37663122 Math Formula Sheet AIEEE

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Transcript of 37663122 Math Formula Sheet AIEEE

MATHS FORMULA - POCKET BOOK

MATHS FORMULA - POCKET BOOK 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 +

QUADRATIC EQUATION & EXPRESSION1. Quadratic expression : A polynomial of degree two of the form ax2 + bx + c, a 0 is called 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 = 3.b + b2 4ac 2a

, then other root

5.

Sum of roots : S=+=

and

=

b b2 4ac 2a

Coefficient of x b = Coefficient of x2 a

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) (ii) (iii) If D > 0 If D = 0 If D < 0

Product of roots : P = =

cons tant term c = Coefficient of x2 a

roots are real and distinct (unequal) roots are real and equal (Coincident)

6.

.ormation of an equation with given roots : x2 Sx + P = 0

roots are imaginary and unequal i.e. non real complex numbers. Suppose a, b, c Q a 0 then 7.

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

(i) (ii)

If D > 0 and D is a perfect square roots are rational & unequal If D > 0 and D is not a perfect square roots are irrational and unequal.

Roots under particular cases : .or the equation ax2 + bx + c = 0, a 0 (i) (ii) (iii) If b = 0 roots are of equal magnitude but of opposite sign. If c = 0 one root is zero and other is b/a If b = c = 0 both roots are zero

.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 # 1E D U C A T

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

, 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, 2405510I O N S

PAGE # 2

MATHS FORMULA - POCKET BOOK (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 roots are +ve. 8. Symmetric function of the roots : If roots of quadratic equation ax2 + bx + c, a 0 are and , then (i) ( ) =

MATHS FORMULA - POCKET BOOK (vi) 4 + 4 = (2 + 2)2 222

={( + )2 2}2 222

.b =G H

2

2ac a2

I J K

2

2c2 a2

both(vii) 4 4 =(2 + 2) (2 2) =

b(b2 2ac) b2 4ac a4 b2 + ac a2

(viii) 2 + + 2 = ( + )2 =

( + ) 4 =

2

b2 4ac a

(ix)

2 + 2 ( + )2 2 + = =

(ii)

2 + 2 = ( + )2 2 =

b2 2ac a2 b b2 4ac a2

(x)

.G IJ H K

2

+

. I G J H K

2

=

4 + 4 [(b2 2ac)2 2a2c2 ] = 2 2 a2c 2

(iii)

2 2 = ( + )

( + ) 4 =

2

9.

Condition for common roots : The equations a1 x2 + b1 x + c1 = 0 and a2x2 + b2x + c2 = 0 have (i) One common root if

b(b2 3ac) (iv) 3 + 3 = ( + )3 3( + ) = a3(v) 3 3 = ( ) [2+ 2 ] =

b1c2 b2c1 c1a2 c2a1 c1a2 c2a1 = a1b2 a2b1

( + )2 4 [2+ 2 ] (b2 ac) b2 4ac a3(ii)

a1 b1 c1 Both roots common if a = b = c 2 2 2

=

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

PAGE # 3E D U C A T I O N S

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

PAGE # 4

MATHS FORMULA - POCKET BOOK 10. Maximum and Minimum value of quadratic expression :

MATHS FORMULA - POCKET BOOK (v) If both the roots lies in the interval (k1, k2) D 0, a.f(k1) > 0, a.f(k2) > 0, k1 < (vi) If k1, k2 lies between the roots

In a quadratic expression ax Where D = b2 4ac (i) If a > 0, quadratic

2

L. b I + bx + c = a MG x + 2a J K MNH

2

D 4a2 ,

OP PQ

b < k2 2a

expression has minimum value

a.f(k1) < 0, a.f(k2) < 0 (vii) will be the repeated root of f(x) = 0 if f() = 0 and f'() = 0 12. .or cubic equation ax3 + bx2 + cx + d = 0 : We have + + =b c d , + + = and = a a a

b 4ac b2 at x = and there is no maximum value. 2a 4a(ii) If a < 0, quadratic expression has maximum valueb 4ac b2 at x = and there is no minimum value. 2a 4a

11. Location of roots : Let f(x) = ax2 + bx + c, a 0 then w.r.to f(x) = 0 (i) (ii) If k lies between the roots then a.f(k) < 0 (necessary & sufficient) If between k1 & k2 their is exactly one root of k1, k2 themselves are not roots f(k1) . f(k2) < 0 (iii) (necessary & sufficient) If both the roots are less than a number k D 0, a.f(k) > 0,

where , , are its roots. 13. .or biquadratic equation ax4 + bx3 + cx2 + dx + e = 0 : We have + + + =

d b , + + + = a ac e and = a a

+ + + + + =

b 0,b >k 2a

(necessary & sufficient)

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

PAGE # 5E D U C A T I O N S

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

PAGE # 6

MATHS FORMULA - POCKET BOOK

MATHS FORMULA - POCKET BOOK * * * * * z z = 2i Im(z) = purely imaginary z z = |z|2

