ N Take  xn  common  &  put 1 + x−n = t  .

x(xn+1)

(vii) ∫ dx(n−1)   n ∈ N  ,    take  xn  common  &  put 1+x−n = tn x x 12( n+ ) n

(viii) take  xx 1 + x )

n ( dx n 1/n n common as  x  and put  1 + x −n = t .

(ix) ∫  dx 2   OR   ∫  +bdcosx 2 x   OR  ∫  asin2 x+bsindxxcosx+ccos2 x a bsin+ x a

Multiply  N..r  &  D. .r  by  sec² x   &   put  tan x = t  .

OR    OR

x

Hint Convert  sines & cosines  into their  respective tangents of half  the  angles , put tan   = t

2

(xi)∫  a.cosx+b.sinx+c dx .  Express   Nr ≡ A(Dr) + B  d (Dr) + c  &  proceed .

.cosx+m.sinx+n dx

(xii) ∫  x x2+1 dx OR 4+xKx2−12+1 dx  where K is any constant  .

4+Kx2+1 x

Hint :  Divide  Nr  &  Dr  by  x²  &  proceed  .

dx dx 2

(xiii)   &

(xv) (x − Ξ±) (Ξ² − x)  ; put  x = Ξ± cos2 ΞΈ + Ξ² sin2 ΞΈ

(x − Ξ±) (x − Ξ²)  ; put  x = Ξ± sec2 ΞΈ − Ξ² tan2 ΞΈ

put  x − Ξ± = t2  or   x − Ξ² = t2 .

DEFINITE    INTEGRAL

b

1. ∫ f(x) dx = F(b) − F(a) where ∫ f(x) dx = F(x) + c

a

b

VERY  IMPORTANT  NOTE  : If  ∫ f(x) dx = 0 ⇒   then  the  equation  f(x) = 0  has  atleast  one  root  lying

a

in  (a , b)  provided  f  is  a  continuous  function  in  (a , b)  .

2. PROPERTIES  OF  DEFINITE  INTEGRAL   :www.bloggermayank.online

b b b a

P−1 ∫ f(x) dx = ∫ f(t) dt   provided  f  is  same P − 2 ∫ f(x) dx =  −∫ f(x) dx

a a a b

b c b

P−3   ∫ f(x) dx = ∫ f(x) dx + ∫ f(x) dx ,  where  c  may lie inside or outside the interval [a, b] . This property to be

a a c

used when  f  is piecewise continuous in  (a, b) .

a

P−4 ∫ f(x) dx = 0    if  f(x)  is  an  odd  function  i.e.  f(x) = − f(−x)  .

−a a  f(x) dx     if  f(x)  is  an  even  function   i.e.  f(x) = f(−x)  .

b b a a

P−5 ∫ f(x) dx = ∫ f(a + b − x) dx ,  In  particular ∫ f(x) dx = ∫ f(a − x)dx

a a 0 0

2a a a a

Pf(x) dx   =  ∫ f(x) dx + ∫ f(2a − x) dx = 2 ∫ f(x) dx     if   f(2a − x) = f(x)

0 0 0

=  0     if   f(2a − x) = − f(x)

a

Pf(x) dx = n  ∫ f(x) dx   ;   where‘a’is the period of the function   i.e. f(a + x) = f(x)

0

b+nT b

P−8 ∫ f(x) dx = ∫ f(x) dx   where  f(x) is periodic with period T &  n ∈ I .

a+nT a

a

P f(x) dx = (n   f(x) dx   if  f(x) is periodic with period 'a' .

b b

P−10 If  f(x) ≤ Ο(x)  for a ≤ x ≤ b   then  ∫ f(x) dx ≤ ∫ Ο (x) dx

a a

bb b

P−11∫f(x)dx  ≤  ∫ f(x)dx .P−12 If f(x) ≥ 0  on the interval  [a, b] ,  then  ∫ f(x) dx ≥ 0.

aa a

3. WALLI’S FORMULA :www.bloggermayank.online

nx . cosmx dx = [(n 1)(n 3)(n 5)....1or2 (m 1)(m 3)....1or− − − ][ − 2] K

(m n)(m n 2)(m n 4)....1or+ + − + − 2

Where K =    if both m and n are even   (m, n ∈ N) ;   = 1    otherwise

4. DERIVATIVE  OF  ANTIDERIVATIVE  FUNCTION  :

If  h(x) & g(x) are differentiable functions of x then ,

h(x) d ∫

f(t) dt = f [h (x)] . h′(x) − f [g (x)] . g′(x)

dx g(x)

5. DEFINITE  INTEGRAL  AS  LIMIT  OF  A SUM :

b

∫ f(x) dx = Limitn→∞ h [f (a) + f (a + h) + f (a + 2h) + ..... + f (a + −n 1 h)]

a

n−1

=  Limith→0 h ∑ f (a + rh)   where  b − a = nh

r=0

n−1

If  a = 0  &  b = 1  then , Limitn→∞ h ∑ f (rh) =  f(x) dx   ;   where  nh = 1 OR

r=0

Limit  1 n 1∑−  f   r  = ∫1 f(x) dx   . n→∞ n r 1= n 0

6. ESTIMATION  OF DEFINITE INTEGRAL :

b

(i) For a monotonic decreasing function in (a , b) ;  f(b).(b − a) < ∫ f(x) dx <  f(a).(b − a)

a b

(ii) For a monotonic increasing  function  in  (a , b) ;  f(a).(b − a)  < ∫ f(x) dx  <  f(b).(b − a)

a

7. SOME  IMPORTANT  EXPANSIONS :www.bloggermayank.online

(i)

(v)

15. AREA UNDER CURVE

(AUC)

THINGS TO REMEMBER :

1. The area bounded by the curve y = f(x) , the x-axis and the ordinates at x = a & x = b is given by,

b b

A = ∫ f (x) dx = ∫ y dx.

a a

2. If the area is below the x−axis then A is negative. The convention is to consider the magnitude only i.e.

b

A =∫ y dx  in this case.

a

3. Area between the curves y = f (x) & y = g (x) between  the ordinates  at  x = a  &  x = b is given by,

b b b

A = ∫ f (x) dx − ∫ g (x) dx = ∫ [ f (x) − g (x) ] dx.

a a a

4. Average value of a function y = f (x)  .r.t. x over an interval

b

a ≤ x ≤ b is defined as : y (av) = 1 ∫ f (x) b−a a

x dA

5. The area functionAxa satisfies the differential equation a = f (x) with initial conditionAaa = 0. dx

Note : If F (x) is any integral of f (x) then ,

Axa = ∫ f (x) dx = F (x) + c Aaa = 0 = F (a) + c  ⇒  c

= − F (a) hence Axa = F (x) − F (a). Finally by taking x = b we get , Aab = F (b) − F (a).

6. CURVE TRACING :

The following outline procedure is to be applied in Sketching the graph of a function y = f (x) which in turn will be extremely useful to quickly and correctly evaluate the area under the curves.

(a) Symmetry : The symmetry of the curve is judged as follows :

(i) If all the powers of y in the equation are even then the curve is symmetrical about the axis of x.

(ii) If all the powers of x are even , the curve is symmetrical about the axis of y.

(iii) If powers of x & y both are even, the curve is symmetrical about the axis of x as well as y.

(iv) If the equation of the curve remains unchanged on interchanging x and y, then the curve is symmetrical about y = x.

(v) If on interchanging the signs of x & y both the equation of  the curve is unaltered then there is symmetry in opposite quadrants.

(b) Find dy/dx & equate it to zero to find the points on the curve where you have horizontal tangents.

(c) Find the points where the curve crosses the x−axis & also the y−axis.

(d) Examine if possible the intervals when f (x) is increasing or decreasing. Examine what happens to ‘y’ when  x → ∞ or − ∞.

7. USEFUL RESULTS :

(i) Whole area of the ellipse, x2/a2 + y2/b2 = 1 is Ο ab.

(ii) Area enclosed between the parabolas y2 = 4 ax & x2 = 4 by is 16ab/3.

(iii) Area included between the parabola y2 = 4 ax & the line y = mx is 8 a2/3 m3.

16. DIFFERENTIAL EQUATION

DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER  AND  FIRST  DEGREE DEFINITIONS :

1. An equation  that  involves  independent  and  dependent  variables  and  the  derivatives  of  the dependent variables  is  called  a  DIFFERENTIAL  EQUATION.www.bloggermayank.online

2. A differential  equation  is  said  to  be  ordinary ,  if  the  differential  coefficients  have  reference to  a single  independent  variable  only  and  it  is  said  to  be  PARTIAL   if  there  are  two  or     more

∂u ∂u ∂u independent  variables .  We are concerned  with  ordinary  differential  equations  only. eg.    + +

∂x ∂y ∂z

= 0  is a partial differential equation.

3. Finding  the unknown function is called  SOLVING  OR  INTEGRATING  the differential  equation . The solution  of  the  differential  equation  is  also  called  its  PRIMITIVE, because  the  differential equation can  be  regarded  as  a  relation  derived  from  it.

4. The  order  of  a differential equation  is  the  order  of  the  highest  differential  coefficient occuring  in  it.

5. The  degree  of  a  differential  equation  which can be written as a polynomial in the derivatives  is  the degree of  the  derivative  of   the  highest  order  occuring  in  it , after  it  has  been  expressed  in a form free from radicals  &  fractions  so far  as  derivatives  are  concerned, thus  the  differential  equation :

 d ym p dm−1 ( )y q

f(x , y)  m  +  Ο (x , y)  m−1  + ....... = 0  is  order  m  &  degree p. Note  that  in  the  differential

d x   dx  equation    ey′′′ − xy′′ + y = 0    order  is  three  but  degree doesn't   apply.

6. FORMATION  OF  A  DIFFERENTIAL  EQUATION :

If  an  equation  in  independent  and  dependent  variables  having  some  arbitrary  constant  is given ,  then a  differential  equation  is  obtained  as  follows  :

Differentiate  the  given  equation   w.r.t.   the  independent  variable  (say x)  as  many times as  the  number  of  arbitrary  constants  in  it .

Eliminate  the  arbitrary  constants  .

The eliminant is the required  differential equation .  Consider forming  a  differential equation  for  y² =  4a(x + b)  where  a  and  b  are  arbitary  constant .

Note : A differential equation represents a family of curves all satisfying some common properties. This

can  be  considered  as  the  geometrical  interpretation  of  the differential equation.

7. GENERAL  AND  PARTICULAR  SOLUTIONS :

The solution of a differential equation which contains a number of independent arbitrary constants equal to the order of the differential equation is called the GENERAL SOLUTION (OR COMPLETE INTEGRAL OR COMPLETE PRIMITIVE) . A solution obtainable from the general solution by giving particular values to the constants is called a PARTICULAR  SOLUTION.

Note that the general solution of a differential equation of the nth order contains ‘n’ & only ‘n’ independent arbitrary constants. The arbitrary constants in the solution of a differential equation are said to be independent, when it is impossible to deduce from the solution an equivalent relation containing fewer arbitrary constants. Thus the two arbitrary constants A, B in the equation  y = A ex + B are not independent since the equation can be written as  y = A eB.  ex = C ex. Similarly the solution y = A sin x + B cos (x + C) appears to contain three arbitrary constants, but they are really equivalent to two only. 8. Elementary  Types  Of  First  Order  &  First  Degree  Differential  Equations  .

TYPE−1. VARIABLES  SEPARABLE : If  the  differential  equation can be expressed as ; f (x)dx + g(y)dy = 0  then  this  is  said  to  be  variable − separable  type.

A  general  solution  of  this  is  given  by    ∫ f(x) dx + ∫ g(y) dy = c ; where  c  is  the  arbitrary  constant  .  consider  the  example  (dy/dx) =  ex−y + x2. e−y.

Note : Sometimes  transformation  to  the  polar  co−ordinates  facilitates  separation of variables.

In  this  connection  it  is  convenient  to  remember  the  following differentials. If  x = r cos ΞΈ ;  y = r sin ΞΈ  then,

(i)  x dx + y dy = r dr    (ii)  dx2 + dy2 = dr2 + r2 dΞΈ2    (iii)  x dy − y dx = r2 dΞΈ

If  x = r sec ΞΈ  &  y = r tan ΞΈ  then  x dx − y dy = r dr   and  x dy − y dx = r2 sec ΞΈ dΞΈ .

TYPE−2 : dy =  f (ax + by + c) ,  b ≠  0. To  solve  this ,  substitute  t = ax + by + c. Then  the  equation dx

reduces  to  separable  type  in the  variable  t  and  x  which  can  be  solved. Consider the

example  (x + y)2  dy = a2 . dx

TYPE−3.  HOMOGENEOUS  EQUATIONS :www.bloggermayank.online

dy f x y( , ) Ο (x , y) are  homogeneous  functions

A differential  equation  of  the  form = where  f (x , y)  &

dx Ο(x y, )

of  x & y , and  of  the  same degree , is  called  HOMOGENEOUS  .  This  equation  may  also  be  reduced  to the  form   dy = g   x & is solved by putting  y = vx  so that the  dependent  variable  y  is  changed  to dx  y

another variable  v, where v  is  some unknown function,  the  differential  equation  is  transformed to  an

dy y x( + y)

equation  with  variables  separable.  Consider + 2 = 0.

dx x

TYPE−4.  EQUATIONS  REDUCIBLE  TO  THE  HOMOGENEOUS  FORM :

If   dydx = a xa x21 ++ b yb y21 ++ cc12   ; where  a1 b2 − a2 b1  ≠  0, i.e.   ab11  ≠  ab22

then  the  substitution  x = u + h,  y = v + k  transform  this  equation  to  a  homogeneous  type in  the  new variables u  and  v  where  h  and  k  are  arbitrary  constants  to  be  chosen so as to  make  the  given equation  homogeneous  which  can  be  solved  by  the  method  as  given in  Type − 3.  If

(i) a1 b2 − a2 b1 = 0 ,  then  a  substitution  u = a1 x + b1 y  transforms  the  differential  equation  to an  equation with  variables  separable.   and

(ii) b1 + a2 = 0 ,  then  a  simple  cross  multiplication  and  substituting  d (xy)  for  x dy + y dx  &  integrating term by term yields the result easily.

Consider  dy = x − 2y + 5  ;   dy = 2x + 3y − 1   &    dy = 2x − y + 1 dx 2x + −y 1 dx 4x + 6y − 5 dx 6x − 5y + 4

(iii) In an equation of the form :  y f (xy) dx + xg (xy)dy = 0  the variables can be separated by the substitution xy = v.

IMPORTANT   NOTE  :

(a) The  function  f (x , y)  is  said  to  be  a  homogeneous  function  of  degree  n  if  for  any real  number  t (≠ 0) ,  we  have  f (tx , ty) =  tn  f(x , y) .

For  e.g.  f(x , y)  =  ax2/3 + hx1/3 . y1/3 + by2/3  is a homogeneous function of  degree 2/3

dy homogeneous

(b) A  differential equation  of  the  form  =  f(x , y)  is  homogeneous  if  f(x , y) is a dx

function  of  degree zero   i.e.   f(tx , ty)  =  t° f(x , y) = f(x , y).  The function  f  does  not  depend on  x &

y x

y  separately  but  only  on  their  ratio or .

x y

LINEAR  DIFERENTIAL  EQUATIONS  :www.bloggermayank.online

A differential equation is said to be linear if the dependent variable  &  its differential   coefficients occur in the first degree only and are not multiplied together The  nth  order  linear  differential  equation is  of  the  form  ;www.bloggermayank.online

d yn dn−1 y

a0 (x)  n  +  a1(x) n−1 + ...... +  an (x) . y  =  Ο (x)  . Where  a0(x) ,  a1(x) ..... an(x) are  called  the dx dx

coefficients  of  the  differential  equation. Note that a linear differential equation is always of  the first degree but every differental equation of  the   first degree need not be

d y2  dy 3

linear. e.g. the differential equation dx2 +  dx +  y2 = 0 is not linear, though  its degree is 1.

