55099833 math formula sheet aieee

7/28/2019 55099833 Math Formula Sheet AIEEE

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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK

E D U C A T I O N S

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510

QUADRATIC EQUATION & EXPRESSION

1. Quadratic expression :

A polynomial of degree two of the form ax2 + bx + c, a 0 iscalled a quadratic expression in x.

2. Quadratic equation :

An equation ax2 + bx + c = 0, a 0, a, b, c R has two and

only two roots, given by

= + b b ac

a

2 4

2and =

b b ac

a

2 4

2

3. Nature of roots :

Nature of the roots of the given equation depends upon the

nature of its discriminant D i.e. b2 4ac.

Suppose a, b, c R, a 0 then

(i) If D > 0 roots are real and distinct (unequal)

(ii) If D = 0 roots are real and equal (Coincident)

(iii) If D < 0 roots are imaginary and unequal i.e.

non real complex numbers.

Suppose a, b, c Q a 0 then

(i) If D > 0 and D is a perfect square roots are rational

& unequal

(ii) If D > 0 and D is not a perfect square roots are

irrational and unequal.

or a quadratic equation their will exist exactly 2 roots real

or imaginary. If the equation ax2 + bx + c = 0 is satisfied for

more than 2 distinct values of x, then it will be an identity &

will be satisfied by all x. Also in this case a = b = c = 0.

PAGE # 2PAGE # 1

4. Conjugate roots :

Irrational roots and complex roots occur in conjugate pairs

i.e.if one root + i, then other root i

if one root + , then other root

5. Sum of roots :

S = + = ba

= Coefficient of xCoefficient of x2

Product of roots :

P = =c

a=

cons t term

Coefficient of x

tan2

6. ormation of an equation with given roots :

x2 Sx + P = 0

x2 (Sum of roots) x + Product of roots = 0

7. Roots under particular cases :

or the equation ax2 + bx + c = 0, a 0

(i) If b = 0 roots are of equal magnitude but of opposite

sign.

(ii) If c = 0 one root is zero and other is b/a

(iii) If b = c = 0 both roots are zero

(iv) If a = c roots are reciprocal to each other.


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E D U C A T I O N S


, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 4PAGE # 3

(v) If a > 0, c < 0 or a < 0, c > 0 roots are of opposite

signs

(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 both

roots are ve

(vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0 both

roots are +ve.

8. Symmetric function of the roots :

If roots of quadratic equation ax2 + bx + c, a 0 are and ,then

(i) ( ) = ( ) + 2 4 = b ac

a

24

(ii) 2 + 2 = ( + )2 2 =b ac

a

2

2

2

(iii) 2 2 = ( + ) ( ) + 2 4 = b b ac

a

2

2

4

(iv) 3 + 3 = ( + )3 3( + ) = b b ac

a

( )2

3

3

(v) 3 3 = ( ) [2+ 2 ]

= ( ) + 2 4 [2+ 2 ]

=( )b ac b ac

a

2 2

3

4

(vi) 4 + 4 = (2 + 2)2 222

={( + )2 2}2 222 = b aca

2

2

2

2.HG IKJ 22

2ca

(vii) 4 4 =(2 + 2) (2 2) = b b ac b ac

a

( )2 2

4

2 4

(viii) 2 + + 2 =( + )2 =b ac

a

2

2

+

(ix)

+

=

2 2+=

( )

+ 2 2

(x)

.

HG

I

KJ

2

+

.

HG I

KJ

2

=

4 4

2 2

+=

[( ) ]b ac a c

a c

2 2 2 2

2 2

2 2

9. Condition for common roots :

The equations a1

x2 + b1

x + c1= 0 and a

2x2 + b

2x + c

2= 0

have

(i) One common root ifb c b cc a c a

1 2 2 1

1 2 2 1

=

c a c aa b a b

1 2 2 1

1 2 2 1

(ii) Both roots common ifa

a1

2=

b

b1

2=

c

c1

2


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E D U C A T I O N S


, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 6PAGE # 5

10. Maximum and Minimum value of quadratic expression :

In a quadratic expression ax2 + bx + c = a xb

a

D

a+.HG

IKJ

L

NMMO

QPP2 4

2

2 ,

Where D = b2 4ac

(i) If a > 0, quadratic expression has minimum value

4

4

2ac b

a

at x =

b

a2and there is no maximum value.

(ii) If a < 0, quadratic expression has maximum value

4

4

2ac b

a

at x =

b

a2and there is no minimum value.

11. Location of roots :

Let f(x) = ax2+ bx + c, a 0 then w.r.to f(x) = 0

(i) If k lies between the roots then a.f(k) < 0

(necessary & sufficient)

(ii) If between k1

& k2

their is exactly one root of k1, k

2

themselves are not roots

f(k1) . f(k

2) < 0 (necessary & sufficient)

(iii) If both the roots are less than a number k

D 0, a.f(k) > 0,b

a2< k (necessary & sufficient)

(iv) If both the roots are greater than k

D 0, a.f(k) > 0,b

a2> k (necessary & sufficient)

(v) If both the roots lies in the interval (k1, k

2)

D 0, a.f(k1) > 0, a.f(k2) > 0, k1 0, xy > 1

(iii) tan1x tan1y = tan1x y

xy

+

.HG

IKJ1 ; if xy > 1

(iv) tan1x tan1y = + tan1x y

xy

+

.HG

IKJ1 ; if x > 0, y < 0, xy < 1

(v) tan1x + tan1y + tan1z = tan1x y z xyz

xy yz zx

+ +

.HG

IKJ1

(vi) sin1x sin1y = sin1 x y y x1 12 2

L

NM

O

QP;

if x,y 0 & x2 + y2 1

(vii) sin1x sin1y = sin1 x y y x1 12 2 LNM

OQP ;

if x,y 0 & x2 + y2 > 1

(viii) cos1x cos1y = cos1 xy x ym 1 12 2

LNMOQP ;

if x,y > 0 & x2 + y2 1

(ix) cos1x cos1y = cos1 xy x ym 1 12 2

LNMOQP ;

if x,y > 0 & x2 + y2 > 1

PAGE # 45 PAGE # 46


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E D U C A T I O N S



6. Inverse trigonometric ratios of multiple angles

(i) 2sin1x = sin1(2x 1 2 x ), if 1 x 1

(ii) 2cos1

x = cos1

(2x2

1), if 1 x 1

(iii) 2tan1x = tan12

1 2x

x

.HG

IKJ= sin1

2

1 2x

x+

.HG

IKJ = cos1

1

1

2

2

+

.HG

IKJ

x

x

(iv) 3sin1x = sin1(3x 4x3)

(v) 3cos1x = cos1(4x3 3x)

(vi) 3tan1x = tan13

1 3

3

2

x x

x

.

HG

I

KJ

PROPERTIES & SOLUTION O. TRIANGLE

Properties of triangle :

1. A triangle has three sides and three angles.In any ABC, we write BC = a, AB = c, AC = b

BB

CC

A

A

a

bc

and BAC = A, ABC = B, ACB = C

2. In ABC :

(i) A + B + C =

(ii) a + b > c, b +c > a, c + a > b

(iii) a > 0, b > 0, c > 0

3. Sine formula :

a

Asin=

b

Bsin=

c

Csin= k(say)

orsinA

a=

sinB

b=

sinC

c= k (say)

4. Cosine formula :

cos A =b c a

bc

2 2 2

2

+

cos B =c a b

ac

2 2 2

2

+

cos C =

a b c

ab

2 2 2

2

+

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E D U C A T I O N S



5. Projection formula :

a = b cos C + c cos B

b = c cos A + a cos C

c = a cos B + b cos A

6. Napier's Analogies :

tanA B

2=

a b

a b

+cot

C

2

tanB C

2=

b c

b c

+cot

A

2

tanC A

2=

c a

c a

+cot

B

2

7. Half angled formula - In any ABC :

(a) sinA

2=

( ) ( )s b s c

bc

sinB

2=

( ) ( )s c s a

ca

sinC

2=

( ) ( )s a s b

ab

where 2s = a + b + c

(b) cosA2

=s s a

bc

( )

cosB

2=

s s b

ca

( )

cosC

2=

s s c

ab

( )

(c) tanA

2=

( ) ( )

( )

s b s c

s s a

tanB

2=

( ) ( )

( )

s c s a

s s b

tanC

2=

( ) ( )

( )

s b s a

s s c

8. , Area of triangle :

(i) =1

2ab sin C =

1

2bc sin A =

1

2ca sin B

(ii) = s s a s b s c( ) ( ) ( )