COMPLEX NUMBER1. Complex Number : A number of the form z = x + iy (x, y R, i =1 ) is called a complex number, where x is called a real part i.e. x = Re(z) and y is called an imaginary part i.e. y = Im(z).

z1 + z2 +....+zn = z 1 + z 2 + .......... + z n z1 z2 = z 1 z 2 z1z2 = z 1 z 2

Modulus |z| =

x2 + y2 ,

amplitude or amp(z) = arg(z) = = tan1 (i) Polar representation : x = r cos, y = r sin, r = |z| = (ii) Exponential form : z = rei , where r = |z|, = amp.(z) (iii) Vector representation :

y . x

* *

.G z IJ Hz K1 2

=

.z I Gz J H K1 2

(provided z2 0)

ez jn

= ( z )n = z

x2 + y2

* *

c zh

If = f(z), then = f( z ) Where = f(z) is a function in a complex variable with real coefficients.

P(x, y) then its vector representation is z = OP 2. Integral Power of lota : i=2 3 4 1 , i = 1, i = i , i = 1

* * 4.

z + z = 0 or z = z z = 0 or z is purely imaginary z= z

z is purely real

Modulus of a complex number : Magnitude of a complex number z is denoted as |z| and is defined as |z| = (i) (ii) (iii)

Hence i4n+1 = i, i4n+2 = 1, i4n+3 = i, i4n or i4(n+1) = 1 3. Complex conjugate of z : If z = x + iy, then z = x iy is called complex conjugate of z * * *E D U C A T I O N S

(Re(z))2 + (Im(z))2 , |z| 0

z z = |z|2 = | z |2 z1 =

z is the mirror image of z in the real axis.|z| = | z | z + z = 2Re(z) = purely real, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 7E D U C A T

z |z|2

|z1 z2|2 = |z1|2 + |z2|2 2 Re (z1 z2 )PAGE # 8

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510I O N S

MATHS FORMULA - POCKET BOOK (iv) |z1 + z2|2 + |z1 z2|2 = 2 [|z1|2 + |z2|2] (v) |z1 z2| |z1| + |z2| (vi) |z1 z2| |z1| |z2| 5. Argument of a complex number : Argument of a complex number z is the made by its radius vector with +ve direction of real axis. arg z = , = , (i) (ii) (iii) z 1st quad. z 3rd quad.th

MATHS FORMULA - POCKET BOOK 6. Square root of a complex no.a + ib =

L M M N L M M N

|z|+a |z|a +i , for b > 0 2 2 |z|+a |z|a i , for b < 0 2 2

O P P Q

=

O P P Q

= , z 2nd quad. = , z 4 quad. arg (any real + ve no.) = 0 arg (any real ve no.) = arg (z z ) = /2

7.

De-Moiver's Theorem : It states that if n is rational number, then (cos + isin)n = cos + isin n and (cos + isin)n = cos n i sin n

8.

Euler's formulae as z = rei, where ei = cos + isin and ei = cos i sin

(iv) arg (z1.z2) = arg z1 + arg z2 + 2 k (v) arg

ei + ei = 2cos and ei ei = 2 isin

.G z I Hz J K1 2

= arg z1 arg z2 + 2 k

9.

nth roots of complex number z1/n = r1/n cos

(vi) arg ( z ) = arg z = arg

. 1I G zJ H K

, if z is non real (i) (ii)

L . 2m + I + i sin. 2m + I O M G n J G n JP , K H KQ N H

= arg z, if z is real (vii) arg ( z) = arg z + , arg z ( , 0] = arg z , arg z (0, ] (viii) arg (zn) = n arg z + 2 k (ix) arg z + arg z = 0 argument function behaves like log function.

where m = 0, 1, 2, ......(n 1) Sum of all roots of z1/n is always equal to zero Product of all roots of z1/n = (1)n1 z

10. Cube root of unity : cube roots of unity are 1, , 2 where =

1 + i 3 and 1 + + 2 = 0, 3 = 1 2PAGE # 10

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

PAGE # 9E D U C A T I O N S

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

MATHS FORMULA - POCKET BOOK 11. Some important result : If z = cos + isin (i) z+

MATHS FORMULA - POCKET BOOK

or

z z1 z z1 z z2 + z z2 = 0z z1 + z 2 2

1 = 2cos z1 = 2 isin z 1 = 2cosn zn

or or

=

|z1 z 2 | 2

(ii)

z

|z z1|2 + |z z2|2 = |z1 z2|2

(iii)

zn +

Where z1, z2 are end points of diameter and z is any point on circle. 13. Some important points : (i) (ii) (b) yz + zx + xy = 0 (d) x + y + z = 3xyz3 3 3

(iv) If x = cos + isin , y = cos + i sin & z = cos + isin and given x + y + z = 0, then (a) (c)

Distance formula PQ