TYPE − 5.  LINEAR  DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER   :

dy + Py = Q , where The most general form of a linear differential equations of first order  is

dx

P& Q  are  functions  of  x . To  solve  such  an  equation  multiply  both  sides  by e∫ Pdx  .

NOTE : (1) The  factor  e∫ Pdx  on  multiplying  by  which  the  left  hand  side  of  the differential equation becomes  the differential coefficient  of  some  function  of  x  &  y ,  is called integrating factor of the differential equation popularly abbreviated as I. F.

(2) It is very important to remember that on multiplying by the integrating factor , the left  hand  side  becomes the  derivative  of  the  product  of  y  and  the  I. F.

(3) Some times a  given differential  equation  becomes  linear  if  we  take  y  as  the independent variable and x  as  the  dependent  variable.   e.g.  the  equation  ; dy 2 + 3   can  be  written  as  (y2 + 3) dx = x + y + 1  which  is  a  linear differential

(x + y + 1) =  y

dx dy

equation.

TYPE−6.  EQUATIONS  REDUCIBLE  TO  LINEAR  FORM   :

dy + py = Q . yn   where  P & Q functions  of  x ,  is reducible to  the  linear form by dividing The  equation dx it by  yn  &  then substituting  y−n+1 = Z .  Its solution  can  be  obtained  as in Type−5. Consider  the example  (x3 y2 + xy) dx = dy.

dy n  is  called  BERNOULI’S  EQUATION. The  equation  + Py = Q . y

dx

9. TRAJECTORIES  :

Suppose we are given the family of plane curves. Ξ¦ (x, y, a) = 0 depending on a single parameter a. A curve making at each of its points a fixed angle Ξ± with the curve of the family passing through that point is called an isogonal trajectory of that family ; if in particular Ξ± = Ο/2, then it is called an orthogonal trajectory.

Orthogonal trajectories : We set up the differential equation of the given family of curves. Let it be of the form F (x, y, y') = 0 The differential equation of the orthogonal trajectories is of the form F x, y, − y1′ =

0 The general integral of this equation

Ξ¦1 (x, y, C) = 0 gives the family of orthogonal trajectories. Note : Following  exact  differentials  must  be  remembered   :

xdy − ydx  y

(i) xdy + y dx = d(xy) (ii) x2 = d x

ydx − xdy  x xdy + ydx

(iii)    2 = d  (iv)      = d(lnxy) y  y xy

dxx ++ dyy xdyxy− ydx = dln yx

(v)    = d(ln(x+y)) (vi)

(vii) ydxxy− xdy = dln  xy (viii) xdyx2 − ydx2 = dtan−1  yx

+ y

(ix) ydxx2 −+ yxdy2 = dtan−1  xy (x)     xdxx2 ++ ydyy2 = dln x2 + y2 

(xi) d − 1  = xdy 2+ ydx2 (xii) d  ex  = ye dxx −2 e dyx

 xy x y  y  y

 ey  xe dyy − e dxy

(xiii) d   = 2 www.bloggermayank.online

 x  x

17. STRAIGHT LINES & PAIR OF STRAIGHT LINES

1. DISTANCE  FORMULA :

The distance between the points A(x1,y1) and B(x2,y2) is  (x1 − x )2 2 + −(y1 y )2 2 .

2. SECTION  FORMULA :

If P(x , y) divides  the line joining  A(x1 , y1)  &  B(x2 , y2) in the ratio m : n,  then  ; mx +nx my +ny

x =  2 1    ;      y = 2 1 If m  is positive, the division is internal, but  if  m is negative, the m n+ m n+ n n division is external .

Note : If  P divides  AB  internally  in  the  ratio m : n  &  Q divides  AB externally  in  the  ratio m : n  then P & Q are said to be harmonic conjugate of each other w.r.t. AB.

Mathematically  ;    2 = 1 + 1  i.e.  AP, AB & AQ are in  H.P

AB AP AQ

3. CENTROID  AND  INCENTRE :

If A(x1, y1),  B(x2, y2),  C(x3, y3) are the vertices of triangle  ABC, whose sides BC, CA, AB are of lengths  a, b, c respectively, then the coordinates of the centroid are : x1+x32+x3,y1+y32+y3  &  the

coordinates of the incentre are :

 ax1+abx+ +b c2+cx3,ay1+a+ +byb c2+cy3Note  that  incentre  divides

the  angle  bisectors  in  the  ratio (b + c) : a  ;   (c + a) : b &  (a + b) : c.

REMEMBER  :(i) Orthocentre , Centroid  & circumcentre  are always  collinear &  centroid divides the line joining orthocentre  &  cercumcentre  in  the  ratio  2 : 1 .

(ii) In an isosceles triangle G, O, I & C lie on  the  same  line  .

4. SLOPE  FORMULA  :

If ΞΈ  is  the angle at which a straight line is inclined to the positive direction of x−axis, & 0° ≤ ΞΈ < 180°,  ΞΈ ≠ 90°,  then the slope of the line, denoted by m, is defined by m = tan ΞΈ. If ΞΈ is 90°,  m does not exist, but  the line is parallel to the y−axis.If ΞΈ = 0, then  m = 0  & the line is parallel to the x−axis. If A (x1, y1) & B (x2, y2),  x1≠ x2, are points on a straight line, then the slope m of the  line is given by: m =

x xy y11−− 22  .

5. CONDITION  OF  COLLINEARITY  OF  THREE  POINTS − (SLOPE FORM)  :Points  A

(x1, y1),  B (x2, y2),  C(x3, y3) are collinear if   xy1−−xy2  =  xy22−−xy33  .

 1 2 

6. EQUATION OF A STRAIGHT LINE IN VARIOUS FORMS  :

(i) Slope − intercept  form:   y = mx + c  is the equation of a straight line whose slope is m &  which makes  an  intercept  c  on  the  y−axis .

(ii) Slope one point form: y − y1 = m (x − x1) is the equation of a straight line whose slope is m & which passes through the point (x1, y1).

(iii) Parametric form : The  equation  of  the  line  in  parametric  form  is  given  by x−x1 y−y1

= = r (say). Where ‘r’ is the distance of any point (x , y) on  the line from the fixed cosΞΈ sinΞΈ

point (x1, y1) on the line. r is positive if the point (x, y) is on the right of (x1, y1) and negative if (x, y) lies on the left of (x1, y1) .www.bloggermayank.online

(iv) Two point form :   y − y1 = xy22 −− xy11 (x − x1)  is  the equation of a  straight  line which passes through the points (x1, y1) &  (x2, y2) .

(v) Intercept form :  x + y = 1  is the equation of a straight line which makes intercepts a & b  on  OX

a b

&  OY  respectively .

(vi) Perpendicular  form :  xcos Ξ± + ysin Ξ± = p  is  the  equation  of  the  straight line where the  length of the perpendicular from the origin O on the line is  p  and this perpendicular makes angle Ξ± with positive side of  x−axis .

(vii) General Form :  ax + by + c = 0  is  the  equation  of  a  straight  line in the general form

7. POSITION  OF  THE  POINT (x1, y1) RELATIVE  TO  THE  LINE ax + by + c = 0 :If  ax1

+ by1 + c  is of the  same sign as  c, then the point (x1, y1) lie  on  the origin side of ax + by + c = 0 . But if the sign of  ax1 + by1 + c  is opposite to that of  c, the point (x1, y1) will lie on the non-origin side of  ax + by + c = 0.

8. THE  RATIO  IN  WHICH  A GIVEN  LINE  DIVIDES  THE  LINE  SEGMENT JOINING TWO POINTS :

Let the given line  ax + by + c = 0  divide the line segment joining  A(x1, y1) & B(x2, y2) in the ratio

m : n, then m = − ax1 + by1 + c . If  A & B are on the same side of the given line then  m is negative n ax2 + by2 + c n

m but  if  A & B are  on opposite  sides of the given line ,  then   is positive n

9. LENGTH OF PERPENDICULAR FROM A POINT ON A LINE  :

The length of perpendicular from  P(x1, y1) on  ax + by + c = 0  is   .

10. ANGLE BETWEEN TWO STRAIGHT LINES IN TERMS OF THEIR SLOPES :

If m1 & m2  are the slopes of two intersecting  straight  lines  (m1 m2 ≠ −1)  &  ΞΈ  is  the acute angle between them, then  tan ΞΈ = m1 − m2 . 1+m m1 2

Note : Let  m1, m2, m3 are the slopes of three lines  L1 = 0 ;  L2 = 0 ;  L3 = 0   where  m1 > m2 > m3 then the interior angles of the  ∆ ABC found by these lines are given by,

m1 − m2  ;   tan B = m2 − m3  &  tan C = m3 − m1

tan A =

1 + m m1 2 1 + m m2 3 1 + m m3 1

11. PARALLEL  LINES  :

(i) When two straight lines are parallel their slopes are equal. Thus any line parallel to  ax + by + c = 0 is of the type  ax + by + k = 0 .  Where  k  is a  parameter.

(ii) The  distance  between  two  parallel   lines   with   equations   ax + by + c1 = 0   &  ax + by + c2 =

c1−c2  Note  that  the  coefficients  of  x &  y  in  both  the  equations  must  be  same. 0  is

a2+b2

p1 2p

(iii) The area of the parallelogram = sinΞΈ , where p1 & p2 are  distances between two pairs of opposite sides & ΞΈ is the angle between any two adjacent sides . Note that area of the    parallelogram bounded  by  the  lines  y = m1x + c1, y = m1x + c2  and  y = m2x + d1 , y = m2x + d2 is given by

(c1 −c ) (d2 1 −d )2 . m1 −m2

12. PERPENDICULAR  LINES  :

(i) When two lines of slopes m1& m2 are at right angles, the product of their slopes is −1, i.e. m1 m2 = −1.  Thus any line perpendicular to  ax + by + c = 0 is of the  form  bx − ay + k = 0, where  k  is any parameter.www.bloggermayank.online

(ii)St. lines  ax + by + c = 0  &  a′ x + b′ y + c′ = 0  are  right angles if & only  if aa′ + bb′ = 0.

13. Equations  of  straight  lines  through (x1 , y1) making  angle Ξ±  with  y = mx + c   are: (y − y1) =  tan (ΞΈ − Ξ±) (x − x1) & (y − y1) =  tan (ΞΈ + Ξ±) (x − x1) ,  where  tan ΞΈ = m.

14. CONDITION  OF  CONCURRENCY :

Three lines a1x + b1y + c1 = 0,  a2x + b2y + c2 = 0  &  a3x + b3y + c3 = 0  are concurrent if

a1 b1 c1

a2 b2 c2 = 0 . Alternatively :  If three constants  A, B & C  can be found such that  A(a1x + b1y a3 b3 c3

+ c1) + B(a2x + b2y + c2) + C(a3x + b3y + c3) ≡ 0 , then  the  three  straight lines  are concurrent.

15. AREA OF A TRIANGLE : If (xi, yi), i = 1, 2, 3  are the vertices of a triangle, then its area

x1 y1 1

1

is equal to x2 y2 1, provided the vertices are considered in the counter clockwise sense. The

2

x3 y3 1 above formula  will give a (−) ve area if the vertices  (xi, yi) , i = 1, 2, 3 are placed in the clockwise sense.

16. CONDITION  OF  COLLINEARITY  OF  THREE  POINTS−(AREA FORM):

x1 y1 1

The points (xi , yi) , i = 1 , 2 , 3 are collinear if  x2 y2 1=0.

x3 y3 1

17. THE  EQUATION  OF  A  FAMILY  OF  STRAIGHT  LINES  PASSING  THROUGH  THE POINTS  OF INTERSECTION OF  TWO  GIVEN  LINES:

The  equation  of  a  family  of   lines  passing  through  the  point  of  intersection  of a1x + b1y + c1 = 0  & a2x + b2y + c2 = 0 is given by (a1x + b1y + c1) + k(a2x + b2y + c2) = 0, where k is an

arbitrary real  number.

Note: If  u1 = ax + by + c ,  u2 = a′x + b′y + d ,  u3 = ax + by + c′ ,  u4 = a′x + b′y + d′ then, u1 = 0;    u2 = 0;    u3 = 0;    u4 = 0   form a  parallelogram.

u2 u3 − u1 u4 = 0  represents the diagonal BD.

Proof : Since it is the first degree equation in  x & y  it is a straight line. Secondly point B  satisfies  the  equation because  the  co−ordinates  of B satisfy u2 = 0 and u1 = 0.

Similarly for the point D.  Hence the result.

On the similar lines  u1u2 − u3u4 = 0  represents the diagonal  AC.

Note: The diagonal AC is also given by u1 + Ξ»u4 = 0    and    u2 + Β΅u3 = 0,  if the two equations are identical for some  Ξ» and  Β΅.

[For getting the values of Ξ» & Β΅ compare the coefficients of x, y & the constant  terms]

18. BISECTORS  OF  THE  ANGLES  BETWEEN  TWO  LINES  :

(i) Equations  of  the  bisectors  of  angles  between  the  lines   ax + by + c = 0   &

ax + by + c a′x + b′y + c′

a′x + b′y + c′ = 0 (ab′ ≠  a′b) are : = ±

a b2 + 2 a′2 +b′2

(ii) To discriminate between the acute angle bisector & the obtuse angle bisector If ΞΈ  be the angle between one of the lines & one of the bisectors, find tan ΞΈ .

If tan ΞΈ < 1, then  2 ΞΈ <  90° so that  this bisector is the acute angle bisector .

If tan ΞΈ > 1, then we get the bisector to be the obtuse angle bisector .

(iii) To discriminate between the bisector of the angle containing the origin & that of the angle not containing  the  origin.  Rewrite  the  equations  ,  ax + by + c = 0   & a′x + b′y + c′ = 0  such  that the constant  terms  c , c′  are  positive.  Then;                    ax + by + c a′x+b′y+c′  gives the  equation

of  the bisector of the angle containing the origin &  gives the equation of

the bisector of the angle not containing the origin.www.bloggermayank.online

(iv) To discriminate between acute angle bisector & obtuse angle bisector proceed as follows Write ax + by + c = 0  &  a′x + b′y + c′ = 0  such that constant terms are positive .

If aa′ + bb′ <  0 ,  then the angle between the lines that contains the origin is acute and the equation of the

ax+by+c a′x + b′y + c′

bisector of this acute angle is = +

a2+b2 a′2 +b′2

therefore ax + by + c = − a′x + b′y + c′  is the equation of other bisector a2 +b2 a′ + ′2 b 2

If, however , aa′ + bb′ > 0 ,  then  the  angle  between  the  lines  that  contains the origin is obtuse & the equation  of  the bisector  of  this  obtuse  angle  is:

ax + by + c a′x + b′y + c′ ax + by + c a′x + b′y + c′

= +  ;   therefore  = − a2 +b2 a′ + ′2 b 2 is the equation of other bisector.