9. tan A2

tan B2

= s cs

tanB

2tan

C

2=

s a

s

tanC

2tan

A

2=

s b

s

10. Circumcircle of triangle and its radius :

(i) R =a

A2sin=

b

B2sin=

c

C2sin

(ii) R =abc

4Where R is circumradius

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E D U C A T I O N S



11. Incircle of a triangle and its radius :

(iii) r =

s

(iv) r = (s a) tanA

2= (s b) tan

B

2= (s c) tan

C

2

(v) r = 4R sinA

2sin

B

2sin

C

2

(vi) cos A + cos B + cos C = 1 +

r

R

(vii) r =

aB C

A

sin sin

cos

2 2

2

=b

A C

B

sin sin

cos

2 2

2

=c

B A

C

sin sin

cos

2 2

2

12. The radii of the escribed circles are given by :

(i) r1

=

s a, r

2=

s b, r

3=

s c

(ii) r1

= s tanA

2, r

2= s tan

B

2, r

3= s tan

C

2

(iii) r1 = 4R sin

A

2 cos

B

2 cos

C

2 ,

r2

= 4R cosA

2sin

B

2cos

C

2,

r3

= 4R cosA

2cos

B

2sin

C

2

(iv) r1

+ r2

+ r3

r = 4R

(v)1

1r+

1

2r+

1

3r=

1

r

(vi)1

12r

+1

22r

+1

32r

+12r

=a b c2 2 2

2

+ +

(vii)1

bc+

1

ca+

1

ab=

1

2Rr

(viii) r1r2 + r2r3 + r3r1 = s

2

(ix) = 2R2 sin A sin B sin C = 4Rr cosA

2cos

B

2cos

C

2

(x) r1

=

aB C

A

cos cos

cos

2 2

2

, r2

=

bC A

B

cos cos

cos

2 2

2

,

r3

=

cA B

C

cos cos

cos

2 2

2

PAGE # 51 PAGE # 52


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E D U C A T I O N S



(ii) d = h (cot cot)

d

h

HEIGHT AND DISTANCE

1. Angle of elevation and depression :

If an observer is at O and object is at P then XOP is

called angle of elevation of P as seen from O.

If an observer is at P and object is at O, then QPO is

called angle of depression of O as seen from P.

2. Some useful result :

(i) In any triangle ABC if AD : DB = m : n

ACD = , BCD = & BDC =

then (m + n) cot = m cot ncot

C

B

B

A

A

Dm n

= ncotA mcotB [m n Theorem]

PAGE # 53 PAGE # 54

MATHS FORMULA POCKET BOOK


28/93


E D U C A T I O N S



POINT

1. Distance formula :

Distance between two points P(x1, y

1) and Q(x

2, y

2) is

given by d(P, Q) = PQ

= ( ) ( )x x y y2 12

2 12 +

= ( ) ( )Difference of x coordinate Difference of ycoordinate2 2+

Note : (i) d(P, Q) 0

(ii) d(P, Q) = 0 P = Q

(iii) d(P, Q) = d(Q, P)

(iv) Distance of a point (x, y) from origin

(0, 0) = x y2 2+

2. Use of Distance ormula :

(a) In Triangle :

Calculate AB, BC, CA

(i) If AB = BC = CA, then is equilateral.

(ii) If any two sides are equal then is isosceles.

(iii) If sum of square of any two sides is equal tothe third, then is right triangle.

(iv) Sum of any two equal to left third they do notform a triangle

i.e. AB = BC + CA or BC = AC + AB

or AC = AB + BC. Here points are collinear.

(b) In Parallelogram :Calculate AB, BC, CD and AD.

(i) If AB = CD, AD = BC, then ABCD is a parallelo-gram.

(ii) If AB = CD, AD = BC and AC = BD, then ABCDis a rectangle.

(iii) If AB = BC = CD = AD, then ABCD is a rhombus.

(iv) If AB = BC = CD = AD and AC = BD, then ABCD

is a square.

(C) or circumcentre of a triangle :

Circumcentre of a triangle is equidistant from verticesi.e. PA = PB = PC.

Here P is circumcentre and PA is radius.

(i) Circumcentre of an acute angled triangle is in-side the triangle.

(ii) Circumcentre of a right triangle is mid point ofthe hypotenuse.

(iii) Circumcentre of an obtuse angled triangle isoutside the triangle.

3. Section formula :

(i) Internally :

AP

BP=

m

n= , Here > 0

A(x , y )1 1 B(x , y )2 2P

m n

Pmx nx

m n

my ny

m n2 1 2 1+

+

+

+

.

H

G I

K

J,

(ii) Externally :

AP

BP=

m

n=

A(x , y )1 1

B(x , y )2 2

P

nm

Pmx nx

m n

my ny

m n2 1 2 1

.HG

IKJ,

(iii) Coordinates of mid point of PQ arex x y y1 2 1 2

2 2

+ +.HG

IKJ,

(iv) The line ax + by + c = 0 divides the line joining thepoints

(x1, y

1) & (x

2, y

2) in the ratio =

( )

( )

ax by c

ax by c1

2 2

+ +

+ +

PAGE # 55 PAGE # 56

MATHS FORMULA POCKET BOOK MATHS FORMULA POCKET BOOK


29/93


E D U C A T I O N S



(v) or parallelogram midpoint of diagonal AC = mid pointof diagonal BD

(vi) Coordinates of centroid G

x x x y y y1 2 3 1 2 3

3 3

+ + + +.

HG

I

KJ,

(vii) Coordinates of incentre I

ax bx cx

a b c

ay by cy

a b c

1 2 3 1 2 3+ +

+ +

+ +

+ +

.HG

IKJ,

(viii) Coordinates of orthocentre are obtained by solving theequation of any two altitudes.

4. Area of Triangle :

The area of triangle ABC with vertices A(x1, y

1), B(x

2, y

2)

and C(x3, y

3).

=1

2

x y

x y

x y

1 1

2 2

3 3

1

1

1

(Determinant method)

=1

2

x y

x y

x y

x y

1 1

2 2

3 3

1 1

=1

2[x

1y

2+ x

2y

3+ x

3y

1 x

2y

1 x

3y

2 x

1y

3]

[Stair method]Note :

(i) Three points A, B, C are collinear if area of triangleis zero.

(ii) If in a triangle point arrange in anticlockwise thenvalue of be +ve and if in clockwise then will beve.

5. Area of Polygon :

Area of polygon having vertices (x1, y

1), (x

2, y

2), (x

3, y

3)

........ (xn, y

n) is given by area

=1

2

x y

x y

x y

x y

x y

n n

1 1

2 2

3 3

1 1

M M. Points must be taken in order.

6. Rotational Transformation :

If coordinates of any point P(x, y) with reference to newaxis will be (x', y') then

xB yBx' cos sin

y' sin cos

7. Some important points :

(i) Three pts. A, B, C are collinear, if area of triangle iszero

(ii) Centroid G ofABC divides the median AD or BE or C inthe ratio 2 : 1

(iii) In an equilateral triangle, orthocentre, centroid,

circumcentre, incentre coincide.(iv) Orthocentre, centroid and circumcentre are always

collinear and centroid divides the line joining orthocentreand circumcentre in the ratio 2 : 1

(v) Area of triangle formed by coordinate axes & the line

ax + by + c = 0 isc

ab

2

2.

PAGE # 57 PAGE # 58


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E D U C A T I O N S



4. Position of a point with respect to a straight line :

The line L(xi, y

i) i = 1, 2 will be of same sign or of opposite

sign according to the point A(x1, y

1) & B (x

2, y

2) lie on same

side or on opposite side of L (x, y) respectively.

5. Equation of a line parallel (or perpendicular) to the line

ax + by + c = 0 is ax + by + c' = 0 (or bx ay + = 0)

6. Equation of st. lines through (x1,y

1) making an angle

with y = mx + c is

y y1=

m

m

tan

tan

1 m(x x

1)

7. length of perpendicular from (x1, y1) on ax + by + c = 0

is| |ax by c

a b

1 1

2 2

+ +

+

8. Distance between two parallel lines ax + by + ci= 0,

i = 1, 2 is| |c c

a b

1 2

2 2

+

9. Condition of concurrency for three straight lines

Li a

ix + b

iy + c

i= 0, i = 1, 2, 3 is

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

= 0

10. Equation of bisectors of angles between two lines :

a x b y c

a b

1 1 1

12

12

+ +

+ =

a x b y c

a b

2 2 2

22

22

+ +

+

11. amily of straight lines :

The general equation of family of straight line will be written

in one parameter

The equation of straight line which passes through point ofintersection of two given lines L

1and L

2can be taken as

L1 + L2 = 0

12. Homogeneous equation : If y = m1x and y = m

2x be the

two equations

represented by ax2 + 2hxy + by2 = 0 , then m1+ m

2= 2h/b

and m1m

2= a/b

13. General equation of second degree :

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent a pair of

straight line if

a h g

h b f

g f c= 0

If y = m1

x + c & y = m2

x + c represents two straight lines

then m1+ m

2=

2h

b, m

1m

2=

a

b.

14. Angle between pair of straight lines :

The angle between the lines represented by

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 or ax2 + 2hxy + by2 = 0

is tan = 22

h aba b

+( )

(i) The two lines given by ax2 + 2hxy + by2 = 0 are

(a) Parallel and coincident iff h2 ab = 0

(b) Perpendicular iff a + b = 0

(ii) The two line given by ax2 + 2hxy + by2 + 2gx + 2fy + c

= 0 are

(a) Parallel if h2 ab = 0 & af2 = bg2

(b) Perpendicular iff a + b = 0

(c) Coincident iff g2 ac = 0

13. Combined equation of angle bisector of the angle between

the lines ax2 + 2hxy + by2 = 0 is

x y

a b

2 2

=

xy

h

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MATHS FORMULA POCKET BOOK MATHS FORMULA POCKET BOOK

E D U C A T I O N S



CIRCLE

1. General equation of a circle :x2 + y2 + 2gx + 2fy + c = 0

where g, f and c are constants(i) Centre of the cirle is (g, f)

i.e. .