(v) Another way of identifying an acute and obtuse angle bisector is as follows : Let  L1 = 0 & L2 = 0  are the given lines &  u1 = 0  and u2 = 0 are the bisectors between L1 = 0 & L2 = 0. Take a point P on any one of the lines L1 = 0 or L2 = 0  and drop perpendicular on  u1 = 0 & u2 = 0 as shown. If , p < q  ⇒  u1 is the acute angle bisector .

p > q  ⇒  u1 is the obtuse angle bisector . u1 = 0

p = q  ⇒  the lines L1 & L2 are perpendicular .

Note : Equation  of  straight lines  passing  through  P(x1, y1) &  equally inclined with the lines a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0  are those which are parallel to the bisectors  between  these  two  lines  &  passing through  the  point  P .

19. A  PAIR  OF  STRAIGHT  LINES  THROUGH  ORIGIN  :

(i) A homogeneous equation of degree two of the type  ax² + 2hxy + by² = 0  always represents a  pair  of straight  lines  passing  through  the  origin  &  if   : (a) h² > ab  ⇒ lines  are  real  &  distinct  .

(b) h² = ab  ⇒ lines are coincident  .

(c) h² < ab  ⇒ lines are imaginary with real point of intersection i.e. (0, 0)

(ii) If y = m1x & y = m2x  be the two equations represented by ax² + 2hxy + by2 = 0, then;

2h a

m1 + m2 = −  b  &  m1 m2 =  b .

(iii) If ΞΈ  is  the  acute  angle  between  the  pair of straight  lines  represented  by, ax2 + 2hxy + by2 = 0,  then;

tan ΞΈ  =   . The condition that these lines are:

(a) At  right  angles  to  each  other  is  a + b = 0.  i.e.  co−efficient  of   x2 + coefficient of  y2 =0.  (b)    Coincident  is  h2 = ab.

(c) Equally  inclined  to  the  axis  of  x  is  h = 0.  i.e.  coeff.  of  xy = 0.

Note: A homogeneous equation of degree  n  represents n straight lines passing through origin.

20. GENERAL EQUATION  OF SECOND DEGREE REPRESENTING A PAIR OF  STRAIGHT LINES:

(i) ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of  straight  lines  if:

a h g

abc + 2fgh − af2 − bg2 − ch2 = 0,   i.e.  if   h b f = 0.

g f c

(ii) The angle ΞΈ  between the two lines representing by a general equation is the same as that between the two lines represented by its homogeneous part only .

21. The joint equation  of  a pair of  straight lines joining origin to the points of intersection of the line given by  lx + my + n = 0 ................(i) &

the 2nd degree curve :  ax² + 2hxy + by² + 2gx + 2fy + c = 0 .......  (ii)

is   ax2 + 2hxy + by2 + 2gx  lx my−+n +2fylx my−+n +clx my−+n 2 = 0  ......  (iii)

(iii)  is obtained by homogenizing (ii) with the help  of  (i),  by writing (i)  in the form: lx my+  = 1.

 −n 

22. The equation to the straight lines bisecting the angle between the straight lines,

ax2 + 2hxy + by2 = 0  is  x2 − y2 = xy .www.bloggermayank.online

a − b h

23. The product of the perpendiculars,  dropped  from  (x1, y1)  to the pair of lines represented  by the equation,

ax 2 +2hx y +by 2

ax² + 2hxy + by² = 0  is  1 1 1 1 .

(a − +b)2 4h2

24. Any second degree curve through the four point of intersection of  f(x y) = 0  &  xy = 0  is given by f (x y) + Ξ» xy = 0  where  f(xy) = 0 is also a second degree curve. 18. CIRCLE

STANDARD  RESULTS :

1. EQUATION OF A CIRCLE IN VARIOUS FORM :

(a)     The circle with centre(h, k) &  radius‘r’has the equation ;(x − h)2  +  (y − k)2  = r2. (b) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre as :

(−g, −f)  &  radius  = g2 + −f2 c .

Remember  that  every  second  degree  equation in  x & y  in  which coefficient of x2 = coefficient of y2  &  there is no xy term always  represents a circle.

If g2 + f 2 − c > 0⇒ real circle. g2 + f 2 − c = 0⇒ point circle. g2 + f 2 − c < 0⇒ imaginary circle.

Note that the general equation of a circle contains three arbitrary constants, g, f & c which corresponds to the fact that a unique circle passes through three non collinear points.

(c) The  equation  of  circle  with  (x1 , y1)  &  (x2 , y2) as its diameter is  :

(x − x1)  (x − x2)  +  (y − y1)  (y − y2)  =  0.

Note that this will be the circle of least radius passing through  (x1 , y1)  &  (x2 , y2).

2. INTERCEPTS MADE BY A CIRCLE ON THE AXES :

The  intercepts  made  by  the  circle  x2 + y2 + 2gx + 2fy + c  =  0  on  the  co-ordinate  axes are  2 g2 − c

&  2 f2 − c  respectively

NOTE  : If g2 − c  > 0 circle  cuts  the  x  axis  at  two  distinct  points.

If g2 = c circle touches the x-axis.

If g2 < c circle  lies  completely  above or below  the  x-axis.

3. POSITION OF A POINT  w.r.t.  A CIRCLE  :

The  point  (x1 , y1) is inside, on or outside the circle x2 + y2 + 2gx + 2fy + c = 0. according  as  x12  +  y12  +  2gx1  +  2fy1  +  c  ⇔  0 . Note : The greatest & the least distance of a point A from a circle with centre C & radius r is  AC + r &  AC − r  respectively.

4. LINE  &  A CIRCLE :

Let  L = 0 be a line &  S = 0 be a circle. If r is  the  radius  of  the circle &  p is the length of the perpendicular from  the  centre  on the line, then :www.bloggermayank.online (i) p >  r ⇔  the line does not meet the circle i. e. passes out side the circle.

(ii) p =  r ⇔  the  line  touches  the  circle.

(iii) p <  r ⇔  the  line  is  a  secant  of  the circle.

(iv) p = 0  ⇒  the  line  is  a  diameter  of  the circle.

5. PARAMETRIC EQUATIONS OF A CIRCLE :

The parametric equations of (x − h)2  +  (y − k)2  =  r2 are :

x =  h + r cos ΞΈ   ;   y = k + r sin ΞΈ   ;   − Ο < ΞΈ ≤ Ο    where (h, k) is the centre,

r  is the radius  &  ΞΈ  is  a  parameter.  Note that equation of a straight line joining two point Ξ± & Ξ² on the

circle x2 + y2 = a2 is x cos   + y sin   = a cos   .

6. TANGENT  &  NORMAL :

(a) The  equation  of  the  tangent  to  the  circle   x2 + y2 = a2  at  its  point  (x1 , y1) is, x x1  +  y y1 =  a2. Hence equation of a tangent at (a cos Ξ±, a sin Ξ±) is ;

x cos Ξ± + y sin Ξ± = a. The point of  intersection of the tangents at the points P(Ξ±) and Q(Ξ²) is Ξ±+Ξ² Ξ±+Ξ² acos 2 ,  asin . Ξ±−Ξ²

cos 2 cos 2

(b) The  equation  of  the  tangent  to  the  circle  x2 + y2 + 2gx + 2fy + c = 0  at  its  point (x1 , y1)  is xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.

(c) y = mx + c  is  always  a tangent  to  the  circle  x2 + y2 = a2   if  c2 = a2 (1 + m2) and the point  of  contact  is

 a m2 a2 

− ,  .

c c 

(d) If  a  line  is  normal / orthogonal  to  a  circle  then  it  must  pass  through  the  centre of  the  circle.  Using this  fact  normal  to the circle x2 + y2  + 2gx + 2fy + c = 0  at (x1 , y1)  is y − y1 = xy11++gf (x − x1).

7. A FAMILY OF CIRCLES   :

(a) The equation of  the family of  circles  passing  through  the  points  of  intersection  of two  circles S1 = 0 &   S2 =  0 is   :      S1 + K S2  =  0        (K ≠ −1).

(b) The  equation  of  the  family  of  circles  passing  through  the  point  of  intersection  of  a  circle S =   0  & a  line  L = 0  is given by  S + KL = 0.www.bloggermayank.online (c) The  equation  of  a  family  of  circles  passing  through two given points (x1 , y1) & (x2 , y2)  can  be  written

x y 1

in  the form :  (x − x1) (x − x2) + (y − y1) (y − y2) + K x y1 1 1 = 0   where  K  is  a parameter

x y2 2 1

(d) The equation of a family of circles touching a fixed line y − y1 = m (x − x1) at the fixed point  (x1 , y1) is (x − x1)2 + (y − y1)2 + K [y − y1 −  m (x − x1)]  = 0 , where K is a parameter.

In  case  the line through (x1 , y1) is parallel to y - axis  the  equation  of  the family of circles touching it at (x1 , y1) becomes  (x − x1)2 +  (y − y1)2 + K (x − x1) = 0.

Also  if  line  is  parallel  to x - axis  the  equation  of  the  family  of  circles  touching it  at (x1 , y1)  becomes   (x − x1)2 +  (y − y1)2  +  K (y − y1)  =  0.

(e) Equation  of  circle  circumscribing  a  triangle  whose sides are given by L1 = 0  ; L2 = 0 & L3 = 0  is given by ;  L1L2 + Ξ» L2L3 + Β΅ L3L1 = 0  provided co-efficient of xy = 0 &  co-efficient of  x2 = co-efficient of y2.

(f) Equation  of  circle  circumscribing  a  quadrilateral  whose  side  in  order are represented  by  the  lines L1 = 0, L2 = 0, L3 = 0  & L4 = 0  is  L1L3 + Ξ» L2L4 = 0  provided co-efficient of  x2 = co-efficient of y2 and co-efficient of  xy = 0.

8. LENGTH OF A TANGENT AND POWER OF A POINT   :

The  length  of  a  tangent  from  an  external  point   (x1 , y1)  to  the  circle   S ≡ x2 + y2 + 2gx + 2fy + c =

0   is  given  by    L = x12 +y12 +2gx1+2f y1 +c = S1 . Square  of  length  of  the tangent  from  the  point

P  is  also  called  THE  POWER  OF  POINT w.r.t. a  circle.  Power  of  a  point  remains  constant  w.r.t.  a  circle. Note that  : power of a point  P is positive,  negative or zero according as the point  ‘P’ is outside,  inside or on the circle respectively.

9. DIRECTOR CIRCLE   :

The locus of the point of intersection of  two perpendicular  tangents is called the DIRECTOR CIRCLE of the given circle. The director circle of a circle is the concentric circle having radius equal to 2 times the original circle.

10. EQUATION OF THE CHORD WITH A GIVEN MIDDLE POINT :

The equation of the chord of the circle  S ≡ x2 + y2 + 2gx + 2fy + c = 0  in  terms  of its  mid point M (x1, y1)  is  y − y1 = − xy11++gf (x − x1).  This  on  simplication  can  be  put in  the  form xx1 + yy1 + g (x + x1) +

f (y + y1) + c = x12 + y12 + 2gx1 + 2fy1 + c which is designated by  T = S1.   Note that  : the  shortest chord  of  a  circle  passing  through  a  point  ‘M’ inside  the circle,  is one  chord  whose  middle  point  is

M.

11. CHORD  OF  CONTACT :

If  two tangents  PT1  &  PT2 are drawn  from  the  point  P (x1, y1)  to the circle

S ≡ x2 + y2 + 2gx + 2fy + c = 0, then  the  equation  of  the  chord  of  contact T1T2  is :   xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.

REMEMBER   : (a) Chord  of  contact  exists  only  if  the  point ‘P’ is not inside .

2LR

(b) Length of chord of contact T1 T2  = R2+L2 .

RL3

(c) Area of the triangle  formed  by  the  pair  of  the  tangents  &  its  chord  of  contact  =

R2+L2 Where  R  is

the  radius  of  the  circle  &  L  is  the  length  of  the  tangent from (x1, y1) on S = 0.

(d) Angle between the pair of tangents from (x1, y1) = tan−1   L22RL−R2  where  R = radius  ;   L = length of

tangent.

(e) Equation  of  the circle circumscribing  the  triangle  PT1 T2  is :

(x − x1)  (x + g)  +  (y − y1)  (y + f)  =  0.

(f) The  joint  equation  of  a  pair  of  tangents  drawn  from  the point  A (x1 , y1) to  the  circle x2 + y2 + 2gx + 2fy + c  =  0    is   :    SS1 = T2. Where  S ≡  x2 + y2 + 2gx + 2fy + c     ;   S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c    T ≡  xx1  +  yy1  +  g(x + x1)  +  f(y + y1)  +  c.

12. POLE  &  POLAR :

(i) If through a  point  P  in the  plane  of  the  circle ,  there  be  drawn  any  straight   line to  meet  the  circle  in

Q and  R,  the  locus  of  the  point  of  intersection  of  the tangents at  Q & R  is  called  the  POLAR  OF THE POINT P  ;   also P is called the  POLE OF THE POLAR.

(ii) The equation to the polar of a point  P (x1 ,  y1)  w.r.t.  the circle  x2 + y2 = a2  is given by xx1 + yy1 = a2, & if the circle is general then the equation of the polar becomes

xx1 + yy1  + g (x + x1) + f (y + y1) + c = 0. Note  that  if  the point (x1 , y1)  be  on  the circle  then  the  chord  of contact,  tangent  &  polar  will be represented by the same equation.

x2 + y2 = a2  is − AaC2 ,− BaC2  . (iii) Pole of a given line  Ax + By + C = 0  w.r.t.  any circle

(iv) If  the  polar  of  a  point  P  pass  through  a point  Q,  then  the  polar  of  Q  passes through  P.

(v) Two lines L1 & L2 are conjugate of each other if Pole of L1 lies on L2 &  vice versa Similarly  two points P & Q  are  said  to be conjugate of  each  other  if  the  polar of  P  passes  through  Q  & vice-versa.

13. COMMON  TANGENTS  TO  TWO  CIRCLES :

(i) Where the two circles  neither  intersect  nor  touch each other , there  are  FOUR common  tangents,  two of  them  are  transverse  &  the  others  are  direct  common tangents.

(ii) When  they intersect there are  two common  tangents, both of them being  direct.

(iii) When they touch each other :www.bloggermayank.online

(a) EXTERNALLY :  there are three common tangents, two direct and one is the tangent  at  the point of contact .

(b) INTERNALLY :  only one common tangent possible at their point of contact.

(iv) Length  of an  external  common  tangent &  internal common tangent to the two circles is given by:

Lext = d2 − −(r1 r )2 2  &  Lint = d2 − +(r1 r )2 2 .

Where  d = distance between the centres of the two circles . r1 & r2 are the radii of the 2  circles.

(v) The  direct  common  tangents  meet  at  a  point  which  divides  the  line joining centre  of  circles externally  in  the ratio of  their  radii.

Transverse  common  tangents  meet  at  a  point  which  divides  the  line joining centre  of  circles internally  in  the  ratio of their  radii.

14. RADICAL  AXIS & RADICAL  CENTRE :

The radical axis of two circles is the locus of points whose powers w.r.t. the two circles are equal. The equation of radical axis of the two circles  S1 = 0 & S2 = 0  is given ;

S1 − S2 = 0    i.e.  2 (g1 − g2) x + 2 (f1 − f2) y + (c1 − c2) = 0.

NOTE  THAT :

(a) If  two  circles  intersect,  then  the  radical  axis  is the common chord of the two circles.

(b) If  two circles  touch  each  other  then  the radical  axis  is  the  common tangent of the  two  circles  at  the common  point  of  contact.