HGIKJ

1

2

1

2coeff of x coeff of y. , .

(ii) Radius is g f c2 2+

2. Central (Centre radius) form of a circle :(i) (x h)2 + (y k)2 = r2 , where (h, k) is circle centre and

r is the radius.

(ii) x2 + y2 = r2 , where (0, 0) origin is circle centre and r is

the radius.

3. Diameter form : If (x1, y

1) and (x

2, y

2) are end pts. of a

diameter of a circle, then its equation is

(x x1) (x x2) + (y y1) (y y2) = 0

4. Parametric equations :

(i) The parametric equations of the circle x2 + y2 = r2 are

x = rcos, y = r sin ,

where point (r cos , r sin )

(ii) The parametric equations of the circle

(x h)2 + (y k)2 = r2 are x = h + rcos, y = k + rsin(iii) The parametric equations of the circle

x2 + y2 + 2gx + 2fy + c = 0 are

x = g + g f c2 2+ cos, y = f + g f c2 2+ sin

(iv) or circle x2 + y2 = a2, equation of chord joining 1 & 2 is

xcos 1 2

2

++ ysin

1 2

2

+= r cos

1 2

2

.

5. Concentric circles : Two circles having same centre C(h, k)

but different radii r1

& r2

respectively are called concentric

circles.

6. Position of a point w.r.t. a circle : A point (x1, y

1) lies

outside, on or inside a circle

S x2 + y2 + 2gx + 2fy + c = 0 according as

S1 x1

2 + y12 + 2gx

1+ 2fy

1+ c is +ve, zero or ve

7. Chord length (length of intercept) = 2 r p2 2

8. Intercepts made on coordinate axes by the circle :

(i) x axis = 2 g c2

(ii) y axis = 2 f c2

9. Length of tangent = S1

10. Length of the intercept made by line : y = mx + c with the

circle x2 + y2 = a2 is

2a m c

m

2 2 2

2

1

1

( )+

+or (1 + m2) |x

1 x

2|

where |x1 x

2| = difference of roots i.e.

D

a

11. Condition of Tangency : Circle x2 + y2 = a2 will touch the

line y = mx + c if c = a 1 2+ m

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E D U C A T I O N S



12. Equation of tangent, T = 0 :

(i) Equation of tangent to the circle

x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y

1) is

xx1

+ yy1

+ g(x + x1) + f(y + y

1) + c = 0

(ii) Equation of tangent to the circle x2 + y2 = a2at any

point (x1, y

1) is xx

1+ yy

1= a2

(iii) In slope form : rom the condition of tangency for

every value of m.

The line y = mx a 1 2+ m is a tangent to the circle

x2 + y2 = a2 and its point of contact is

+

+

.HG

IKJ

am

m

a

m1 12 2,

(iv) Equation of tangent at (a cos , a sin ) to the

circle x2 + y2 = a2 is x cos + y sin = a.

13. Equation of normal :

(i) Equation of normal to the circle

x2 + y2 + 2gx + 2fy + c = 0 at any point P(x1, y

1) is

y y1 =

y f

x g

1

1

+

+ (x x1)

(ii) Equation of normal to the circle x2 + y2 = a2at any

point (x1, y

1) is xy

1 x

1y = 0

14. Equation of pair of tangents SS1= T2

15. The point of intersection of tangents drawn to the circle x2

+ y2 = r2 at point 1 & 2 is given as

r rcos

cos

,sin

cos

1 2

1 2

1 2

1 2

2

2

2

2

+

+

.

H

GGG

I

K

JJJ

16. Equation of the chord of contact of the tangents drawn

from point P outside the circle is T = 0

17. Equation of a chord whose middle pt. is given by T = S1

18. Director circle : Equation of director circle for x2 + y2 = a2 is

x2 + y2 = 2a2. Director circle is a concentric circle whose

radius is 2 times the radius of the given circle.

19. Equation of polar of point (x1, y

1) w.r.t. the circle S = 0 is T = 0

20. Coordinates of pole : Coordinates of pole of the line

lx + my + n = 0 w.r.t the circle x2 + y2 = a2 are .

HGIKJ

a l

n

a m

n

2 2

,

21. amily of Circles :

(i) S +S' = 0 represents a family of circles passing through

the pts. of intersection of

S = 0 & S' = 0 if 1

(ii) S + L = 0 represent a family of circles passing through

the point of intersection of S = 0 & L = 0

(iii) Equation of circle which touches the given straight line

L = 0 at the given point (x1, y

1) is given as

(x x1)2+ (y y1)2 + L = 0.

PAGE # 65 PAGE # 66



34/93

E D U C A T I O N S



(iv) Equation of circle passing through two points A(x1, y

1)

& B(x2, y

2) is given as

(x x1) (x x

2) + (y y

1) (y y

2) +

x yx y

x y

11

1

1 1

2 2

= 0.

22. Equation of Common Chord is S S1= 0.

23. The angle of intersection of two circles with centres C1

& C2and radii r

1& r

2is given by

cos =r r d

r r

12

12 2

1 22

+ , where d = C

1C

2

24. Position of two circles : Let two circles with centres C1, C

2

and radii r1, r

2.

Then following cases arise as

(i) C1C

2> r

1+ r

2 do not intersect or one outside the

other, 4 common tangents.

(ii) C1C

2= r

1+ r

2 Circles touch externally, 3 common

tangents.

(iii) |r1 r2| < C1 C2 < r1 + r2 Intersection at two realpoints, 2 common tangents.

(iv) C1C

2= |r

1 r

2| internal touch, 1 common tangent.

(v) C1

C2< |r

1+ r

2| one inside the other, no tangent.

Note : Point of contact divides C1

C2

in the ratio r1: r

2

internally or externally as the case may be

PAGE # 67 PAGE # 68

25. Equation of tangent at point of contact of circle is

S1 S

2= 0

26. Radical axis and radical centre :

(i) Equation of radical axis is S S1= 0

(ii) The point of concurrency of the three radical axis of

three circles taken in pairs is called radical centre of

three circles.

27. Orthogonality condition :

If two circles S x2

+ y2

+ 2gx + 2fy + c = 0and S' = x2 + y2 + 2g'x + 2f'y + c' = 0 intersect each other

orthogonally, then 2gg' + 2ff' = c + c'.



35/93

E D U C A T I O N S



PARABOLA

1. Standard Parabola :

Imp. Terms y2 = 4ax y2 = 4ax x2 = 4ay x2 = 4ay

Vertex (v) (0, 0) (0, 0) (0, 0) (0, 0)

ocus (f) (a, 0) (a, 0) (0, a) (0, a)

Directrix (D) x = a x = a y = a y = a

Axis y = 0 y = 0 x = 0 x = 0

L.R. 4a 4a 4a 4a

ocal x + a a x y + a a y

distance

Parametric (at2, 2at) ( at2, 2at) (2at, at2) (2at, at2)

Coordinates

Parametric x = at2 x = at2 x = 2at x = 2at

Equations y = 2at y = 2at y = 2at2 y = at2

y2 = 4ax

y2 = 4ax

x2 = 4ay

x2

= 4ay

PAGE # 69 PAGE # 70



36/93

E D U C A T I O N S



2. Special orm of Parabola

* Parabola which has vertex at (h, k), latus rectum l

and axis parallel to x-axis is

(y k)2 = l (x h)

axis is y = k and focus at h k+.HG

IKJ

l

4,

* Parabola which has vertex at (h, k), latus rectum l and

axis parallel to y-axis is

(x h)2 = l (y k)

axis is x = h and focus at h k, +

.HG

IKJ

l

4

* Equation of the form ax2 + bx + c = y represents

parabola.

i.e. y 4

4

2ac b

a

= a x

b

a+

.HG

IKJ22

,with vertex

.

HGIKJ

b

a

ac b

a2

4

4

2

, and axes parallel to y-axis

Note : Parametric equation of parabola (y k)2

= 4a(x h) are x = h + at2, y = k + 2at

3. Position of a point (x1, y

1) and a line w.r.t. parabola

y2 = 4ax.

* The point (x1, y1) lies outside, on or inside theparabola y2 = 4ax

according as y12 4ax

1>, = or < 0

* The line y = mx + c does not intersect, touches,

intersect a parabola y2 = 4ax according as

c > = < a/m

Note : Condition of tangency for parabola y2 = 4ax, we

have c = a/m and for other parabolas check disc. D = 0.

4. Equations of tangent in different forms :

(i) Point orm / Parametric form

Equations of tangent of all other standard parabolas

at (x1, y1) / at t (parameter)Equation Tangent at Parametric Tangent of 't'

of parabola (x1, y

1) coordinates't'

y2=4ax yy1=2a(x+x

1) (at2, 2at) ty=x+at2

y2=4ax yy1=2a(x+x

1) (at2, 2at) ty=x+at2

x2=4ay xx1=2a(y+y

1) (2at, at2) tx=y + at2

x2=4ay xx1=2a(y+y

1) (2at, at2) tx =y+at2

(ii) Slope form

Equations of tangent of all other parabolas in slope

form

Equation Point of Equations Condition of

of contact in of tangent Tangency

parabolas terms of in terms of

slope(m) slope (m)

y2 = 4ax am

am22,.HG IKJ y = mx + am c = am

y2 = 4ax .HG

IKJ

a

m

a

m22

, y = mx a

mc =

a

m

x2 = 4ay (2am, am2) y = mx am2 c = am2

x2 = 4ay (2am, am2) y = mx + am2 c = am2

5. Point of intersection of tangents at any two points

P(at12, 2at

1) and Q(at

22, 2at

2) on the parabola y2 = 4ax

is (at1t2, a(t

1+ t

2)) i.e. (a(G.M.)2, a(2A.M.))

6. Combined equation of the pair of tangents drawn from a

point to a parabola is SS' = T2, where S = y2 4ax,

S' = y12 4ax

1and T = yy

1 2a(x + x

1)

PAGE # 71 PAGE # 72



37/93

E D U C A T I O N S



7. Equations of normal in different forms

(i) Point orm / Parametric form

Equations of normals of all other standard parabolas

at (x1, y

1) / at t (parameter)

Eqn. of Normal Point Normals

parabola at (x1, y

1) 't' at 't'

y2 = 4ax yy1

=y

a1

2(xx

1) (at2, 2at) y+tx = 2at+at3

y2 = 4ax yy1

= ya1

2(xx

1) (at2, 2at) ytx = 2at+at3

x2 = 4ay yy1

= 2

1

a

x (xx1) (2at, at2) x+ty = 2at+at3

x2 = 4ay yy1

=2

1

a

x (xx1) (2at, at2) xty = 2at+at3

(ii) Slope form

Equations of normal, point of contact, and condition

of normality in terms of slope (m)

Eqn. of Point of Equations Condition of

parabola contact of normal Normality

y2 = 4ax (am2, 2am) y = mx2amam3 c = 2amam3

y2 = 4ax (am2, 2am) y = mx+2am+am3 c = am+am3

x2 = 4ay .HG

IKJ

22

a

m

a

m, y = mx+2a+

a

m2c = 2a+

a

m2

x2 = 4ay2

2

a

m

a

m,

.HG

IKJ y = mx2a

a

m2c = 2a

a

m2

Note :

(i) In circle normal is radius itself.

(ii) Sum of ordinates (y coordinate) of foot of normals

through a point is zero.

(iii) The centroid of the triangle formed by taking the foot

of normals as a vertices of concurrent normals of

y2 = 4ax lies on x-axis.

8. Condition for three normals from a point (h, 0) on x-axis

to parabola y2 = 4ax

(i) We get 3 normals if h > 2a

(ii) We get one normal if h 2a.

(iii) If point lies on x-axis, then one normal will be x-axis

itself.

9. (i) If normal of y2 = 4ax at t1

meet the parabola again

at t2

then t2

= t1

2

1t

(ii) The normals to y2

= 4ax at t1 and t2 intersect eachother at the same parabola at t

3, then

t1t2

= 2 and t3

= t1

t2

10. (i) Equation of focal chord of parabola y2 = 4ax at t1

is

y =2

1

1

12

t

t (x a)

If focal chord of y

2

= 4ax cut (intersect) at t1 andt2

then t1t2

= 1 (t1

must not be zero)

(ii) Angle formed by focal chord at vertex of parabola is

tan =2

3|t

2 t

1|

(iii) Intersecting point of normals at t1

and t2

on the

parabola y2 = 4ax is

(2a + a(t1

2

+ t2

2 + t1

t2

), at1

t2

(t1

+ t2

))

PAGE # 73 PAGE # 74



38/93

E D U C A T I O N S



11. Equation of chord of parabola y2 = 4ax which is bisected

at (x1, y

1) is given by T = S

1

12. The locus of the mid point of a system of parallel chords

of a parabola is called its diameter. Its equation is y =2a

m.

13. Equation of polar at the point (x1, y

1) with respect to

parabola y2 = 4ax is same as chord of contact and is given

by

T = 0 i.e. yy1

= 2a(x + x1

)

Coordinates of pole of the line l x + my + n = 0 w.r.t. the

parabola y2 = 4ax isn am

l l,

.HG

IKJ

2

14. Diameter : It is locus of mid point of set of parallel chords

and equation is given by T = S1

15. Important results for Tangent :

(i) Angle made by focal radius of a point will be twice

the angle made by tangent of the point with axis of

parabola

(ii) The locus of foot of perpendicular drop from focus to

any tangent will be tangent at vertex.(iii) If tangents drawn at ends point of a focal chord are

mutually perpendicular then their point of intersection

will lie on directrix.

(iv) Any light ray travelling parallel to axis of the parabola

will pass through focus after reflection through

parabola.

PAGE # 75 PAGE # 76

(v) Angle included between focal radius of a point and

perpendicular from a point to directrix will be bisected

of tangent at that point also the external angle will

be bisected by normal.

(vi) Intercepted portion of a tangent between the point

of tangency and directrix will make right angle at

focus.

(vii) Circle drawn on any focal radius as diameter will

touch tangent at vertex.

(viii) Circle drawn on any focal chord as diameter will touch

directrix.



39/93

E D U C A T I O N S



ELLIPSE

1. Standard Ellipse (e < 1)

x

a

y

b

2

2

2

21+ =

RS|T|

UV|W|

or a > b or b > a

Centre (0, 0) (0, 0)

Vertices (a, 0) (0, b)

Length of major axis 2a 2b

Length of minor axis 2b 2a

oci (ae, 0) (0, be)

Equation of directrices x = a/e y = b/e

Relation in a, b and e b2 = a2(1 e2) a2 = b2(1 e2)

Length of latus rectum 2b2/a 2a2/b

Ends of latus rectum .HG

IKJ

aeb

a,

2

.HG

IKJ

a

bbe

2

,

Parametric coordinates (a cos , b sin ) (a cos , b sin )

0 < 2

ocal radii SP = a ex1 SP = b ey1

S'P = a + ex1

S'P = b + ey1

Sum of focal radii SP + S'P = 2a 2b

Distance btn foci 2ae 2be

Distance btn directrices 2a/e 2b/e

Tangents at the vertices x = a, x = a y = b, y = b

Note : If P is any point on ellipse and length of perpendiculars

from to minor axis and major axis are p1

& p2, then |x

p|

= p1

, |yp| = p

2

p

a

p

b

12

222

2+ = 1

a > b

b > a

PAGE # 77 PAGE # 78

Imp. terms

Ellipse



40/93

E D U C A T I O N S



2. Special form of ellipse :

If the centre of an ellipse is at point (h, k) and the

directions of the axes are parallel to the coordinate axes,

then its equation isx h

a

c h22

+( )y k

b

2

2 = 1.

3. Auxillary Circle : The circle described by taking centre of

an ellipse as centre and major axis as a diameter is called

an auxillary circle of the ellipse.

If xa

2

2 +yb

2

2 = 1 is an ellipse then its auxillary circle is

x2 + y2 = a2.

Note : Ellipse is locus of a point which moves in such a

way that it divides the normal of a point on diameter of

a point of circle in fixed ratio.

4. Position of a point and a line w.r.t. an ellipse :

* The point lies outside, on or inside the ellipse if

S1

=x

a

12

2 +y

b

12

2 1 > , = or < 0

* The line y = mx + c does not intersect, touches,

intersect, the ellipse if

a2m2+ b2 < = > c2

5. Equation of tangent in different forms :

(i) Point form : The equation of the tangent to the

ellipsex

a

2

2 +y

b

2

2 = 1 at the point (x1, y1) is

xx

a

1

2 +yy

b

1

2 = 1.

(ii) Slope form : If the line y = mx + c touches the

ellipsex

a

2

2 +y

b

2

2 = 1, then c2 = a2m2 + b2. Hence, the

straight line y = mx a m b2 2 2+ always represents

the tangents to the ellipse.

Point of contact :

Line y = mx a m b2 2 2+ touches the ellipse

xa

2

2 +yb

2

2 = 1 at

+

+.HG IKJ

a ma m b

ba m b

2

2 2 2

2

2 2 2, .

(iii) Parametric form : The equation of tangent at any

point (a cos , b sin ) is

x

acos +

y

bsin = 1.

6. Equation of pair of tangents from (x1, y

1) to an ellipse

x

a

2

2 +y

b

2

2 = 1 is given by SS1 = T2

7. Equation of normal in different forms :

(i) Point form : The equation of the normal at (x1, y1)

to the ellipsex

a

2

2 +y

b

2

2 = 1 is

a x

x

2

1

b x

y

2

1= a2 b2.

PAGE # 79 PAGE # 80



41/93

E D U C A T I O N S



(ii) Parametric form : The equation of the normal to the

ellipsex

a

2

2 +y

b

2

2 = 1 at (a cos , b sin ) is

ax sec by cosec = a2 b2.

(iii) Slope form : If m is the slope of the normal to the

ellipsex

a

2

2 +y

b

2

2 = 1, then the equation of normal

is y = mx m a b

a b m

( )2 2

2 2 2

+.

The co-ordinates of the point of contact are

+

+

.HG

IKJ

a

a b m

mb

a b m

2

2 2 2

2

2 2 2, .

Note : In general three normals can be drawn from a point

(x1, y

1) to an ellipse

x

a

2

2 +y

b

2

2 = 1.