(c) Radical  axis  is always  perpendicular to the line joining the centres of  the 2circles.

(d) Radical  axis  need  not  always  pass  through  the mid  point  of  the  line  joining  the  centres of the two circles.

(e) Radical  axis  bisects  a  common  tangent  between  the  two  circles.

(f) The  common  point  of  intersection  of  the  radical  axes  of  three  circles  taken  two  at  a  time  is  called the  radical  centre  of  three  circles.

(g) A  system  of  circles ,  every  two  which  have  the  same  radical  axis,  is  called a  coaxal  system.

(h) Pairs of circles which do not have radical axis are concentric.

15. ORTHOGONALITY OF TWO CIRCLES :

Two circles S1= 0  &  S2= 0  are  said  to  be orthogonal  or  said  to intersect  orthogonally if the tangents at  their  point  of  intersection  include  a  right  angle.  The  condition  for  two  circles to be  orthogonal is  :   2 g1 g2 + 2 f1 f2 =  c1 + c2  . Note :

(a) Locus  of  the centre of a variable circle orthogonal to two fixed circles is the radical axis between the two fixed circles .

(b) If  two  circles  are  orthogonal,  then the polar of a point 'P' on first circle w.r.t. the second circle passes through the point Q which is the other end of the diameter through P . Hence locus of a point which moves such that its polars w.r.t. the circles  S1 = 0 ,  S2 = 0 & S3 = 0  are concurrent in a circle which is orthogonal 19to all the three circles..CONIC SECTION PARABOLA

1. CONIC SECTIONS:

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixedpoint is in a constant ratio to its perpendicular distance from a fixed straight line.

The fixed point is called the FOCUS.

The fixed straight line is called the DIRECTRIX.

The constant ratio is called the ECCENTRICITY denoted by e.

The line passing through the focus & perpendicular to the directrix is called the AXIS. A point of intersection of a conic with its axis is called a VERTEX.

2. GENERAL EQUATION OF A CONIC : FOCAL DIRECTRIX PROPERTY :

The general equation of a conic with focus (p, q) & directrix lx + my + n = 0 is :

(l2 + m2) [(x − p)2 + (y − q)2] = e2 (lx + my + n)2 ≡ ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

3. DISTINGUISHING BETWEEN THE CONIC :

The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also upon the value of the eccentricity e. Two different cases arise.

CASE (I) : WHEN THE FOCUS LIES ON THE DIRECTRIX.

In this case D ≡ abc + 2fgh − af2 − bg2 − ch2 = 0 & the general equation of a conic represents a pair of

straight lines if :www.bloggermayank.online e > 1 the lines will be real & distinct intersecting at S. e = 1 the lines will coincident. e < 1 the lines will be imaginary.

CASE (II) : WHEN THE FOCUS DOES NOT LIE ON DIRECTRIX.

a parabola    an ellipse    a hyperbola    rectangular hyperbola

e = 1 ; D ≠ 0,   0 < e < 1 ; D ≠ 0 ;    e > 1 ; D ≠ 0 ; e > 1 ; D ≠ 0 h² = ab    h² < ab    h² > ab      h² > ab ; a + b = 0

4. PARABOLA : DEFINITION :

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus)is equal to its perpendicular distance from a fixed straight line (directrix).

Standard equation of a parabola is y2 = 4ax. For this parabola :

(i) Vertex is (0, 0) (ii) focus is (a, 0) (iii) Axis is y = 0   (iv) Directrix is x + a = 0

FOCAL DISTANCE :The distance of a point on the parabola from the focus is called the FOCAL DISTANCE OF THE POINT.

FOCAL CHORD  :

A chord of the parabola, which passes through the focus is called a FOCAL CHORD.

DOUBLE ORDINATE : A chord of the parabola perpendicular to the axis of the symmetry is called a DOUBLE ORDINATE.

LATUS RECTUM :

A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the LATUS RECTUM. For y2 = 4ax.

Length of the latus rectum = 4a.    ends of the latus rectum are L(a, 2a) & L' (a, − 2a).

Note that: (i) Perpendicular distance from focus on directrix = half the latus rectum.

(ii)Vertex is middle point of the focus & the point of intersection of directrix & axis. (iii) Two parabolas are laid to be equal if they have the same latus rectum.

Four standard forms of the parabola are y2 = 4ax ;  y2 = − 4ax ; x2 = 4ay ; x2 = − 4ay

5. POSITION OF A POINT RELATIVE TO A PARABOLA :

The point (x1 y1) lies outside, on or inside the parabola y2 = 4ax according as the expression y12 − 4ax1 is positive, zero or negative.

6. LINE & A PARABOLA :

The line y = mx + c meets the parabola y2 = 4ax in two points real, coincident or imaginary according as a

a<> c m  ⇒   condition of tangency is, c =  m .

 2  a(1 m

7. Length of the chord intercepted by the parabola on the line y = m x + c is :   4  + m )(a2 −mc) .

Note: length of the focal chord making an angle Ξ± with the x− axis is 4aCosec² Ξ±.

8. PARAMETRIC REPRESENTATION :

The simplest & the best form of representing the co−ordinates of a point on the parabola is (at2, 2at). The equations x = at² & y = 2at together represents the parabola y² = 4ax, t being the parameter. The equation of a chord joining t1 & t2 is 2x − (t1 + t2) y + 2 at1 t2 = 0.

Note: If chord joining  t1, t2 & t3, t4 pass a through point (c, 0) on  axis, then t1t2 = t3t4 = − c/a.

9. TANGENTS TO THE PARABOLA y2 = 4ax :

(i) y y1 = 2 a (x + x1) at the point (x1, y1)  ; (ii) y = mx +  a (m ≠ 0) at  a2 ,2ma  m m

(iii) t y = x + a t²  at  (at2, 2at).

Note : Point of intersection of the tangents at the point t1 & t2 is [ at1 t2, a(t1 + t2) ].

10. NORMALS TO THE PARABOLA  y2 = 4ax :

(i) y − y1 =− y1 (x − x1) at (x1, y1)   ; (ii) y = mx − 2am − am3 at (am2, − 2am)

2a

(iii) y + tx = 2at + at3 at (at2, 2at).

Note : Point of intersection of normals at t1 & t2 are, a (t 12 + t 22 + t1t2 + 2) ; − a t1 t2 (t1 + t2).

11. THREE VERY IMPORTANT RESULTS :

(a) If t1 & t2 are the ends of a focal chord of the parabola y² = 4ax then t1t2 = −1. Hence the co-ordinates at the

extremities of a focal chord can be taken as (at2, 2at) &  a2 ,− 2ta  .

t

(b) If the normals to the parabola y² = 4ax at the point t1, meets the parabola again at the point t2, then

t2 =− +t1  t21.

(c) If the normals to the parabola y² = 4ax at the points t & t intersect again on the parabola at the point 't ' then t1 t2 = 2 ; t3 = − (t1 + t2) and the line joining t1 & t1 2 passes through a fixed point (2 −2a, 0). 3

General Note :

(i) Length of subtangent at any point P(x, y) on the parabola y² = 4ax equals twice the abscissa of the point P. Note that the subtangent is bisected at the vertex.

(ii) Length of subnormal is constant for all points on the parabola & is equal to the semi latus rectum.

(iii) If a family of straight lines can be represented by an equation Ξ»2P + Ξ»Q + R = 0 where Ξ» is a parameter and P, Q, R are linear functions of x and y then the family of lines will be tangent to the curve Q2 = 4 PR.

12. The equation to the pair of tangents which can be drawn from any point (x1, y1) to the parabola y² = 4ax is given by :  SS1 = T2 where  :

S ≡ y2 − 4ax ; S1 = y12 − 4ax1; T ≡ y y1 − 2a(x + x1).

13. DIRECTOR CIRCLE :

Locus of the point of intersection of the perpendicular tangents to the parabola y² = 4ax is called the DIRECTOR CIRCLE. It’s equation is x + a = 0 which is parabola’s own directrix.

14. CHORD OF CONTACT : www.bloggermayank.online

Equation to the chord of contact of tangents drawn from a point P(x1, y1) is     yy1 = 2a (x + x1). Remember that the area of the triangle formed by the tangents from the point (x1, y1) & the chord of contact is (y12 − 4ax1)3/2 ÷ 2a. Also note that the chord of contact exists only if the point P is not inside.

15. POLAR & POLE :

(i) Equation of the Polar of the point P(x1, y1) w.r.t. the parabola y² = 4ax is,

(ii) The pole of the line y y1= 2a(x + xlx + my + n = 0 w.r.t. the parabola y² = 4ax is 1)  n ,− 2am. Note:  1 1 

(i) The polar of the focus of the parabola is the directrix.

(ii) When the point (x1, y1) lies without the parabola the equation to its polar is the same as the equation to the chord of contact of tangents drawn from (x1, y1) when (x1, y1) is on the parabola the polar is the same as the tangent at the point.

(iii) If the polar of a point P passes through the point Q, then the polar of Q goes through P.

(iv) Two straight lines are said to be conjugated to each other w.r.t. a parabola when the pole of one lies on the other.

(v) Polar of a given point P w.r.t. any Conic is the locus of the harmonic conjugate of P w.r.t. the two points is which any line through P cuts the conic.

16. CHORD WITH A GIVEN MIDDLE POINT  :

Equation of the chord of the parabola  y² = 4ax whose middle point is

(x1, y1) is y − y1 =  2a (x − x1). This reduced to   T = S1 y1

where T ≡ y y1 − 2a (x + x1) & S1 ≡ y12 − 4ax1.

17. DIAMETER :

The locus of the middle points of a system of parallel chords of a Parabola is called a DIAMETER. Equation to the diameter of a parabola is y = 2a/m, where m = slope of parallel chords.

Note:

(i) The tangent at the extremity of a diameter of a parabola is parallel to the system of chords it bisects.

(ii) The tangent at the ends of any chords of a parabola meet on the diameter which bisects the chord.

(iii) A line segment from a point P on the parabola and parallel to the system of parallel chords is called the ordinate to the diameter bisecting the system of parallel chords and the chords are called its double ordinate.

18. IMPORTANT HIGHLIGHTS  :

(a) If the tangent & normal at any point ‘P’ of the parabola intersect the axis at T & G then ST = SG = SP where ‘S’ is the focus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP & the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection.

(b) The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus.www.bloggermayank.online

(c) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and hence a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P (at2, 2at) as diameter touches the tangent at the vertex and intercepts a chord of length a 1+ t2 on a normal at the point P.

(d) Any tangent to a parabola & the perpendicular on it from the focus meet on the tangtent at the vertex.

(e) If the tangents at P and Q meet in T, then : TP and TQ subtend equal angles at the focus S.

ST2 = SP. SQ   & The triangles SPT and STQ are similar.

(f) Tangents and Normals at the extremities of the latus rectum of a parabola y2 = 4ax constitute a square, their points of intersection being (−a, 0) & (3 a, 0).

(g) Semi latus rectum of the parabola y² = 4ax, is the harmonic mean between segments of any focal chord of

2bc 1 1 1

the parabola is ; 2a =   i.e. + = . b+ c b c a

(h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

(i) The orthocentre of any triangle formed by three tangents to a parabola y2 = 4ax lies on the directrix & has the co-ordinates − a, a (t1 + t2 + t3 + t1t2t3).

(j) The area of the triangle formed by three points on a

parabola is twice the area of the triangle formed by the tangents at these points.

(k) If normal drawn to a parabola  passes through a point P(h, k) then k = mh − 2am − am3  i.e. am3 + m(2a − h) + k = 0. Then gives m1 + m2 + m3 = 0 ; m1m2 + m2m3

2a − h

m3m1 = a ; m1 m2 m3 = −

where m1, m2, & m3 are the slopes of the three concurrent normals.  Note that the algebraic sum of the: slopes of the three concurrent normals is zero. ordinates of the three conormal points on the

parabola is zero.

Centroid of the ∆ formed by three co-normal points lies on the x-axis.

(l) A circle circumscribing the triangle formed by three co−normal points passes through the vertex of the parabola and its equation is,  2(x2 + y2) − 2(h + 2a)x − ky = 0

Suggested problems from S.L.Loney: Exercise-25 (Q.5, 10, 13, 14, 18, 21), Exercise-26 (Important) (Q.4, 6, 7,

16, 17, 20, 22, 26, 27, 28, 34, 38),  Exercise-27 (Q.4, 7), Exercise-28 (Q.2, 7, 11, 14, 17, 23), Exercise-29 (Q.7, 8, 10, 19, 21, 24, 26, 27), Exercise-30 (2, 3, 13, 18, 20, 21, 22, 25, 26, 30) Note: Refer to the figure on Pg.175 if necessary.

ELLIPSE

Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is x2 + y2 =1.

a2 b2

Where a > b & b² = a²(1 − e²) ⇒  a2 − b2 = a2 e2. Where e = eccentricity (0 < e < 1).   FOCI : S ≡ (a e, 0) & 4. S′ ≡ (− a e, 0).

EQUATIONS OF DIRECTRICES :

a a . x =    &  x =−  e e

5.

VERTICES :

A′ ≡ (− a, 0)  &  A ≡ (a, 0) .

MAJOR AXIS :

The line segment A′ A in which the foci  S′ & S lie is of length 2a & is called the major axis (a > b) of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (z).

MINOR AXIS :www.bloggermayank.online

The y−axis intersects the ellipse in the points B′ ≡ (0, − b) & B ≡ (0, b). The line segment B′B of length 2b (b < a) is called the Minor Axis of the ellipse.

PRINCIPAL AXIS :

The major & minor axis together are called Principal Axis of the ellipse. 6.

CENTRE :

The point which bisects every chord of the conic drawn through it is called the centre of the conic.  C ≡ (0, (i)

2 2

on the ellipse (0 ≤ ΞΈ < 2 Ο). Note that  (PN) = =b Semi  minor  axis  Hence  “ If from each point of a (QN) a Semi  major  axis

circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle”.

PARAMETRIC REPRESENTATION :

x2 y2

The equations x = a cos ΞΈ & y = b sin ΞΈ  together represent the ellipse  2 + =2 1.

a b

Where ΞΈ is a parameter. Note that if  P(ΞΈ) ≡ (a cos ΞΈ, b sin ΞΈ) is on the ellipse then ;

Q(ΞΈ) ≡ (a cos ΞΈ, a sin ΞΈ) is on the auxiliary circle.

LINE AND AN ELLIPSE :

x2 y2

The line y = mx + c meets the ellipse 2 + 2 =1  in two points real, coincident or imaginary according

a b

x2

as c2 is < = or > a2m2 + b2. Hence y = mx + c is tangent to the ellipse 2 + y22 =1  if c2 = a2m2 + b2.

a b

The equation to the chord of the ellipse joining two points with eccentric angles Ξ±& Ξ² is given by x Ξ±+Ξ² y Ξ±+Ξ² Ξ±−Ξ²  cos +  sin = cos

a 2 b 2 2

TANGENTS :

0) the origin is the centre of the ellipse x + y =1.Note :The figure formed by the tangents at the extremities of latus rectum is rhoubus of area

x x21 + yy21 =1 is tangent to the ellipse at (x1, y1). a b

a2 b2

DIAMETER :(ii)

A chord of the conic which passes through the centre is called a diameter of the conic.

FOCAL CHORD : A chord which passes through

(iii) a focus is called a focal chord.

DOUBLE ORDINATE :(iv)

A chord perpendicular to the major axis is called a double ordinate.