8. Properties of tangents & normals :

(i) Product of length of perpendicular from either focii to

any tangent to the ellipse will be equal to square of

semi minor axis.

(ii) The locus of foot of perpendicular drawn from either

focii to any tangent lies on auxillary circle.

(iii) The circle drawn on any focal radius as diameter will

touch auxillary circle.

(iv) The protion of the tangent intercepted between the

point and directrix makes right angle at corresponding

focus.

(v) Sum of square of intercept made by auxillary circle on

any two perpendicular tangents of an ellipse will be

constant.

(vi) If a light ray originates from one of focii, then it willpass through the other focus after reflection from

ellipse.

9. Equation of chord of contact of the tangents drawn from

the external point (x1, y

1) to an ellipse is given by

xx

a12 +

yy

b12 = 0 i.e. T = 0.

10. The equation of a chord of an ellipsex

a

2

2 +y

b

2

2 = 1 whose

mid point is (x1, y

1) is T = S

1.

11. Equation of chord joining the points (a cos , b sin ) and

(a cos , b sin ) on the ellipsex

a

2

2 +y

b

2

2 = 1 is

x

a

cos +

2

+y

b

sin +

2

= cos

2

(i) Relation between eccentric angles of focal chord

tan12

, tan22

=

e

e

1

1

(ii) Sum of feet of eccentric angles is odd .

i.e. 1 + 2 + 3 + 4 = (2n + 1) .

PAGE # 81 PAGE # 82



42/93

E D U C A T I O N S



12. Equation of polar of the point (x1, y

1) w.r.t. the ellipse

x

a

2

2 +y

b

2

2 = 1 is given byxx

a

1

2 +yy

b

1

2 = 0 i.e. T = 0.

The pole of the line l x + my + n = 0 w.r.t. the ellipse

x

a

2

2 +y

b

2

2 = 1 is .

HGIKJ

a

n

b n

n

2 2l, .

13. Eccentric angles of the extremities of latus rectum of the

ellipse

x

a

2

2 +

y

b

2

2 = 1 are tan1

.HG

IKJ

b

ae .

14. (i) Equation of the diameter bisecting the chords of

slope in the ellipsex

a

2

2 +y

b

2

2 = 1 is

y = b

a m

2

2 x

(ii) Conjugate Diameters : The straight lines y = m1x,

y = m2x are conjugate diameters of the ellipse

x

a

2

2 +y

b

2

2 = 1 if m1m2 = b

a

2

2 .

(iii) Properties of conjugate diameters :

(a) If CP and CQ be two conjugate semi-diameters

of the ellipse xa

2

2 + yb

2

2 = 1, then

CP2 + CQ2 = a2 + b2

(b) If and are the eccentric angles of the

extremities of two conjugate diameters, then

=

2

(c) If CP, CQ be two conjugate semi-diameters of

the ellipsex

a

2

2 +y

b

2

2 = 1 and S, S' be two foci

of the ellipse, then SP.S'P = CQ2

(d) The tangents at the ends of a pair of conjugate

diameters of an ellipse form a parallelogram.

15. The area of the parallelogram formed by the tangents at

the ends of conjugate diameters of an ellipse is constant

and is equal to the product of the axis i.e. 4ab.

16. Length of subtangent and subnormal at p(x1, y

1) to the

ellipse

x

a

2

2 +y

b

2

2 = 1 isa

x

2

1 x

1& (1 e2) x

1

PAGE # 83 PAGE # 84


HYPERBOLA


43/93

E D U C A T I O N S



HYPERBOLA

1. Standard Hyperbola :

x

a

2

2 y

b

2

2 = 1 x

a

2

2 +y

b

2

2 = 1

orx

a

2

2 y

b

2

2 = 1

Centre (0, 0) (0, 0)

Length of

transverse axis 2a 2bLength of

conjugate axis 2b 2a

oci (ae, 0) (0, be)

Equation of

directrices x = a/e y = b/e

Eccentricity e =

a b

a

2 2

2

+.

HG

I

KJ e =a b

b

2 2

2

+.

HG

I

KJLength of L.R. 2b2/a 2a2/b

Parametric

co-ordinates (a sec , b tan ) (b sec , a tan )

0 < 2 0 < 2

ocal radii SP = ex1

a SP = ey1

b

S'P = ex1 + a S'P = ey1 + bS'P SP 2a 2b

Tangents at

the vertices x = a, x = a y = b, y = b

Equation of the y = 0 x = 0

transverse axis

Equation of the x = 0 y = 0

conjugate axis

Hyperbola

Conjugate Hyperbola

PAGE # 85 PAGE # 86

Imp. terms

Hyperbola


f f


44/93

E D U C A T I O N S



2. Special form of hyperbola :

If the centre of hyperbola is (h, k) and axes are parallel

to the co-ordinate axes, then its equation is

( )x h

a

22

( )y k

b

22 = 1.

3. Parametric equations of hyperbola :

The equations x = a sec and y = b tan are known

as the parametric equations of hyperbolax

a

2

2

y

b

2

2= 1

4. Position of a point and a line w.r.t. a hyperbola :

The point (x1, y

1) lies inside, on or outside the hyperbola

x

a

2

2 y

b

2

2 = 1

according asx

a

12

2 y

b

12

2 1 is +ve, zero or ve.

The line y = mx + c does not intersect, touches, intersect

the hyperbola

according as c2 a2m2 b2.

5. Equations of tangents in different forms :

(a) Point form : The equation of the tangent to the

hyperbolax

a

2

2 y

b

2

2 = 1

at (x1, y

1) is

xx

a

1

2 yy

b

1

2 = 1.

(b) Parametric form : The equation of tangent to the

hyperbola

x

a

2

2

y

b

2

2 = 1 at (a sec , b tan ) is

x

asec

y

btan = 1.

(c) Slope form : The equations of tangents of slope m

to the hyperbola

x

a

2

2

y

b

2

2 = 1 are y = mx a m b2 2 2 and the

co-ordinates of points of contacts are

.HG

IKJ

a m

a m b

b

a m b

2

2 2 2

2

2 2 2, .

6. Equation of pair of tangents from (x1, y1) to the hyperbola

x

a

2

2 y

b

2

2 = 1 is given by SS1 = T2

7. Equations of normals in different forms :

(a) Point form : The equation of normal to the hyperbola

xa

2

2 yb

2

2 = 1 at (x1, y1) isa xx

2

1+ b y

y

2

1= a2 + b2.

(b) Parametric form : The equation of normal at

(a sec , b tan ) to the hyperbola

x

a

2

2 y

b

2

2 = 1 is ax cos + by cot = a2+ b2

PAGE # 87 PAGE # 88


( ) Sl f Th i f h l h


45/93

E D U C A T I O N S



(c) Slope form : The equation of the normal to the

hyperbolax

a

2

2 y

b

2

2 = 1 in terms of the slope m of

the normal is y = mx m m a b

a b m

( )2 2

2 2 2

+

(d) Condition for normality : If y = mx + c is the normal

ofx

a

2

2 y

b

2

2 = 1,

then c = m m a b

a b m

( )2 2

2 2 2

+

or c2 =

m a b

a m b

( )

( )

2 2 2

2 2 2

+

, which

is condition of normality.

(e) Points of contact : Co-ordinates of points of contact

are

.HG

IKJ

a

a b m

mb

a b m

2

2 2 2

2

2 2 2, m .

8. The equation of director circle of hyperbola

x

a

2

2 y

b

2

2 = 1 is x2 + y2

= a2 b2.

9. Equation of chord of contact of the tangents drawn from

the external point (x1, y

1) to the hyperbola is given by

xx

a

1

2 yy

b

1

2 = 1.

10. The equation of chord of the hyperbolax

a

2

2 y

b

2

2 = 1

whose mid point is (x1, y

1) is T = S

1.

11. Equation of chord joining the points P(a sec 1, b tan

1)

and Q(a sec 2, b tan

2) is

xa

cos 1 22.HG IKJ yb sin

1 22+.HG IKJ = cos

1 2

2+.HG IKJ.

12. Equation of polar of the point (x1, y

1) w.r.t. the hyperbola

is given by T = 0.

The pole of the line l x + my + n = 0 w.r.t.

xa

2

2 yb

2

2 = 1 is .HG IKJan b mn2 2l,

13. The equation of a diameter of the hyperbola

x

a

2

2 y

b

2

2 = 1 is y =b

a m

2

2 x.

14. The diameters y = m1x and y = m

2x are conjugate if

m1m

2=

b

a

2

2

15. Asymptotes of a hyperbola :

* The equations of asymptotes of the hyperbola

x

a

2

2 y

b

2

2 = 1 are y = b

ax.

Asymptote to a curve touches the curve at infinity.

* The asymptote of a hyperbola passes through the

centre of the hyperbola.

PAGE # 89 PAGE # 90


* The combined equation of the asymptotes of the * Equation of tangent at (x y ) to xy


46/93

E D U C A T I O N S



The combined equation of the asymptotes of the

hyperbolax

a

2

2 y

b

2

2 = 1 isx

a

2

2 y

b

2

2 = 0.

* The angle between the asymptotes of

x

a

2

2 y

b

2

2 = 1 is 2 tan1

y

b

2

2 or 2 sec1 e.

* A hyperbola and its conjugate hyperbola have the

same asymptotes.

* The bisector of the angles between the asymptotes

are the coordinate axes.

* Equation of hyperbola Equation of asymptotes =

Equation of asymptotes Equation of conjugate

hyperbola = constant.

16. Rectangular or Equilateral Hyperbola :

* A hyperbola for which a = b is said to be rectangular

hyperbola, its equation is x2 y2 = a2

* xy = c2 represents a rectangular hyperbola with

asymptotes x = 0, y = 0.

* Eccentricity of rectangular hyperbola is 2 and angle

between asymptotes of rectangular hyperbola is 90.

* Parametric equation of the hyperbola xy = c2

are

x = ct, y =c

t, where t is a parameter.

* Equation of chord joining t1, t

2on xy = c2 is

x + y t1t2

= c(t1

+ t2)

* Equation of tangent at (x1, y

1) to xy

= c2 isx

x1+

y

y1= 2.

Equation of tangent at t is x + yt2 = 2ct

* Equation of normal at (x1, y

1) to xy = c2 is

xx1

yy1

= x12 y

12

* Equation of normal at t on xy = c2 is

xt3 yt ct4 + c = 0.

(This results shows that four normal can be drawn

from a point to the hyperbola xy = c2)

* If a triangle is inscribed in a rectangular hyperbola

then its orthocentre lies on the hyperbola.

* Equation of chord of the hyperbola xy = c2 whose

middle point is given is T = S1

* Point of intersection of tangents at t1

& t2

to the

hyperbola xy = c2 is2 21 2

1 2 1 2

c t t

t t

c

t t+ +

.HG

IKJ

,

PAGE # 91 PAGE # 92


MEASURES O. CENTRAL TENDENCY 3 Geometric Mean :


47/93

E D U C A T I O N S



MEASURES O. CENTRAL TENDENCY

AND DISPERSION

1. Arithmetic mean :(i) or ungrouped data (individual series) x =

x x x

n no of termsn1 2+ + +......

( . )=

i

n

ix

n

=1

(ii) or grouped data (continuous series)

(a) Direct method x =

i

n

i i

i

n

i

fx

f

=

=

1

1

, where xi, i = 1 .... n

be n observations and fibe their corresponding

frequencies

(b) short cut method : x = A +

fd

f

i i

i

,

where A = assumed mean, di= x

i A = deviation

for each term

2. Properties of A.M.

(i) In a statistical data, the sum of the deviation of items

from A.M. is always zero.

(ii) If each of the n given observation be doubled, thentheir mean is doubled

(iii) If x is the mean of x1, x2, ...... xn. The mean of ax1, ax2.....ax

nis a x where a is any number different from

zero.

(iv) Arithmetic mean is independent of origin i.e. it is x

effected by any change in origin.

3. Geometric Mean :

(i) or ungrouped data

G.M. = (x1x

2x

3.....x

n)1/n

or G.M. = antilog1

1n

xii

n

log=

.HGIKJ

(ii) or grouped data

G.M. = x x xf f nf Nn

1 2

1

1 2....e j , where N = fii

n

=

1

= antilog

f x

f

i

i

n

i

i

i

n

=

=

.

H

GGGG

I

K

JJJJ

1

1

log

4. Harmonic Mean - Harmonic Mean is reciprocal of arith-

metic mean of reciprocals.

(i) or ungrouped data H.M. =n

xii

n1

1=

(ii) or grouped data H.M. =

f

f

x

i

i

n

i

ii

n

=

=

.HG IKJ1

1

5. Relation between A.M., G.M and H.M.

A.M. G.M. H.M.

Equality holds only when all the observations in the series

are same.

PAGE # 93 PAGE # 94


6. Median : (ii) or series in descending order


48/93

E D U C A T I O N S



6. Median :

(a) Individual series (ungrouped data) : If data is raw,

arrange in ascending or descending order and n be

the no. of observations.

If n is odd, Median = Value ofn

th+.

HGIKJ

1

2observation

If n is even, Median =1

2[Value of

nth

2

.HG

IKJ + value of

n

th

2 1+.HG

IKJ ] observation.

(b) Discrete series : irst find cumulative frequencies of

the variables arranged in ascending or descending

order and

Median =n

th+.

HG I

KJ1

2

observation, where n is cumulative

frequency.

(c) Continuous distribution (grouped data)

(i) or series in ascending order

Median = l +

NC

f

2

.HG

IKJ

i

Where l = Lower limit of the median class.

f = requency of the median class.

N = Sum of all frequencies.

i = The width of the median class

C = Cumulative frequency of the

class preceding to median class.

g

Median = u -

NC

f

2

.HG

IKJ

i

where u = upper limit of median class.

7. Mode :

(i) or individual series : In the case of individual series,

the value which is repeated maximum number of times

is the mode of the series.

(ii) or discrete frequency distribution series : In the caseof discrete frequency distribution, mode is the value of

the variate corresponding to the maximum frequency.

(iii) or continuous frequency distribution : first find the

model class i.e. the class which has maximum frequency.

or continuous series

Mode = l 1 +

f f

f f f1 0

1 0 22

LNM

OQP i

Where l 1 = Lower limit of the model class.

f1

= requency of the model class.

f0

= requency of the class preceding model

class.

f2

= requency of the class succeeding model

class.i = Size of the model class.

8. Relation between Mean, Mode & Median :

(i) In symmetrical distribution : Mean = Mode = Median

(ii) In Moderately symmetrical distribution : Mode = 3 Me-

dian 2 Mean

PAGE # 95 PAGE # 96


Measure of Dispersion : fi= requency of the corresponding x

i


49/93

E D U C A T I O N S



The degree to which numerical data tend to spread about

an average value is called variation or dispersion.

Popular methods of measure of dispersion.

1. Mean deviation : The arithmetic average of deviations

from the mean, median or mode is known as mean deviation.

(a) Individual series (ungrouped data)

Mean deviation =| |x S

n

Where n = number of terms, S = deviation of variate

from mean mode, median.(b) Continuous series (grouped data).

Mean deviation =

f x s

f

| |=

f x s

N

| |

Note : Mean deviation is the least when measured from the

median.

2. Standard Deviation :S.D. () is the square root of the arithmetic mean of the

squares of the deviations of the terms from their A.M.

(a) or individual series (ungrouped data)

=( )x x

N

2where x = Arithmetic mean of

the series

N = Total frequency

(b) or continuous series (grouped data)

(i) Direct method =f x x

Ni i( )

2

Where x = Arithmetic mean of series

xi

= Mid value of the class

i i

N = f = Total frequency

(ii) Short cut method

= fd

N

fd

N

2 2

.HGIKJ or =

d

N

d

N

2 2

.HGIKJ

Where d = x A = Derivation from assumed mean A

f = requency of item (term)

N = f = Total frequency.

Variance Square of standard direction

i.e. variance = (S.D.)2 = ()2

Coefficient of variance = Coefficient of S.D. 100

=

x 100

PAGE # 97 PAGE # 98


MATRICES AND DETERMINANTS 4. Trace of a matrix : Sum of the elements in the principal


50/93

E D U C A T I O N S



MATRICES AND DETERMINANTS

MATRICES :

1. Matrix - A system or set of elements arranged in a rectan-gular form of array is called a matrix.

2. Order of matrix : If a matrix A has m rows & n columns then

A is of order m n.

The number of rows is written first and then number of col-

umns. Horizontal line is row & vertical line is column

3. Types of matrices : A matrix A = (aij)

mn

A matrix A = (aij)

mxnover the field of complex numbers is

said to be

Name Properties

A row matrix if m = 1

A column matrix if n = 1

A rectangular matrix if m n

A square matrix if m = n

A null or zero matrix if aij= 0 i j. It is denoted by O.

A diagonal matrix if m = n and aij

= 0 for i j.

A scalar matrix if m = n and aij

= 0 for i j

= k for i = j

i.e. a11

= a22

....... = ann

= k (cons.)

Identity or unit matrix if m = n and aij

= 0 for i j

= 1 for i = j

Upper Triangular matrix if m = n and aij

= 0 for i > j

Lower Triangular matrix if m = n and aij

= 0 for i < j

Symmetric matrix if m = n and aij

= aji

for all i, j

or AT = A

Skew symmetric matrix if m = n and aij

= aji i, j

or AT = A

p p

diagonal is called the trace of a matrix.

trace (A B) = trace A trace B

trace kA = k trace A

trace A = trace AT

trace In= n when I

nis identity matrix.

trace On= O O

nis null matrix.

trace AB trace A trace B.

5. Addition & subtraction of matrices : If A and B are twomatrices each of order same, then A + B (or A B) is defined

and is obtained by adding (or subtracting) each element of B

from corresponding element of A

6. Multiplication of a matrix by a scalar :

KA = K (aij)

mn= (Ka)

mnwhere K is constant.

Properties :

(i) K(A + B) = KA + KB

(ii) (K1K

2)A = K

1(K

2A) = K

2(K

1A)

(iii) (K1+ K

2)A = K

1A + K

2A

7. Multiplication of Matrices : Two matrices A & B can be

multiplied only if the number of columns in A is same as the

number of rows in B.

Properties :

(i) In general matrix multiplication is not commutative i.e.

AB BA.