LATUS RECTUM : The focal chord(v) perpendicular to the major axis is called the latus rectum. Length of latus rectum  (LL′) =

7.

2b2 = (minor axis)2 = 2a(1 e− 2)= 2e (distance

a major  axis (i) from focus to the corresponding directrix)

NOTE : (ii)

(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence distance of focus from the extremity of a minor axis is equal to semi major axis. i.e. BS = CA. (iii)

(ii) If the equation  of the ellipse is given as x2 + y2 =1 & nothing is mentioned, then the rule is to assume

2 2

a b that  a > b. 8.

2. POSITION OF A POINT w.r.t. AN ELLIPSE :

The point P(x1, y1) lies outside, inside or on the ellipse according as ; x122 +y122 −1 > < or = 0.

a b

3. AUXILIARY CIRCLE / ECCENTRIC ANGLE :

9.

A circle described on major axis as diameter is called the auxiliary circle.

10.

Let Q be a point on the auxiliary circle x2 + y2 = a2 such that QP produced is perpendicular to the x-axis then P & Q are called as the CORRESPONDING

y = mx ± a m2 2 + b2  is tangent to the ellipse for all values of m.

Note that there are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction.

xcosΞΈ+ ysinΞΈ=1 is tangent to the ellipse at the point (a cos ΞΈ, b sin ΞΈ). a b

The eccentric angles of point of contact of two parallel tangents differ by Ο. Conversely if the difference between the eccentric angles of two points is p then the tangents at these points are parallel.

Ξ±+Ξ² sin

Point of intersection of the tangents at the point Ξ± & Ξ² is  a coscos Ξ±−Ξ²22 , b cos Ξ±−Ξ²2 .

NORMALS :www.bloggermayank.online

2 2

Equation of the normal at (x1, y1) is a x − b y = a² − b² = a²e².

x1 y1

Equation of the normal at the point (acos ΞΈ, bsin ΞΈ) is ;  ax. sec ΞΈ − by. cosec ΞΈ = (a² − b²).

Equation of a normal in terms of its slope 'm' is y = mx −

(a2 −b )m2

. a2 + b2 2m

DIRECTOR CIRCLE :

Locus of the point of intersection of the tangents which meet

at right angles is called the Director Circle. The equation to this locus is  x² + y² = a² + b²  i.e. a circle whose centre is the centre of the ellipse & whose radius is the length of the line joining the ends of the major & minor axis.

Chord of contact, pair of tangents, chord with a given middle point, pole & polar are to be interpreted as they are in parabola.

DIAMETER :

The locus of the middle points of a system of parallel chords with slope 'm' of an ellipse is a straight line

b2

passing through the centre of the ellipse, called its diameter and has the equation y = −   x.

2

a m

x2 y2

11. IMPORTANT  HIGHLIGHTS : Refering to an ellipse + =1.

2 2

H − 1 If P be any point on the ellipse with S & S′ as its foci thena b

(SP) + (S′P) = 2a.

H − 2 The product of the length’s of the perpendicular segments from the foci on any tangent to the ellipse is b2 and the feet of these perpendiculars Y,Y′ lie on its auxiliary circle.The tangents at these feet to the auxiliary circle meet on the ordinate of P and that the locus of their point of intersection is a similiar ellipse as that of the original one. Also the lines joining centre to the feet of the perpendicular Y and focus to the point of contact of tangent are parallel.

H − 3 If the normal at any point P on the ellipse with centre C meet the major & minor axes in G & g respectively, & if

CF be perpendicular upon this normal, then

(i) PF. PG = b2 (ii) PF. Pg = a2 (iii) PG. Pg = SP. S′ P (iv) CG. CT = CS2

(v) locus of the mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse. [where S and S′ are the focii of the ellipse and T is the point where tangent at P meet the major axis]

H − 4 The tangent & normal at a point P on the ellipse bisect the external & internal angles between the focal distances of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus & vice−versa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point P meet on the normal PG and bisects it where G is the point where normal at P meets the major axis.

H − 5 The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.www.bloggermayank.online H − 6 The circle on any focal distance as diameter touches the auxiliary circle.

H − 7 Perpendiculars from the centre upon all chords which join the ends of any perpendicular diameters of the ellipse are of constant length.

H − 8 If the tangent at the point P of a standard ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then,

(i) T t. PY = a2 − b2 and (ii) least value of  Tt  is  a + b.  HYPERBOLA

The HYPERBOLA is a conic whose eccentricity is greater than unity. (e > 1).

1. STANDARD EQUATION & DEFINITION(S)

x2 y2

Standard equation of the hyperbola is =1.

a2 b2

Where b2 = a2 (e2 − 1)

b2

or a2 e2 = a2 + b2  i.e.  e2 = 1 +  2

a

2

= 1 +   C.A 

 T.A 

FOCI :

S ≡ (ae, 0)   & S′ ≡ (− ae, 0).

a a

EQUATIONS OF DIRECTRICES :       x =   & x = −   . e e

2b2 (C.A.)2

VERTICES : A ≡ (a, 0)&   A′ ≡ (− a, 0).  l (Latus rectum) = = = 2a (e2 − 1).

a T.A.

Note : l (L.R.) = 2e (distance from focus to the corresponding directrix)

TRANSVERSE AXIS  : The line segment A′A of length 2a in which the foci S′ & S both lie is called the

T.A. OF THE HYPERBOLA.

CONJUGATE AXIS  : The line segment B′B between the two points B′ ≡ (0, − b) & B ≡ (0, b) is called as the

C.A. OF THE HYPERBOLA.

The T.A.  &  the C.A. of the hyperbola are together called the Principal axes of the hyperbola.

2. FOCAL PROPERTY :

The difference of the focal distances of any point on the hyperbola is constant and equal to transverse axis i.e. PS − PS′ = 2a . The distance SS' = focal length.

3. CONJUGATE HYPERBOLA :

Two hyperbolas such that transverse & conjugate axes of one hyperbola are respectively the conjugate &

x2 y2

the transverse axes of the other are called CONJUGATE HYPERBOLAS of each other. eg.  − =1   &

2 2

a b

x2 y2

+ =1 are conjugate hyperbolas of each. a2 b2

Note  :(a)If e1& e2 are the eccentrcities of the hyperbola & its conjugate then e1−2 + e2−2 = 1.

(b)The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

(c) Two hyperbolas are said to be similiar if they have the same eccentricity.

4. RECTANGULAR OR EQUILATERAL HYPERBOLA :

The particular kind of hyperbola in which the lengths of the transverse & conjugate axis are equal is called an EQUILATERAL HYPERBOLA. Note that the eccentricity of the rectangular hyperbola is 2 and the length of its latus rectum is equal to its transverse or conjugate axis.

5. AUXILIARY CIRCLE :www.bloggermayank.online A circle drawn with centre C & T.A. as a diameter is called the AUXILIARY CIRCLE of the hyperbola. Equation of the auxiliary circle is x2 + y2 = a2.

Note from the figure that

P & Q are called the

"CORRESPONDING POINTS " on the hyperbola & the auxiliary  circle. 'ΞΈ' is called the eccentric angle of the point 'P' on the hyperbola. (0 ≤ ΞΈ <2Ο).

Note : The equations x = a sec ΞΈ & y = b tan ΞΈ together represents the hyperbola  x2 − =y2 1

2 2

a b

where ΞΈ is a parameter. The parametric equations : x = a cos h Ο, y = b sin h Ο also represents the same hyperbola.

General Note : Since the fundamental equation to the hyperbola only differs from that to the ellipse in having – b2 instead of  b2 it will be found that many propositions for the hyperbola are derived from those for the ellipse by simply changing the sign of b2.

6. POSITION OF A POINT 'P' w.r.t. A HYPERBOLA :

The quantity x122 − y122 =1 is positive, zero or negative according as the point (x1, y1) lies within, upon or a b without the curve.

7. LINE AND A HYPERBOLA :2 2 The straight line y = mx + c is a secant, a tangent or passes outside x y 2 > = < a2 m2 − b2.

the hyperbola + =1 according as:  c

a2 b2

8. TANGENTS AND NORMALS : TANGENTS :

(a) Equation of the tangent to the hyperbola x2 y2 xx1 −yy1 =1.

a2 − =b2 1 at the point (x1, y1) is a2 b2

Note: In general two tangents can be drawn from an external point (x1 y1) to the hyperbola and they are y − y1 = m1(x

− x1) & y − y1 = m2(x − x2), where m1 & m2 are roots of the equation  (x12 − a2)m2 − 2 x1y1m + y12 + b2 = 0. If D < 0, then no tangent can be drawn from (x1 y1) to the hyperbola.

(b)Equation of the tangent to the hyperbola x2 − y2 =1 at the point (a sec ΞΈ, b tan ΞΈ) is xsecΞΈ− ytanΞΈ=1. a2 b2 a b

cos ΞΈ1 −ΞΈ2 sin ΞΈ1 +ΞΈ2

Note : Point of intersection of the tangents at ΞΈ1 & ΞΈ2 is x = a cos ΞΈ +ΞΈ2 , y = b cos ΞΈ +ΞΈ2

2 2

2 2 −b2 can be taken as the tangent to the hyperbola x2 − y2 =1. Note that there are

(c) y = mx ± a m

a b

two parallel tangents having the same slope m.

(d) Equation of a chord joining Ξ± & Ξ² is  x cos Ξ± −Ξ² −  y sin Ξ± +Ξ² = cos Ξ± +Ξ² a 2 b 2 2

NORMALS:

(a) The equation of the normal to the hyperbola  x22 − =y22 1 at the point P(x1, y1) on it is a x2 + b y2 = a2 − b2 a b x1 y1

= a2 e2. 2 2

ΞΈ, b tanΞΈ) on the hyperbola x − =y 1 is

(b) The equation of the normal at the point P (a sec

2 2

a x + by 2 + b2 = a2 e2. a b

= a

secΞΈ tanΞΈ

(c) Equation to the chord of contact, polar, chord with a given middle point, pair of tangents from an external point is to be interpreted as in ellipse.www.bloggermayank.online

9. DIRECTOR CIRCLE :

The locus of the intersection of tangents which are at right angles is known as the DIRECTOR CIRCLE of the hyperbola. The equation to the director circle is :  x2 + y2 = a2 − b2.

If b2 < a2  this circle is real;  if b2 = a2  the radius of the circle is zero  &  it reduces to a point circle at the origin. In this case the centre is the only point from which the tangents at right angles can be drawn to the curve.  If b2 > a2, the radius of the circle is imaginary, so that there is no such circle & so no tangents at right angle can be drawn to the curve.

10. HIGHLIGHTS ON TANGENT AND NORMAL :

2 2

H−1 Locus of the feet of the perpendicular drawn from focus of the hyperbola x − y =1 upon any tangent is a2 b2

its auxiliary circle i.e. x2 + y2 = a2 & the product of the feet of these perpendiculars is b2 · (semi C ·A)2

H−2 The portion of the tangent between the point of contact & the directrix subtends a right angle at the corresponding focus.

H−3 The tangent & normal at any point of a hyperbola bisect the angle between the focal radii. This spells the reflection property of the hyperbola as "An incoming light ray " aimed towards one focus is

reflected from the outer surface of the hyperbola towards the other focus. It follows that if an ellipse and a hyperbola have the same foci, they cut at right angles at any of their common point.

x2 y2 x2 y2

Note that the ellipse + =1 and the hyperbola =1(a > k > b > 0) Xare confocal and a2 b2 a2−k2 k2−b2

therefore orthogonal.

H−4 The foci of the hyperbola and the points P and Q in which any tangent meets the tangents at the vertices are concyclic with PQ as diameter of the circle.

11. ASYMPTOTES :Definition : If the length of the perpendicular let fall from a

point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the Hyperbola.

To find the asymptote of the hyperbola :

2 2

Let y = mx + c is the asymptote of the hyperbola x − y =1.

a2 b2

Solving these two we get the quadratic as

(b2− a2m2) x2− 2a2 mcx − a2 (b2 + c2) = 0 ....(1)

In order that y = mx + c be an asymptote, both roots of equation (1) must approach infinity, the conditions for which are :

coeff of x2 = 0 & coeff of x = 0. b2 − a2m2 = 0 or m =±  b  &

a

2 mc = 0 ⇒ c = 0. ∴ equations of asymptote are x + =y 0   and x − =y 0.

a

a b a b x2 y2

combined equation to the asymptotes  2 − =2 0.

a b

PARTICULAR CASE :

When b = a the asymptotes of the rectangular hyperbola. x2 − y2 = a2 are, y = ±  x which are at right angles.

Note : (i) Equilateral hyperbola ⇔ rectangular hyperbola.

(ii) If a hyperbola is equilateral then the conjugate hyperbola is also equilateral.

(iii) A hyperbola and its conjugate have the same asymptote.

(iv) The equation of the pair of asymptotes differ the hyperbola & the conjugate hyperbola by the same constant only.

(v) The asymptotes pass through the centre of the hyperbola & the bisectors of the angles between the asymptotes are the axes of the hyperbola.www.bloggermayank.online

(vi) The asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through the extremities of each axis parallel to the other axis.

(vii) Asymptotes are the tangent to the hyperbola from the centre.

(viii) A simple method to find the coordinates of the centre of the hyperbola expressed as a general equation of degree 2 should be remembered as:

Let f (x, y) = 0 represents a hyperbola.

∂f ∂f ∂f ∂f

Find & . Then the point of  intersection of = 0 & = 0 gives the centre of the

∂x ∂y ∂x ∂y hyperbola.

12. HIGHLIGHTS ON ASYMPTOTES:

H−1 If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to the square of the semi conjugate axis.

H−2 Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix & the common points of intersection lie on the auxiliary circle.

2 2

H−3 The tangent at any point P on a hyperbola x − y =1 with centre C, meets the asymptotes in Q and R and a2 b2

cuts off a ∆ CQR of constant area equal to ab from the asymptotes & the portion of the tangent intercepted between the asymptote is bisected at the point of contact. This implies that locus of the centre of the circle circumscribing the ∆ CQR in case of a rectangular hyperbola is the hyperbola itself & for a standard hyperbola the locus would be the curve, 4(a2x2 − b2y2) = (a2 + b2)2.

H−4 If the angle between the asymptote of a hyperbola x2 − y2 =1 is 2ΞΈ then e = secΞΈ.

a2 b2

13.RECTANGULAR HYPERBOLA :Rectangular hyperbola referred to its asymptotesas

axis of coordinates.(a)Eq. is xy = c2 with parametric representation x = ct, y = c/t, t ∈ R – {0}.

+ t2) with slope m = – 1 .

(b) Eq.of a chord joining the points (t1) & (t2) is x + t1t2y = c(t1

t t1 2

(x1, y1) is x + y = 2 & at P (t) is x + ty = 2c.

(c) Equation of the tangent at P

x1 y1 t

c 2(x – ct) (d) Equation of normal :  y – = t

t

(e) Chord with a given middle point as (h, k) is  kx + hy = 2hk.

20.BINOMIAL  EXPONENTIAL  & LOGARITHMIC  SERIES

1. BINOMIAL  THEOREM  : The  formula  by  which  any  positive  integral  power  of  a  binomial expression  can  be expanded  in  the  form  of  a  series  is  known  as  BINOMIAL  THEOREM  . If  x , y ∈ R  and  n ∈ N,  then  ;

n

(x + y)n = nC0 xn + nC1 xn−1 y + nC2 xn−2y2 + ..... + nCr xn−ryr + ..... + nCnyn  = ∑ nCr xn – r yr.

r=0

This  theorem  can  be  proved  by  Induction  .www.bloggermayank.online OBSERVATIONS  :(i)The number of terms in the expansion is (n + 1)  i.e. one or more than the index .(ii) The  sum  of  the  indices  of  x & y  in  each  term  is  n  (iii) The binomial coefficients of the terms  nC0 , nC1 ....  equidistant from the beginning and the end  are equal.

2. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE  :

(i) General  term (ii) Middle  term(iii)Term  independent  of  x    & (iv) Numerically  greatest term

(i) The  general  term  or  the (r + 1)th  term  in  the  expansion  of  (x + y)n  is  given  by ; Tr+1 = nCr xn−r . yr

(ii) The  middle  term(s)  is  the  expansion  of  (x + y)n  is (are)  :

(a) If  n  is  even ,  there  is  only  one  middle  term  which  is  given  by  ;

T(n+2)/2 = nCn/2 . xn/2 . yn/2

(b) If  n  is  odd ,  there  are  two  middle  terms  which  are  :

T(n+1)/2    &    T[(n+1)/2]+1

(iii) Term independent of  x  contains no  x ;  Hence find the value  of  r  for which the exponent  of  x  is zero.

(iv) To  find  the  Numerically  greatest  term  is  the  expansion  of  (1 + x)n ,  n ∈ N find

Tr 1+ = n nCrxr 1r− = n −rr +1x . Put the absolute value of x & find the value of r  Consistent  with  the

Tr Cr−1x

Tr 1+

inequality  > 1.

Tr

Note  that  the  Numerically  greatest  term  in  the  expansion  of  (1 − x)n , x > 0 , n ∈ N  is  the same  as the  greatest  term  in  (1 + x)n  .

3.If ( A +B)n = I + f,  where   I & n  are  positive  integers,  n  being  odd  and  0 < f < 1, then

(I + f) . f = Kn  where  A − B2 = K > 0  & A − B < 1. If  n  is  an  even  integer,  then  (I + f) (1 − f) = Kn.

4.(ii) BINOMIAL  COEFFICIENTS  :C0 + C2 + C4 + ....... = C1 + C3 + C5 + ....... = 2(i) Cn0− + C1 1 + C2 + ....... + Cn = 2n

(iii) C ² + C ² + C ² + .... + C ² = 2nC = (2n) !

0 1 2 n n n! n!

(iv) C0.Cr + C1.Cr+1 + C2.Cr+2 + ... + Cn−r.Cn = (n + r)(n −r)!

REMEMBER  : (i) (2n)! = 2n . n! [1. 3. 5 ...... (2n − 1)] 5. BINOMIAL  THEOREM  FOR  NEGATIVE  OR  FRACTIONAL  INDICES

If  n ∈ Q ,  then  (1 + x)n = 1+ n x + n(n −1) x2+ n(n −1)(n −2) x3+......∞  Provided | x | <  1.

2! 3!

Note  : (i)When the  index  n  is  a  positive  integer  the  number  of  terms  in  the  expansion of (1 + x)n  is  finite  i.e.  (n + 1)  &  the  coefficient  of  successive  terms  are  :

nC0 , nC1 , nC2 , nC3 ..... nCn

(ii) When the index  is  other than  a  positive integer  such  as negative integer  or fraction,  the  number  of terms  in  the  expansion  of  (1 + x)n  is   infinite  and  the symbol  nCr  cannot  be  used  to  denote  the Coefficient  of  the  general  term .

(iii) Following  expansion  should  be  remembered  (x < 1).

(a)  (1 + x)−1 = 1 − x + x2 − x3 + x4 − .... ∞(b)  (1 − x)−1 = 1 + x + x2 + x3 + x4 + .... ∞

(c)  (1 + x)−2 = 1 − 2x + 3x2 − 4x3 + .... ∞(d)  (1 − x)−2 = 1 + 2x + 3x2 + 4x3 + ..... ∞

(iv) The expansions in ascending powers of  x  are only valid if  x  is ‘small’.  If  x  is large i.e. | x | > 1  then  we

1

may  find   it  convinient   to   expand  in  powers of  ,  which  then  will  be  small. x

n = 1 + nx + n(n −1) x² + n(n −1)(n −2) x3 .....

6. APPROXIMATIONS :(1 + x)

1.2 1.2.3

If  x < 1,  the  terms of  the above expansion go on decreasing and if  x  be very small, a stage may  be reached when we may neglect the terms containing higher powers of  x  in the  expansion. Thus,  if  x  be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx,  approximately. This  is  an  approximate  value  of  (1 + x)n.

7. EXPONENTIAL  SERIES :www.bloggermayank.online

x

(i) ex = 1 + 1! + x22! + x3!3 +.......∞ ;  where  x  may be any real or complex &  e = Limitn→∞ 1 +  n1n

2 3

(ii) ax = 1 +  x ln a +  x ln a2 +  x ln a3 +.......∞  where  a > 0

1! 2! 3!

Note : (a) e = 1 +

(b) e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is

2.7182818284.

(c) e + e−1 = 2 1+ 1 + 1 + 1 +.......∞ (d) e − e−1 = 2 1+ 1 + 1 + 1 +.......∞

2! 4! 6! 3! 5! 7!

(e) Logarithms to the base ‘e’ are known as the Napierian system, so named after Napier,  their inventor.  They are also called  Natural Logarithm.

8. LOGARITHMIC  SERIES :

(i) ln (1+ x) = x    where −1 < x ≤ 1

(ii) ln (1− x) = − x − x2 − x3 − x4 +.......∞  where  −1 ≤ x < 1

2 3 4

(1+ x)

(iii) ln  (1− x) = 2 x + x33 + x55 +......∞ x < 1

REMEMBER  : (a)   +... ∞ = ln 2 (b) eln x = x

(c) ln2 = 0.693 (d) ln10 = 2.303 21. VECTOR & 3-D

1. DEFINITIONS:

A VECTOR may be described as a quantity having both magnitude & direction. A vector is generally

represented by a directed line segment, say AB. A is called the initial point & B is called the terminal

point. The magnitude of vector AB is expressed by AB.

ZERO VECTOR  a vector of zero magnitude i.e.which has the same initial & terminal point, is called a ZERO VECTOR. It is denoted by O.

UNIT VECTOR a vector of unit magnitude in direction of a vector a is called unit vector along a and is

denoted by aΛ symbolically  aΛ=a . EQUAL VECTORS  two vectors are said to be equal if they have the same a

magnitude, direction & represent the same physical quantity.  COLLINEAR VECTORS  two vectors are said to be collinear if their  directed line segments are parallel disregards to their direction. Collinear vectors are also called PARALLEL VECTORS. If they have the same direction they are named as like vectors otherwise

unlike vectors. Simbolically,  two  non  zero  vectors a and b are collinear if and only if,  a Kb= ,  where

K ∈ R  COPLANAR VECTORS a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that  “TWO VECTORS ARE ALWAYS COPLANAR”.  POSITION VECTOR  let O

be a fixed origin, then the position vector of a point P is the vector OP. If  a &b & position vectors of

two point A and B, then  , AB= b − a =  pv of B −  pv of A

2. VECTOR ADDITION  :

If two vectors a &b are represented by OA&OB , then their sum a + b is a vector represented by

OC, where OC is the diagonal  of the parallelogram OACB.

a + b=b + a  (commutative)   (a + b) + c = a + (b + c)   (associativity)

a + 0= =a 0 + a   a + −( a)=0= −( a)+a

3. MULTIPLICATION  OF  VECTOR  BY  SCALARS  :

If a is a vector & m is a scalar, then ma is a vector parallel to a whose modulus is m times that  of a. This multiplication is called SCALAR

MULTIPLICATION. If a &b are vectors & m, n are scalars, then: m(a)=(a)m=ma m(na)=n(ma)=(mn)a

(m + n)a=ma+na m(a + b)=ma+mb

4. SECTION  FORMULA  :

If a & b are the position vectors of two points A & B then the p.v. of a point

which divides AB in the ratio m : n is given by :   r= na + mb .  Note p.v

m + n

a + b

of mid point of AB =

2

5. DIRECTION  COSINES   :www.bloggermayank.online

Let  a = a1Λi +a2Λj+a3kΛ the angles which this  vector makes  with the +ve directions OX,OY & OZ are

a1    ,    cosΞ²=a2 called DIRECTION ANGLES & their cosines are called the DIRECTION COSINES .   cosΞ±=

a a

,    cosΞ=a3  . Note that, cos² Ξ± + cos² Ξ² + cos² Ξ = 1 a

6. VECTOR EQUATION OF A LINE  :

Parametric vector  equation of a line passing through two point A(a) & B(b) is given by, r = a + t(b−a) where t is a parameter. If the line passes  through  the  point  A(a)  & is parallel to the vector b then its equation is, r = a + tb

Note that the equations of the bisectors of the angles between the lines r = a + Ξ» b & r = a + Β΅ c is :  r

= a + t (b c+ )  &  r = a + p (c−b).

7. TEST OF COLLINEARITY  :

Three points A,B,C with position vectors a,b,c respectively are collinear,  if & only if there exist scalars x , y , z  not all zero simultaneously such that  ;  xa + yb+ zc = 0, where x + y + z = 0.

8. SCALAR  PRODUCT  OF  TWO  VECTORS  :

a.b = a b cos (0ΞΈ ≤ΞΈ≤Ο),  note that if ΞΈ is acute then a.b > 0  &  if ΞΈ is obtuse  then a.b < 0

a.a a= =2 a ,a.b b.a2 = (commutative)

a . (b + c) = a.b + a.c (distributive) a.b = 0 ⇔ ⊥a b    (a ≠ 0 b ≠ 0)

Λi.Λi = Λj.Λj= k.Λ kΛ =1;  Λi.Λj = Λj.kΛ = k.Λ Λi = 0 projection of   a onb =  a.b .

Note: That vector component of a along b =   a ⋅ b b

 b2 

and perpendicular to b =  a –   a ⋅ bb.www.bloggermayank.online

 b2 

the angle Ο between a & b is given by  cosΟ = a b.       0 ≤ Ο ≤ Ο

a b

if  a = a1Λi +a2Λj+a3kΛ   &    b = b1Λi + b2Λj+ b3kΛ then a.b = a1b1 + a2b2 + a3b3

a= a12 +a22 +a32 , b = b12 + b22 + b32

Note : (i) Maximum  value  of  a . b =  a  b 

(ii) Minimum  values  of    a . b =  a . b =  − a  b 

(iii) Any  vector  a  can  be  written  as  ,  a = (a . i i) + (a . j j) + (a . k k) .

(iv) A vector in the direction of the bisector of the angle between the two vectors  a&b is  + . Hence bisector  of  the  angle  between the two vectors a&b is Ξ» (a + b), where Ξ» ∈ R+. Bisector of the exterior angle between a&b  is  Ξ» (a − b) , Ξ» ∈ R+  .

9. VECTOR  PRODUCT  OF  TWO  VECTORS  :

(i) If a&b are two vectors   &   ΞΈ   is   the  angle  between  them  then  a×b = a bsinΞΈn , where n

is the unit vector perpendicular to both a&b such  that  a,b&n forms a  right  handed  screw  system

.

(ii) Lagranges Identity : for any two vectors a & b axb;( )2 = a 2 b 2 −( . )a b 2 =a a a b. . a b b b. .

(iii) Formulation of vector product in terms of scalar product:

The vector product  axb is the vector c , such that

(i) |c| = a b2 2 − (a b⋅ )2 (ii) c ⋅ a = 0;  c ⋅ b =0 and(iii) a, b, c form a right handed system

(iv) a×b=0⇔a&b are parallel (collinear) (a ≠ 0,b ≠ 0) i.e. a = Kb , where K is a scalar  a ×b ≠ b×a (not commutative)

(ma)×b = a×(mb) = m(a×b)    where m is a scalar .    a×(b+c) =(a×b)+(a×c) (distributive)

Λi× = × =Λi Λj Λj kΛ ×kΛ = 0 Λi× =Λj k,Λ Λj×kΛ = Λi, kΛ × =Λi Λj

Λi Λj kΛ

(v) If a = a1Λi +a2Λj+a3kΛ   &    b = b1Λi + b2Λj+ b3kΛ   then  a×b =a1 a2 a3

b1 b2 b3

(vi) Geometrically  a×b = area  of  the  parallelogram  whose  two  adjacent  sides are represented by

a&b .

(vii) Unit vector perpendicular to the plane of  a & b isnΛ =± a×b

a×b

A vector of magnitude ‘r’ & perpendicular to the palne of  a&b is± r a( ×b)

a×b

If ΞΈ is the angle between a&bthensinΞΈ=

(viii) Vector area If a,b&c are the pv’s of 3 points  A, B & C then the vector area of triangle ABC

=   [axb + bxc + cxa]. The points A, B & C are collinear if axb + bxc + cxa = 0

1  Area of  any quadrilateral whose diagonal vectors are d &d1 2 is given by 2 d xd1 2

10. SHORTEST  DISTANCE  BETWEEN  TWO  LINES  :

If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called SKEW LINES. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance

vector would be equal to that of the projection of AB along the direction of the line of shortest distance,

→→ →→

LM is parallel to pxq   i.e.   LM =Projection of AB on LM = Projection of AB on pxq

(b−a) . (pxq)

p x q

AB . (pxq)

= =www.bloggermayank.online p x q

1. The  two  lines  directed  along p & q will  intersect  only  if  shortest distance = 0  i.e.

(b − a).(pxq) = 0  i.e. (b − a) lies in the plane containing p & q . ⇒ [(b−a p q) ] = 0

bx a( 2 − a1)

b

2. If two lines are given by r1 = a1 + Kb & r2 = a2 + Kb  i.e.  they are parallel then , d =

11. SCALAR  TRIPLE  PRODUCT / BOX PRODUCT / MIXED PRODUCT :

The  scalar  triple  product  of  three  vectors a,b&c  is  defined  as :

axb.c=a b c sinΞΈ cosΟ where ΞΈ is the angle between a & b  & Ο  is  the angle between a b&c× . It

is also defined as [abc] , spelled as box product .

Scalar triple product geometrically represents the volume of the parallelopiped  whose three  couterminous edges are represented by a,b&ci.e.V [abc]=

In a scalar triple product the position of dot & cross can be interchanged i.e.

a.(bxc)=(axb).c OR[abc]=[bca]=[cab]

a . (bx c) =− a .(cx b) i.e. [a b c] = − [a c b]

If a = +a1Λi a2Λj a+ 3kΛ  ; b = b1Λi + b2Λj+ b3kΛ & c = c1Λi +c2Λj+c3kΛ then  .

In general  ,  if  a = a l1 + a m2 + a n3  ;  b = b l1 + b m2 + b n3   &   c = c l1 + c m2 + c n3

a1 a2 a3 then  [a b c] =b1 b2 b3[l mn]   ;  where , m & n  are non coplanar vectors .

c1 c2 c3

If  a , b , c are coplanar ⇔ [a b c] = 0 .

Scalar product of three vectors, two of which are equal or parallel is 0 i.e. [a b c] = 0 ,

Note : If a , b , c are  non − coplanar  then [a b c] > 0 for  right  handed  system  & [a b c] < 0 for left handed system .

[i j k] = 1     [Ka b c] = K[a b c]        [(a + b) c d] = [a c d] + [ b c d]

The volume of the tetrahedron OABC with O as origin & the pv’s of A, B and C being  a , b & c respectively

is given by V =  [a b c]

The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices are a , b , c & d are

given by   [a + + +b c d].

Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron .

Remember that  :  [a − b b− c c−a] = 0     &     [a + b b+ c c+a] = 2 [a b c] .

*12. VECTOR  TRIPLE  PRODUCT  :

Let  a,b,c be any three vectors, then the expression a ×(b× c) is a vector & is called a vector triple product .www.bloggermayank.online

GEOMETRICAL INTERPRETATION OF  a ×(b× c)

Consider the expression a ×(b× c) which itself is a vector, since it is a cross  product  of  two  vectors a & (bx c) . Now a x ( b x c) is  a vector perpendicular to the plane containing a & (bx c) but  b x c is a vector perpendicular to the plane b&c , therefore a x ( b x c) is  a vector lies in the plane of  b&c and

perpendicular to a . Hence we can express a x ( b x c) in terms of b & c

i.e. a x ( b x c) = xb + yc where x & y are scalars .

a x ( b x c) = (a . c b) − (a . b c)    (a x b) x c = (a . c b) − (b . c a)

(a x b) x c ≠ a x (b x c)

13. LINEAR  COMBINATIONS / Linearly Independence and Dependence of Vectors : Given a finite set of vectors a ,b,c,...... then the vector r=xa+yb+zc ........+ is called a linear combination of a ,b,c,...... for any  x, y, z ...... ∈ R. We have the following results :

(a) FUNDAMENTALTHEOREM IN PLANE :  Let a,b be  non zero ,  non collinear vectors . Then any vector r coplanar with a,b can be expressed uniquely as a linear combination of  a,b i.e. There exist some unique x,y ∈ R such that xa + yb r= .

(b) FUNDAMENTAL THEOREM IN SPACE : Let a,b,c be non−zero, non−coplanar vectors in space. Then  any vector r , can be uniquily expressed as a linear combination of a,b,c i.e.  There exist some unique x,y ∈

R such that  xa+yb+ zc=r .

(c) If  x ,x ,......x1 2 n  are  n  non zero vectors,  & k1, k2, .....kn  are n  scalars & if the linear  combination

k x1 1 + k x2 2 +........k xn n = 0 ⇒ k1 = 0,k2 = 0.....kn = 0 then we say that  vectors  x ,x ,......x1 2 n are

LINEARLY INDEPENDENT VECTORS .

(d) If  x ,x ,......x1 2 n  are  not LINEARLY INDEPENDENT  then  they  are said  to be LINEARLY DEPENDENT vectors

.  i . e . i f  k x1 1+k x2 2+........ k x+ n n = 0  &  if  there  exists at least  one kr ≠ 0 then x ,x ,......x1 2 n are said to be LINEARLY DEPENDENT . Note :

If  a = 3i + 2j + 5k then a is expressed as a LINEAR COMBINATION of vectors Λi, Λj, kΛ .  Also , a , Λi, Λj, kΛ form a linearly dependent set of vectors. In general , every set of four vectors is  a linearly dependent system. Λi, Λj, kΛ are LINEARLY INDEPENDENT set of vectors. For

K1Λi + K2Λj+ K3kΛ = 0  ⇒  K1 = 0 = K2 = K3.

Two vectors a&b are linearly dependent ⇒ a is parallel to b  i.e. axb= 0 ⇒ linear dependence of a&b .  Conversely if axb 0≠ then a&b are linearly independent  .

If  three vectors a,b,c are linearly dependent, then they are coplanar i.e. [a, b c, ] = 0 , conversely, if

[a, b c, ] ≠ 0 , then the vectors are linearly independent.

14. COPLANARITY OF  VECTORS  :

Four points A, B, C, D with position vectors a,b,c,d respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that  xa+ yb+zc+wd=0 where,  x + y + z + w = 0.

15. RECIPROCAL  SYSTEM  OF  VECTORS  :

If  a,b,c & a',b',c' are two sets of non coplanar vectors such that a.a'=b.b'=c.c'=1 then the two systems are called Reciprocal System of vectors. Note : a'=bx c ; b'= cxa ; c'= a x b a b c] [a b c] [a b c]

16. EQUATION OF A PLANE  :www.bloggermayank.online

(a) The  equation  (r − r ).n0 =0 represents  a  plane  containing  the  point  with  p.v.  r wheren0 is a vector normal to the plane . r.n =d is the general equation of a plane.

(b) Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.

17. APPLICATION OF VECTORS  :

(a) Work done against a constant force F over adisplacement s is defined as W F.s=

(b) The  tangential  velocity V of  a  body moving  in a circle is given by V = w× r where r is the pv of the point P.

(c) The moment of F about ’O’ is defined as M = r ×Fwherer  is the

pv  of  P  wrt  ’O’.  The direction  of M is  along  the  normal  to  the plane OPN   such   that  r,F&M  form  a

right  handed system.

(d) Moment of the couple =(r − r ) F× wherer & r are  pv’s of the point of the application of the forces

1 2 1 2

F&− F.

3 -D COORDINATE GEOMETRY

USEFUL RESULTS A General :

(1) Distance (d) between two points  (x1 , y1 , z1) and (x2 , y2 , z2)

d =

(2) Section Fomula

m2 1x + m1 2x m2 1y + m1 2y m2 1z + m1 z2 x = m1 +m2   ;    y = m1 +m2  ;  z =  m1 +m2

( For external division take –ve sign )

Direction Cosine and direction ratio's of a line

(3) Direction cosine of a line has the same meaning as d.c's of a vector.

(a) Any three numbers a, b, c proportional to the direction cosines are called the direction ratios i.e. l m n 1

= = =±

a b c a2 +b2 +c2 same sign either +ve or –ve should be taken through out.

note that d.r's of a line joining  x1 , y1 , z1  and  x2 , y2 , z2 are proportional to x2 – x1  , y2 – y1 and z2 – z1

(b) If ΞΈ is the angle between the two lines whose d.c's are l , m , n  and l , m , n cosΞΈ = l1 l2 + m1 m2+n1 n2  hence if lines are perpendicular then  1 1 1l1 l2 + m2 1m22+ n2 n  = 0

if lines are parallel then  l1 = 1 = n1 l2n2

l1 m1 n1

note that if three lines are coplanar then   l2 m2 n2 = 0

l3 m3 n3

(4)Projection of  join of 2 points on line with d.c's  l, m, n are  l (x2 – x1) + m(y2 – y1) + n(z2 – z1)

B PLANEwww.bloggermayank.online

(i) General equation of degree one in  x, y, z   i.e. ax + by + cz + d = 0  represents a plane.

(ii) Equation of a plane passing through (x1 , y1 , z1) is a (x – x1) + b (y – y1) + c (z – z1) = 0   where a, b, c are the direction ratios of the normal to the plane. x y z

+ + =

(iii) Equation of a plane if its intercepts on the co-ordinate axes are  x1 , y1 , z1  is x1 y1 z1 1.

(iv) Equation of a plane if the length of the perpendicular from the origin on the plane is p and d.c's of the perpendicular as  l , m, , n is      l x + m y + n z = p

(v) Parallel and perpendicular planes – Two planes   a1 x + b1 y + c1z + d1 = 0  and  a2x + b2y + c2z + d2 = 0 are  perpendicular if a1 a2 + b1 b2 + c1 c2 = 0

a1 = b1 = c1 a1 = b1 = c1 = d1

parallel if       and    coincident if

a2 b2 c2 a2 b2 c2 d2

(vi) Angle between a plane and a line is the compliment of the angle between the normal to the plane and the line .  If    LinePlane : r.n: r = +Ξ»a=d b then cos(90−ΞΈ =) sinΞΈ=  | b |.| n |b.n .

where ΞΈ is the angle between the line and normal to the plane.

(vii) Length of the perpendicular from a point (x1 , y1 , z1) to a plane ax + by + cz + d = 0 is

ax1 +by1 +cz1 +d

p = a2 + b2 +c2

(viii) Distance between two parallel planes ax + by + cz + d1 = 0 and  ax+by + cz + d2 = 0  is

(ix) Planes bisecting the angle between two planes a1x + b1y + c1z + d1 = 0  and  a2 + b2y + c2z + d2 = 0  is

given by a x1 +a2b y+1 b+12c z+1c1+2 d1 a x2 +ab y222+ b+22c z+2c22+d2  = ±

1

Of these two bisecting planes , one bisects the acute and the other obtuse angle between the given planes. 4.

(x) Equation of a plane through the intersection of two planes P1and P2is given by P1+Ξ»P2=0

C STRAIGHT LINE IN SPACE

(i) Equation of a line through A (x1 , y1 , z1) and having direction cosines  l ,m , n  are

x −x1 = y− y1 = z−z1 and the lines through  (x1 , y1 ,z1) and (x2 , y2 ,z2) l m n

xx2−−xx11 = yy2−−yy11 = zz2−−zz11 5.

(ii) Intersection  of two planes    a1x + b1y + c1z + d1 = 0   and   a2x + b2y + c2z + d2 = 0 together represent the unsymmetrical form of the straight line.

x − x

(iii) General equation of the plane containing the line    1 = y− y1 = z −z1  is

l m n

A (x – x1) + B(y – y1) + c (z – z1) = 0    where  Al + bm + cn = 0 . 6.

LINE OF GREATEST SLOPEwww.bloggermayank.online

AB is the line of intersection of G-plane and H is the horizontal plane. Line

of greatest slope on a given plane, drawn through a given point on the plane, 7.

is the line through the point 'P' perpendicular to the line of intersetion of the given plane with  any horizontal plane.

22. TRIGONOMETRY-1 (COMPOUND ANGLE)

1 . BASIC  TRIGONOMETRIC  IDENTITIES  :

(a) sin2ΞΈ + cos2ΞΈ  = 1 ;     −1 ≤ sin ΞΈ  ≤ 1  ; −1 ≤ cos ΞΈ  ≤ 1    ∀   ΞΈ ∈ R

(b) sec2ΞΈ − tan2ΞΈ   = 1 ; sec ΞΈ ≥ 1     ∀    ΞΈ ∈ R (c) cosec2ΞΈ − cot2ΞΈ = 1 ; cosec ΞΈ ≥ 1   ∀    ΞΈ ∈ R

2. IMPORTANT  T′  RATIOS:

(a) sin n Ο = 0 ; cos n Ο = (-1)n   ; tan n Ο = 0 where  n ∈ I

(b) sin = (−1)n    & cos= 0    where  n ∈ I

8.

3 1−

(c) sin 15°  or   sin    =    =  cos 75°  or   cos ;

2 2

Ο 3 1+ 5Ο

cos 15°  or   cos    =    =  sin 75°   or     sin ;

12 2 2 12

3 1−3 1+

tan 15° = = 2− 3 = cot 75°  ; tan 75° = = 2+ 3 = cot 15°

3+13−1 Ο 2− 2 Ο 2+ 2 Ο3Ο

(d) sin =  ;     cos =   ;    tan = 2 1− ;     tan   = 2 1+ 9.

8 2 8 2 88

(e) sin    or   sin 18° =        &     cos 36°   or   cos

3. TRIGONOMETRIC  FUNCTIONS  OF  ALLIED  ANGLES  :

If  ΞΈ  is  any  angle, then  − ΞΈ,  90 ± ΞΈ,  180 ± ΞΈ,  270 ± ΞΈ,  360 ± ΞΈ  etc. are called  ALLIED ANGLES.

(a) sin (− ΞΈ) =  − sin ΞΈ ; cos (− ΞΈ) =  cos ΞΈ

(b) sin (90°- ΞΈ) = cos ΞΈ ; cos (90° − ΞΈ)  = sin ΞΈ

(c) sin (90°+ ΞΈ) = cos ΞΈ ; cos (90°+ ΞΈ)  = − sin ΞΈ

(d) sin (180°− ΞΈ) = sin ΞΈ ; cos (180°− ΞΈ) = − cos ΞΈ

(e) sin (180°+ ΞΈ) = − sin ΞΈ; cos (180°+ ΞΈ) = − cos ΞΈ

(f) sin (270°− ΞΈ) = − cos ΞΈ ; cos (270°− ΞΈ) = − sin ΞΈ

(g) sin (270°+ ΞΈ) = − cos ΞΈ ; cos (270°+ ΞΈ) = sin ΞΈ

TRIGONOMETRIC  FUNCTIONS  OF  SUM  OR  DIFFERENCE  OF  TWO  ANGLES  :

(a) sin (A ± B) = sinA cosB ± cosA sinB

(b) cos (A ± B) = cosA cosB ∓ sinA sinB

(c) sin²A − sin²B = cos²B − cos²A = sin (A+B) . sin (A− B)

(d) cos²A − sin²B = cos²B − sin²A = cos (A+B) . cos (A − B)

(e) tan (A ± B) =  tanA ± tanB         (f)      cot (A ± B) =  cotA cotB ∓ 1

1 ∓ tanA tanB cotB ± cotA

FACTORISATION  OF  THE  SUM  OR  DIFFERENCE  OF  TWO  SINES  OR  COSINES  :

C+D C−D C+D C−D

(a)  sinC + sinD = 2 sin cos      (b)  sinC − sinD = 2 cos

2 2

C+D C−D 2 sin C+D sin C−D

(c)  cosC + cosD = 2 cos cos     (d)  cosC − cosD = −

2 2 2 2

TRANSFORMATION  OF  PRODUCTS  INTO  SUM  OR  DIFFERENCE  OF  SINES  &  COSINES

(a)  2 sinA cosB = sin(A+B) + sin(A−B)      (b)  2 cosA sinB = sin(A+B) − sin(A−B)

(c)  2 cosA cosB = cos(A+B) + cos(A−B)   (d)  2 sinA sinB = cos(A−B) − cos(A+B)

MULTIPLE  ANGLES  AND  HALF  ANGLES  :www.bloggermayank.online

(a) sin 2A = 2 sinA cosA   ;   sin ΞΈ = 2 sin

(b) cos2A = cos2A − sin2A = 2cos2A − 1 = 1 − 2 sin2A ; cos ΞΈ = cos2   − sin²  = 2cos2   − 1 = 1 − 2sin2   .

2 cos2A = 1 + cos 2A ,  2sin2A = 1 − cos 2A  ; tan2A = 1 −cos2A

1+cos2A 2 cos2   = 1 + cos ΞΈ  ,  2 sin2  ΞΈ.

(c) tan 2A =    ;   tan

1−2tantan2AA ΞΈ = 1−2tantan(2ΞΈ(ΞΈ22))

2tanA 1−tan A2 − 4 sin3A

(d) sin 2A =    ,    cos 2A = (e)     sin 3A = 3 sinA

1+tan2 A 1+tan2A −tan A3

(f) cos 3A = 4 cos3A − 3 cosA (g) tan 3A =  3tanA

1−3tan2A THREE  ANGLES :

(A+B+C) = tanA+tanB+tanC−tanAtanBtanC

(a) tan

1−tanAtanB−tanBtanC−tanCtanA

NOTE  IF : (i)   A+B+C = Ο   then   tanA + tanB + tanC = tanA tanB tanC

(ii)  A+B+C =    then   tanA tanB + tanB tanC + tanC tanA = 1

(b) If   A + B + C = Ο    then  :   (i)   sin2A + sin2B + sin2C =  4 sinA sinB sinC

A B C

(ii) sinA + sinB + sinC = 4 cos

MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS:

(a) Min. value of a2tan2ΞΈ + b2cot2ΞΈ = 2ab where ΞΈ ∈ R

(b) Max. and Min. value of acosΞΈ + bsinΞΈ are  a2 +b2 and – a2 + b2

(c) If f(ΞΈ) = acos(Ξ± + ΞΈ) + bcos(Ξ² + ΞΈ) where a, b, Ξ± and Ξ² are known quantities then – a2 + +b2 2abcos(Ξ±−Ξ²) < f(ΞΈ) < a2 + +b2 2abcos(Ξ±−Ξ²)

 Ο

(d) If Ξ±,Ξ² ∈ 0, 2 and Ξ± + Ξ² = Ο (constant) then the maximum values of the expression  cosΞ± cosΞ², cosΞ± + cosΞ², sinΞ± + sinΞ² and sinΞ± sinΞ²  occurs when Ξ± = Ξ² = Ο/2.