(ii) A(BC) = (AB)C [Associative law]

PAGE # 99 PAGE # 100


(iii) A.(B + C) = AB + AC [Distributive law] DETERMINANT:


51/93

E D U C A T I O N S



( ) ( ) [ ]

(iv) If AB = AC / B = C

(v) If AB = 0, then it is not necessary A = 0 or B = 0

(vi) AI = A = IA

(vii) Matrix multiplication is commutative for +ve integral

i.e. Am+1 = Am A = AAm

8. Transpose of a matrix :

A' or A

T

is obtained by interchanging rows into columns orcolumns into rows

Properties :

(i) (AT)T = A

(ii) (A B)T = AT BT

(iii) (AB)T = BTAT

(iv) (KA)T

= KAT

(v) IT = I

9. Some special cases of square matrices :A square matrix

is called

(i) Orthogonal matrix : if AAT = In= ATA

(ii) Idempotent matrix : if A2

= A(iii) Involutory matrix : if A2 = I or A1 = A

(iv) Nilpotent matrix : if p N such that Ap = 0

(v) Hermitian matrix : if A = A i.e. aij= a

ji

(vi) Skew - Hermitian matrix : if A = A

DETERMINANT :

1. Minor & cofactor : If A = (aij)

33, then minor of a

11is

M11

= a aa a22 23

32 33and so.

cofactor of an element aijis denoted by C

ijor

ijand is equal

to (1)i+j Mij

or Cij= M

ij, if i = j

= Mij, if i j

Note : |A| = a1111 + a12 12 + a13 13

and a11

21

+ a12

22

+ a13

23

= 0

2. Determinant : if A is a square matrix then determinant of

matrix is denoted by det A or |A|.

expansion of determinant of order 3 3

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

= a1

b c

b c

2 2

3 3b

1a c

a c

2 2

3 3+ c

1a b

a b

2 2

3 3

or = a2b c

b c

1 1

3 3+ b

2a c

a c

1 1

3 3 c

2a ba b

1 1

3 3

Properties :(i) |AT| = |A|

(ii) By interchanging two rows (or columns), value of de-

terminant differ by ve sign.

(iii) If two rows (or columns) are identical then |A| = 0

(iv) |KA| = Kn det A, A is matrix of order n n

PAGE # 101 PAGE # 102


(v) If same multiple of elements of any row (or column) of Properties :


52/93

E D U C A T I O N S



a determinant are added to the corresponding elements

of any other row (or column), then the value of the

new determinant remain unchanged.

(vi) Determinant of :

(a) A nilpotent matrix is 0.

(b) An orthogonal matrix is 1 or 1

(c) A unitary matrix is of modulus unity.

(d) A Hermitian matrix is purely real.

(e) An identity matrix is one i.e. |In| = 1, where In is aunit matrix of order n.

(f) A zero matrix is zero i.e. |0n| = 0, where 0

nis a zero

matrix of order n

(g) A diagonal matrix = product of its diagonal elements.

(h) Skew symmetric matrix of odd order is zero.

3. Multiplication of two determinants :

Multiplication of two second order determinants is defined as

follows.

a b

a b

1 1

2 2

l

l

1 1

2 2

m

m =a b a m b ma b a m b m

1 1 1 2 1 1 1 2

2 1 2 2 2 1 2 2

l l

l l

+ ++ +

If order is different then for their multiplication, express themfirstly in the same order.

MATRICES AND DETERMINANTS :

1. Adjoint of a matrix :

adj A = (Cij)T , where C

ijis cofactor of a

ij

(i) A(adj A) = (adjA) A = |A|In

(ii) |adj A| = |A|n1

(iii) (adjAB) = (adjB) (adjA)

(iii) (adj AT) = (adjA)T

(iv) adj(adjA) = |A|n2

(v) (adj KA) = Kn1(adj A)

2. Inverse of a matrix :

(i) A1 exists if A is non singular i.e. |A| 0

(ii) A1 =adjA

A| |, |A| 0

(iii) A1A = In= A A1

(iv) (AT)1 = (A1)T

(v) (A1

)1

= A

(vi) |A1| = |A|1 =1

| |A

(vii) If A & B are invertible square matrices then

(AB)1 = B1 A1

3. Rank of a matrix :

A non zero matrix A is said to have rank r, if

(i) Every square sub matrix of order (r + 1) or more is

singular

(ii) There exists at least one square submatrix of order r

which is non singular.

PAGE # 103 PAGE # 104


4. Homogeneous & non homogeneous system of linear(b) Solution of homogeneous system of linear equa-


53/93

E D U C A T I O N S



equations :

A system of equations Ax = B is called a homogeneous sys-

tem if B = 0. If B 0, then it is called non homogeneous

system equations.

5. (a) Solution of non homogeneous system of linear

equations :

(i) Cramer's rule : Determinant method

The non homogeneous system Ax = B, B 0 of n

equations in n variables is -

Consistent (with unique solution) if |A| 0 and

for each i = 1, 2, ........ n,

xi=

det

det

A

Ai, where A

iis the matrix obtained

from A by replacing ith column with B.

Inconsistent (with no solution) if |A| = 0 and

at least one of the det (A i) is non zero.Consistent (With infinite many solution), if

|A| = 0 and all det (Ai) are zero.

(ii) Matrix method :

The non homogeneous system Ax = B, B 0 of

n equations in n variables is -

Consistent (with unique solution) if |A| 0 i.e.if A is non singular, x = A1 B.

Inconsistent (with no solution), if |A| = 0 and

(adj A) B is a non null matrix.

Consistent (with infinitely many solutions), if

|A| = 0 and (adj A) B is a null matrix.

PAGE # 105 PAGE # 106

(b) Solution of homogeneous system of linear equa

tions :

The homogeneous system Ax = B, B = 0 of n equations

in n variables is

(i) Consistent (with unique solution) if |A| 0 and

for each i = 1, 2, ......... n

xi= 0 is called trivial solution.

(ii) Consistent (with infinitely many solution),

if |A| = 0

(a) |A| = |Ai| = 0 (for determinant method)

(b) |A| = 0, (adj A) B = 0 (for matrix method)

NOTE : A homogeneous system of equations is never

inconsistent.


.UNCTIONProperties :


54/93

E D U C A T I O N S



1. Modulus function :

|x| =

x xx xx

,,

,

> 0

|x| = a x = a

|x| = a no solution

|x| > a x < a or x > a

|x| a a x a

|x| < a No solution.

|x| > a x R

2. Logarithmic unction :

(i) logba to be defined a > 0, b > 0, b 1

(ii) logab = c b = ac

(iii) logab > c

b > ac, a > 1

or b < ac, 0 < a < 1

(iv) logab > log

ac

b > c, if a > 1

or b < c, if 0 < a < 1

(i) loga1 = 0

(ii) logaa = 1

(iii) a a blog = b

if k > 0, k = b bklog

(iv) logab

1+ log

ab

2+ ...... + log

ab

n= log

a(b

1b

2........b

n)

(v) loga

b

c

.HG

IKJ = loga b loga c

(vi) Base change formulae

logab =

log

logc

c

b

a or loga b =1

logb a

(vii) logam

bn =n

mlog

ab

(viii) loga

1

b

.HG

IKJ = loga b = log1/a b

(ix) log1/a

b

c

.HG

IKJ = loga

c

b

.HG

IKJ

(x)a

b clog =c

b alog

3. Greatest Integer function :

f(x) = [x], where [.]denotes greatest integer function equal

or less than x.

i.e., defined as [4.2] = 4, [4.2] = 5

Period of [x] = 1

PAGE # 107 PAGE # 108


Properties :

(i) 1 [ ]

6. Definition :

d b d f h l


55/93

E D U C A T I O N S



(i) x 1 < [x] x

(ii) [x + I] = [x] + I

[x + y] [x] + [y]

(iii) [x] + [x] = 0, x I

= 1, x I

(iv) [x] = I, where I is an integer x [I, I + 1)

(v) [x] I, x [I, )

(vi) [x] I, x ( , I + 1]

(vii) [x] > I, [x] I + 1, x [I + 1, )

(viii) [x] < I, [x] I 1, x ( , I)

4. ractional part function :

f(x) = {x} = difference between number & its integral part

= x [x].

Properties :

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

(ii) {x + I} = {x}

{x + y}

{x} + {y}

(iii) {x} + {x} = 0, x I

= 1, x I

(iv) [{x}] = 0, {{x}} = {x}, {[x]} = 0

5. Signum function :

f(x) = sgn (x) =

=

R

S||T||

+

1

0 0

1

,

,

,

x R

x

x R

or f(x) =| |x

x, x 0

= 0, x = 0

Let A and B be two given sets and if each element a A is

associated with a unique element b B under a rule f, then

this relation (mapping) is called a function.

Graphically - no vertical line should intersect the graph of

the function more than once.

Here set A is called domain and set of all f images of the

elements of A is called range.

i.e., Domain = All possible values of x for which f(x) exists.

Range = or all values of x, all possible values of f(x).

Table :Domain and Range of some standard functions -

unctions Domain Range

Polynomial function R R

Identity function x R R

Constant function K R (K)

Reciprocal function

1

x R0 R0

x2, |x| (modulus function) R R+{x}

x3, x|x| R R

Signum function| |x

xR {-1, 0, 1}

x +|x| R R+{x}

x -|x| R R-{x}

[x] (greatest integer function) R 1

x - {x} R [0, 1]

x [0, ) [0, ]

ax (exponential function) R R+

log x (logarithmic function) R+ R

PAGE # 109 PAGE # 110


Trigonometric Domain Range7. Kinds of functions :

(i) One one (injection) function f : A B is one one if


56/93

E D U C A T I O N S



unctions

sin x R [-1, 1]

cos x R [-1, 1]

tan x R- RST

UVW

2

3

2, ,... R

cot x R- {0, , 2 ,...} R

sec x R -

R

ST

U

VW

2

3

2, ,...

R - (-1,1)

cosec x R- {0, , 2} R - (-1,1)

Inverse Domain Range

Trigo unctions

sin -1 x (-1, 1] LNM OQP

2 2,

cos-1 x [-1,1] [0, ]

tan-1 x R.

HGIKJ

2 2,

cot-1 x R (0, )

sec-1 x R -(-1,1) [0, ]-

2

RSTUVW

cosec-1 x R - (-1,1) LNM

OQP

2 2, - {0}

(i) One-one (injection) function - f : A B is one-one if

f(a) = f(b) a = b

or a b f(a) f(b), a, b A

Graphically-no horizontal line intersects with the graph

of the function more than once.

(ii) Onto function (surjection) - f : A B is onto if

R (f) = B i.e. if to each y B x A s.t. f(x) = y

(iii) Many one function : f : A B is a many one function

if there exist x, y A s.t. x y

but f(x) = f(y)

Graphically - atleast one horizontal line intersects with

the graph of the function more than once.

(iv) Into function : f is said to be into function if R(f) < B

(v) One-one-onto function (Bijective) - A function which

is both one-one and onto is called bijective function.

8. Inverse function : f1 exists iff f is one-one & onto both

f1 : B A, f1(b) = a f(a) = b

9. Transformation of curves :

(i) Replacing x by (x a) entire graph will be shifted parallel

to x-axis with |a| units.If a is +ve it moves towards right.

a is ve it moves toward left.

Similarly if y is replace by (y a), the graph will be

shifted parallel to y-axis,

upward if a is +ve

downward if a is ve.

PAGE # 111 PAGE # 112


57/93


unction Period 15. Composite function :


58/93

E D U C A T I O N S


, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510PAGE # 115 PAGE # 116

sin x, cos x 2

sec x, cosec x

tan x, cot x

sin (x/3) 6

tan 4x /4

cos 2x 1

|cos x|

sin4 x + cos4 x /2

2 cosx .

HGIKJ

36

sin3 x + cos3 x 2/3

sin3 x + cos4 x 2

sinsin

xx5

2

tan2 x cot2 x

x [x] 1

[x] 1

NON PERIODIC UNCTIONS :

x , x2, x3, 5

cos x2

x + sin x

x cos x

cos x

If f : X Y and g : Y Z are two function, then the

composite function of f and g, gof : X Z will be defined as

gof(x) = g(f(x)), x X

In general gof fog

If both f and g are bijective function, then so is gof.


LIMIT

lim

5. Limit of the greatest integer function :

Let c be any real number


59/93

E D U C A T I O N S



1. Limit of a function : limx a f(x) = l (finite quantity)

2. Existence of limit : limx a f(x) exists ifflim

x a f(x) = lim

x a +f(x) =l

3. Indeterminate forms :0

0,

, , 0, 0 , 0 , 1

4. Theorems on limits :

(i) limx a (k f(x)) = klimx a f(x), k is a constant.

(ii) limx a

(f(x) g(x)) = limx a

f(x) Limx a

g(x)

(iii) limx a f(x).g(x) =limx a f(x).

Limx a g(x)

(iv) limx af x

g x

( )

( )=

lim ( )

lim ( )

x a

x a

f x

g x

, provided limx a g(x) 0

(v) limx a f(g(x)) = flim ( )x a

g x

.H

IK, provided value of

g(x) function f(x) is continuous.

(vi) limx a [f(x) + k] = limx a f(x) + k

(vii) limx a log(f(x)) = loglim ( )x a

f x

.H

IK

(viii) limx a (f(x))g(x) = lim ( )

lim ( )

x a

g x

f xx a

LNM

OQP

Case I : If c is not an integer, then limx c [x] = [c]

Case II: If c is an integer, then limx c [x] = c 1, limx c + [x] = c

and limx c

[x] = does not exist

6. Methods of evaluation of limits :

(i) actorisation method : If limx a

f x

g x

( )

( )is of

0

0form

then factorize num. & devo. separately and cancel the

common factor which is participating in making0

0form.

(ii) Rationalization method : If we have fractional powers

on the expression in num, deno or in both, we rationalize

the factor and simplify.

(iii) When x :Divide num. & deno. by the highest power

of x present in the expression and then after removing

the indeterminate form, replace1

x,

12x

,.. by 0.

(iv) limx a

x a

x a

n n

= nan1

(v) By using standard results (limits) :

(a) limx 0 sinx

x= 1 =

limx 0

x

xsin

(b) limx 0 tanx

x= 1 = limx 0

x

xtan

(c) limx 0 sinx = 0

(d) limx 0 cosx =limx 0

1

cosx= 1

PAGE # 117 PAGE # 118


(e) limx 0

sinx

x

0

=

180

(vi) By substitution :

(a) If x a, then we can substitute

+ t t


60/93

E D U C A T I O N S



x 180

(f) limx 0

sin1x

x= 1 = lim

x 0

x

xsin1

(g) limx 0

tan

1x

x= 1 = lim

x 0

x

xtan1

(h) limx 0 a

x

x 1

= logea

(i)limx 0

e

x

x 1

= 1

(j) limx 0

log( )1 + x

x= 1

(k) limx0

log ( )a x

x

1 +=

1

loga

(l) limx 0( )1 1+ x

x

n

= n

(m) limx sinx

x= limx

cosx

x= 0

(n) limx

sin1

1x

x

= 1

(o) limx0

(1 + x)1/x = e = limx

11

+.HG

IKJx

x

(p) limx 0 (1 + ax)1/x = ea = limx 1 +

.HG

IKJ

a

x

x

x = a + t t = x a

If x a, t 0.

(b) When x substitute x = t t

(c) When x substitute t =1

x t 0+

(vii) By using some expansion :

ex = 1 + x +x2

2!+

x3

3!+ .....

e

x

= 1 x +

x2

2!

x3

3! + .....

log(1 + x) = x x

2

2+

x3

3 ......

log(1 x) = x x2

2

x3

3.....

ex ln a = ax = 1 + xlogea +

( log )

!

x ae2

2

+( log )

!

x ae3

3

+ ......

sinx = x

x3

3!+

x5

5!.......

cosx = 1 x

2

2!+

x4

4!......

tanx = x +x3

3+

2

15x5 + .....

(1 + x)n = 1 + nx +n n( )

!

1

2x2 + .....

7. Sandwich Theorem : In the neighbour hood of x = a

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

limx a

f(x) = limx a

h(x) = l, then limx a

g(x) = l.

l < limx a g(x) < l.

PAGE # 119 PAGE # 120


DI..ERENTIATION 2. UNDAMENTAL RULES OR DIERENTIATION :


61/93

E D U C A T I O N S



1. SOME STANDARD DIERENTIATION :

unction Derivative unction Derivative

A cons. (k) 0 xn nxn 1

loga

x1

x aeloglog

ex

1

x

ax ax loge

a ex ex

sin x cos x cos x sin x

tan x sec2 x cot x cosec2 x

cosec x cosec x cot x sec x sec x tan x

sin1 x1

1 2 x,1


62/93

E D U C A T I O N S



i.e if y = un

dy

dx = nun1

du

dx

OR

(viii) Differentiation of composite functions

Suppose a function is given in form of fog(x) or

f[g(x)], then differentiate applying chain rule

i.e.,d

dx

f[g(x)] = f'g(x) . g'(x)

(ix)d

dx

1

u

.HG

IKJ =

12u

du

dx, u 0

(x)d

dx|u| =

u

u| |

du

dx, u 0

(xi) Logarithmic Differentiation : If a functionis in the

form (f(x))g(x) orf x f x

g x g x1 2

1 2

( ) ( )....

( ) ( ).... We first take log on

both sides and then differentiate.

(a) loge

(mn) = logem + log

en

(b) logem

n= logm log

en

(c) loge(m)n = nlog

em (d) log

nm log

mn = 1

(e) logan

xm =m

nlog

ax (f) aloga x = x

(g) logee = 1 (h) log

nm =

log

loge

e

m

n

dy

dx

at one side and finddy

dx

.

[The relation f(x, y) = 0 in which y is not expressible

explicitly in terms of x are called implicit functions]

(xiii) Differentiation of parametric functions : If x = f(t)

and y = g(t), where t is a parameter, then

dy

dx=

dy

dtdx

dt

=g t

f t

'( )

'( )

(xiv) Differentiation of a function w.r.t. another func-

tion : Let y = f(x) and z = g(x), then differentiationof y w.r.t. z is

dy

dz=

dy dx

dz dx

/

/=

f x

g x

'( )

'( )

(xv) Differentiation of inverse Trigonometric functions

using Trigonometrical Transformation : To sol

55099833 math formula sheet aieee

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