(e) If Ξ±,Ξ² ∈ 0, Ο and Ξ± + Ξ² = Ο(constant) then the minimum values of the expression  secΞ± + secΞ²,

 2

tanΞ± + tanΞ², cosecΞ± + cosecΞ² occurs when Ξ± = Ξ² = Ο/2.

(f) If A, B, C are the angles of a triangle then maximum value of sinA + sinB + sinC  and  sinA sinB sinC occurs when A = B = C = 600

(g) In case a quadratic in sinΞΈ  or cosΞΈ is given then the maximum or minimum values can be interpreted by making a perfect square.

10. Sum of sines or cosines of  n angles,

nΞ²

sin Ξ± + sin (Ξ± + Ξ²) + sin (Ξ± + 2Ξ² ) + ...... + sin (Ξ±+ − Ξ²n 1 ) = sinΞ±+n 1− Ξ²

sin2 2 

nΞ²

cos Ξ± + cos (Ξ± + Ξ²) + cos (Ξ± + 2Ξ² ) + ...... + cos (Ξ±+ − Ξ²n 1 ) = sinsinΞ²22 cosΞ±+n 12− Ξ²

23. TRIGONO-2 (TRIGONOMETRIC EQUATIONS

& INEQUATIONS)

THINGS  TO  REMEMBER  :

1. If  sin ΞΈ = sin Ξ±  ⇒  ΞΈ = n Ο + (−1)n Ξ±  where  Ξ±  ∈ −Ο2 , Ο2 , n ∈ I  .

2. If  cos ΞΈ = cos Ξ±  ⇒  ΞΈ = 2 n Ο  ±  Ξ±  where  Ξ±  ∈ [0 ,  Ο] ,  n ∈ I  .

3. If  tan ΞΈ = tan Ξ±  ⇒  ΞΈ = n Ο  +  Ξ±  where Ξ± ∈ −Ο2 , Ο2 , n ∈ I  .

4. If  sin² ΞΈ  =  sin² Ξ± ⇒  ΞΈ  =  n Ο  ± Ξ±.

5. cos² ΞΈ  =  cos² Ξ±  ⇒  ΞΈ  = n Ο  ±  Ξ±.

6. tan² ΞΈ  =  tan² Ξ±  ⇒  ΞΈ  =  n Ο  ±  Ξ±.      [ Note :  Ξ±  is  called  the  principal  angle ]

7. TYPES  OF  TRIGONOMETRIC  EQUATIONS :

(a) Solutions  of  equations  by  factorising .  Consider  the  equation  ;

(2 sin x − cos x) (1 + cos x) = sin² x  ;  cotx – cosx = 1 – cotx cosx

(b) Solutions  of  equations  reducible  to  quadratic  equations. Consider the equation

(3 cos² x − 10 cos x + 3 = 0    and    2 sin2x + 3 sinx + 1 = 0

(c) Solving  equations  by  introducing  an  Auxilliary  argument .  Consider  the  equation : sin x + cos x = 2  ;  3 cos x + sin x = 2  ; secx – 1 = ( 2 – 1) tanx

(d) Solving  equations  by  Transforming  a  sum  of  Trigonometric  functions  into  a product.

Consider  the  example  :     cos 3 x + sin 2 x − sin 4 x = 0 ; sin2x + sin22x  + sin23x + sin24x  = 2 ;   sinx + sin5x = sin2x + sin4x

(e) Solving  equations  by transforming a  product of  trigonometric  functions  into  a  sum. Consider sin6x

the  equation  :sin 5 x . cos 3 x =  sin 6 x .cos 2 x  ;  8 cosx cos2x cos4x =   ;  sin3ΞΈ = 4sinΞΈ  sin2ΞΈ sin x

sin4ΞΈwww.bloggermayank.online (f) Solving  equations  by  a  change  of  variable  :

(i) Equations  of  the form  of   a . sin x + b . cos x + d = 0 ,  where  a , b & d  are  real  numbers &   a , b ≠ 0  can be solved by changing  sin  x & cos  x  into their corresponding tangent of half the angle.  Consider the equation   3 cos x + 4 sin x = 5.

(ii) Many  equations  can  be  solved  by  introducing  a  new variable . eg. the equation sin4 2

 − 1 = 0    by  substituting ,  sin 2 x . cos 2 x = y

x + cos4 2 x = sin 2 x . cos 2 x  changes to   2 (y + 1) y 2

(g) Solving  equations  with  the  use  of  the  Boundness  of  the  functions   sinx & making two perfect squares. Consider the equations : sin x cos  x4 − 2sinx + 1+sin x4−2cosx . cos x = 0  ;

sin2x + 2tan2x +   tanx –  sinx +   = 0

8. TRIGONOMETRIC INEQUALITIES : There is no general rule to solve a Trigonometric inequations and the same rules of algebra are valid except the domain and range of trigonometric functions should be kept in mind.

Consider the examples : log2sin  x  < – 1 ; sin x cosx +  1  < 0 ; 5−2sin2x ≥6sin x−1

2  2

24. TRIGONO-3(SOLUTIONS OF TRIANGLE)

I. SINE  FORMULA  : In  any  triangle  ABC , a = b = c   .

sinA sinB sinC

b2+c2−a2

II. COSINE  FORMULA  : (i)  cos A =    or   a² = b² + c² − 2bc. cos A 2bc

c2+a2−b2 a2+ −b2 c2

(ii)  cos B = (iii)  cos C =

2ca 2ab

III. PROJECTION   FORMULA  : (i)  a = b cos C + c cos B

(ii) b = c cos A + a cos C (iii)  c = a cos B + b cos A

IV. NAPIER’S  ANALOGY − TANGENT  RULE  : (i)  tan B−C = bb+−cc cot  A2

2

C−A c−a B A−B a−b C (ii)  tan =  cot   (iii)  tan =  cot

2 c+a 2 2 a+b 2

V. TRIGONOMETRIC  FUNCTIONS  OF  HALF  ANGLES  :

A (s b)(s c− − ) B (s c)(s a− − )

(i) sin =   ;  sin

2 bc 2 ca

A s(s a)− B s(s b)−

(ii) cos =   ;  cos =   ;   cos

2 bc 2 ca

A (s b)(s c− − ) a+b+c

(iii) tan 2 = s(s a)−   = s(s−a)   where  s = 2 &  ∆ =  area  of  triangle.

(iv) Area  of  triangle = s(s a)(s b)(s c)− − − .

VI. M − N  RULE  :   In  any  triangle ,

(m + n) cot ΞΈ = mcotΞ± −ncotΞ²

= ncotB−mcotC

VII.   ab sin C =   bc sin A =   ca sin B = area  of  triangle  ABC  .

a b c

= = = 2R sinA sinB sinC

a b c

Note  that   R = ∆   ;  Where  R is  the  radius  of  circumcircle  &  ∆ is  area  of

4 trianglewww.bloggermayank.online

VIII. Radius  of  the  incircle  ‘r’  is  given  by  :

a + b + c − a) tan A = (s − b) tan B = (s − c)

(a)  r =    where  s = (b)  r = (s

s 2 2 2

C tan

2

a sin B2 sin C2 A B C

(c)  r =   &  so on (d)  r = 4R sin sin sin cos A2 2 2 2

IX. Radius  of  the  Ex− circles   r1 , r2 & r3  are  given  by  :

∆ ;  r2 = s −∆ b ;  r3 = s ∆− c (b)     r1 = s tan A2 ;   r2 = s tan B2 ;   r3 = s tan C2

(a) r1 = s − a

a cos B2 cos C2 A

(c)r1 = cos A2   &  so  on (d) r1 = 4 R sin 2 . cos

B A C C

r2 = 4 R sin  2 . cos 2 . cos 2     ; r3 = 4 R sin  2 . cos

X.LENGTH  OF  ANGLE BISECTOR  &  MEDIANS :If ma  and Ξ²  are the lengths of a median and an angle bisector from the angle A then, ma =

2bc cos A and  Ξ²a = b + c 2

Note that  m2a + m2b + m2c =   (a2 + b2 + c2)

XI. ORTHOCENTRE  AND  PEDAL  TRIANGLE :The triangle KLM which is formed by joining the feet of the altitudes is called the pedal triangle.−the distances of the orthocentre from the angular points of the∆ ABC are  2 R cosA ,  2 R cosB and  2 R cos− the distances of P from sides are 2 R cosB cosC, 2 R cosC cosA and  2 R cosA cosB− the sides of the pedal triangle are  a cosA (= R sin 2A), b cosB (= R sin 2B) and  c cosC (= R sin 2C) and its angles areΟ − 2A,  Ο − 2B and  Ο − 2C.

circumradii of the triangles PBC, PCA, PAB and ABC are equal .

XII EXCENTRAL  TRIANGLE :The triangle formed by joining the three excentres I1, I2 and I3 of ∆ ABC is called the excentral or excentric triangle.

Note that :− Incentre I of ∆ ABC is the orthocentre of the excentral  ∆ I1I2I3 .

∆ ABC is the pedal triangle of the ∆ I1I2I3 .− the sides of the excentral triangle are

R cos A ,  4 R cos B and 4 R cos C and its angles are  Ο− A ,  Ο B Ο C

4 −  and  − .

2 2 2 2 2 2 2 2 2

I I1 = 4 R sin  A2 ;   I I2 = 4 R sin  B2 ;   I I3 = 4 R sin  C2 .

XIII. THE DISTANCES  BETWEEN  THE  SPECIAL  POINTS :www.bloggermayank.online

(a) The distance between circumcentre and orthocentre is = R . 1 − 8 cosA cosB cosC

(b) The distance between circumcentre and incentre is = R2 − 2Rr

(c) The distance between incentre and orthocentre is  2r2 − 4R2 cosA cosB cosC

XIV. Perimeter (P) and area (A) of a regular polygon of  n  sides inscribed in a circle of radius  r are given by P Ο 1 2Ο

= 2nr sin and A =   nr2 sin Perimeter and area of a regular polygon of n sides circumscribed about n 2 n

Ο Ο

a given circle of radius r is given by P = 2nr tan and     A = nr2 tan n n

XV. In many kinds of trignometric calculation, as in the solution of triangles, we often require the logarithms of trignometrical ratios . To avoid the trouble and inconvenience of printing the proper sign to the logarithms of the trignometric functions, the logarithms as tabulated are not the true logarithms, but the true logarithms increased by 10 . The symbol L is used to denote these "tabular logarithms" . Thus :

L sin 15ΒΊ 25′ = 10 + log10 sin 15ΒΊ 25′ and L tan 48ΒΊ 23′ = 10 + log10 tan 48ΒΊ 23′

IIT JEE ADVANCED  Physics Syllabus

General: Units and dimensions, dimensional analysis; least count, significant figures; Methods of measurement and error analysis for physical quantities pertaining to the following experiments: Experiments based on using Vernier calipers and screw gauge (micrometer), Determination of g using simple pendulum, Young?s modulus by Searle?s method, Specific heat of a liquid using calorimeter, focal length of a concave mirror and a convex lens using u-v method, Speed of sound using resonance column, Verification of Ohm?s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box.

Mechanics: Kinematics in one and two dimensions (Cartesian coordinates only), projectiles; Uniform Circular motion; Relative velocity.

Newton's laws of motion; Inertial and uniformly accelerated frames of reference; Static and dynamic friction; Kinetic and potential energy; Work and power; Conservation of linear momentum and mechanical energy.

Systems of particles; Centre of mass and its motion; Impulse; Elastic and inelastic collisions.? Law of gravitation; Gravitational potential and field; Acceleration due to gravity; Motion of planets and satellites in circular orbits; Escape velocity.

Rigid body, moment of inertia, parallel and perpendicular axes theorems, moment of inertia of uniform bodies with simple geometrical shapes; Angular momentum; Torque; Conservation of angular momentum; Dynamics of rigid bodies with fixed axis of rotation; Rolling without slipping of rings, cylinders and spheres; Equilibrium of rigid bodies; Collision of point masses with rigid bodies.

Linear and angular simple harmonic motions.

Hooke?s law, Young?s modulus.

Pressure in a fluid; Pascal?s law; Buoyancy; Surface energy and surface tension, capillary rise; Viscosity

(Poiseuille?s equation excluded), Stoke?s law; Terminal velocity, Streamline flow, equation of continuity, Bernoulli?s theorem and its applications.

Wave motion (plane waves only), longitudinal and transverse waves, superposition of waves; Progressive and stationary waves; Vibration of strings and air columns;Resonance; Beats; Speed of sound in gases; Doppler

effect (in sound).www.bloggermayank.online

Thermal physics: Thermal expansion of solids, liquids and gases; Calorimetry, latent heat; Heat conduction in one dimension; Elementary concepts of convection and radiation; Newton?s law of cooling; Ideal gas laws; Specific heats (Cv and Cp for monoatomic and diatomic gases); Isothermal and adiabatic processes, bulk modulus of gases; Equivalence of heat and work; First law of thermodynamics and its applications (only for ideal gases);? Blackbody radiation: absorptive and emissive powers; Kirchhoff?s law; Wien?s displacement law, Stefan?s law.

Electricity and magnetism: Coulomb?s law; Electric field and potential; Electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field; Electric field lines; Flux of

electric field; Gauss?s law and its application in simple cases, such as, to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Capacitance; Parallel plate capacitor with and without dielectrics; Capacitors in series and parallel; Energy stored in a capacitor.

Electric current; Ohm?s law; Series and parallel arrangements of resistances and cells; Kirchhoff?s laws and simple applications; Heating effect of current.

Biot'Savart's law and Ampere?s law; Magnetic field near a current-carrying straight wire, along the axis of a circular coil and inside a long straight solenoid; Force on a moving charge and on a current-carrying wire in a uniform magnetic field.

Magnetic moment of a current loop; Effect of a uniform magnetic field on a current loop; Moving coil galvanometer, voltmeter, ammeter and their conversions.

Electromagnetic induction: Faraday?s law, Lenz?s law; Self and mutual inductance; RC, LR and LC circuits with D.C. and A.C. sources.

Optics: Rectilinear propagation of light; Reflection and refraction at plane and spherical surfaces; Total internal reflection; Deviation and dispersion of light by a prism; Thin lenses; Combinations of mirrors and thin lenses; Magnification.?

Wave nature of light: Huygen?s principle, interference limited to Young?s double-slit experiment. Modern physics: Atomic nucleus; Alpha, beta and gamma radiations; Law of radioactive decay;? Decay constant; Half-life and mean life; Binding energy and its calculation; Fission and fusion processes; Energy calculation in these processes.

Photoelectric effect; Bohr?s theory of hydrogen-like atoms; Characteristic and continuous X-rays, Moseley?s law; de Broglie wavelength of matter waves.