must see

These are copy pasted guys.I think u may found it useful .

a)If A is an orthogonal matrix , then A-1 equals At (t - transpose)

b)If A,B are 2 square matrices , such that A.B = A & B.A = B , then A & B both are independent or idempotent matrix...

c) The inverse of a symmetric matrix is a symm. matrix , that of a diagonal matrix is a diagonal matrix

d)If A,B are symm matrix of same order , then AB - BA is a skew symm matrix....

e)Trace of a matrix = sum of elements of primary diagonal...

f)Trace of a skew symm matrix is zero

g)rank of a null matrix doesnt exist....

h)If A is an invertible matrix , then

|A-1| = 1 / |A|

2)whenever , roots of polynomial are distinct ,

f(0)*f(1) < 0

this is knwn as DISCART's method....

like in qs...

x3 - 3x + a = 0 has distinct roots in (0,1) , then find "a"...

well we cant find the exact value , but we can find the range....

a(a-2) < 0,

so 0 < a < 2....

1) To find the greatest term in the expansion of (1 + x)n

i) calculate .. [ |x| (n+1)] / |x| + 1

if m comes out to be an integer then , Tm & Tm+1 are equal and both are greatest term

if m is nt an integer , then T[m+1] is the greatest term
, where [.] is the greatest integral part

take and example

greatest term in (2 + 3x)9 wen x = 3/2 is....

now just calculate the value

(2 + 3x)9 = 29 [1 + 3x/2]9

as x = 3/2

= [1 + (9/4)]9calculate m now

m = |9/4| (9+1) / [|9/4) + 1]

m = 90/13

wich is nt an integer

so greatest term = Tm+1 = T6+1 = T7....

2) greatest term in the expansion (x + y)n= ( 1 + (y/x))n

3)If n is even , greatest coeff = nCn/2

if n is odd , greatest coeff are , nC(n-1)/2 & nC(n+1)/2

4)The sum of binomial coeff in the expansion (1 + x)n is 2n

5)The sum of coeff of odd terms in the exp (1 + x)n is = to sum of coeff of even terms and each = to 2n-1

6)the coeff of a1n1, a2n2............amnm in the exp of ( a1 + a2 + .......am)n is
= n! / n1! n2!....nm!

7)If (1 + x)n = Co + C1x + C2x2 + .......... + Cnxn....

u can use integration also here

i) if sum contains Co , C1 ,C2...........Cn are all +ive signs , then integrate b/w limits 0 to 1

ii)If sum contains alternative signs ( + and -) then integrate b/w limits 0 and -1

iii) if sum contains odd coeff Co C2...etc then integrate b/w -1 to +1

iv) if sum contains even coeff C1 C3...then subtracting (ii) frm (i) and then dividing by 2

some imp. pts abt quadratic equations.....

1) Remember the foll. points while solving a quadratic equation..

a) x2 + y2 = (x+y)2 - 2xy

b) x3 + y3 = (x+y)3 - 3xy(x+y)

c) x4 + y4 = (x2+y2)2 - 2x2y2

d) x5 + y5 = (x2 + y2)(x3 + y3) - x2y2(x+y)

e) x - y = ( (x+y)2 - 4xy)1/2

2) If ax2 + bx + c is satisfied by more than 2 values , its an identity and a = b = c = 0

3)Common roots ->

if ax2 + bx + c = 0 & px2 + qx + r = 0 have a common root(@) , then

@2/br - qc = @/pc - ar = 1/aq - bp

eliminating @ , we get

(pc - ar)2 = (br - qc)(aq - pb)..

b)The equations , x2 + ax + b = 0 and x2 + bx + a = 0 have a common root if a + b + 1 = 0

c)By common root we means only 1 root is common , but if both the roots are common , then co-eff. of like terms are propotional..
p/a = q/b = r/c (equations are same i.e on pt a))

3)The roots of ax2 + bx + c = 0 are both = in magnitude but opposite sign if b=0

4)Roots are = to zero , if b = 0 , c = 0

roots are = to infinity , if b = 0 , a = 0

5)Roots are opposite in sign , if a and c are of opp. sign and roots are reciprocal of each other if a = c...

6)Both roots are +ive , if a and c have same sign and opp. to that of b

7) a)An equation of degree n has n roots , real or imaginary...

b)Surds and imaginary roots always occurs in pairs , i.e if p + iq is a root then p - iq is also a root

c)An odd degree equation has atleast 1 real root , whose sign is opp. to that of its last term provided that coeff of highest degree term is +ive

d)Every equation of an even degree whose constant term is -ive and highest degree term is +ive has atleast 2 real roots , 1 +ive and 1 -ive

e)If all terms of an equation are +ive and the equation involves odd no. of
powers of x , then its all roots are complex...

8)If a > 0 , then the min. value of ax2 + bx + c is (4ac - b2)/ 4a

b)If a < 0 , then max. value of ax2 + bx + c is (4ac - b2)/4a

9)The condition that 1 root of ax2 + bx + c is m times the other...is

mb2 = ac (m+1)2

10)If 1 root of ax2 + bx + c is square of another , then

b3 + a2c + ac2 = 3abc

11)If the roots of the equation ax2 + bx + c are in ratio m : n , then

mnb2 = (m+n)2 ac

z = a + ib its sq root = +- [( (| Z | + a )/ 2) + i ( ( | Z | - a) / 2)]...(if b > 0)

if b < 0 then its -ive sign before i

2) | z - z1|/ |z - z2| = k

then locus is a circle if k =/= 1

and its a straight line if k = 1

3)|z - z1| - |z - z2| = 2a , then locus =

(i) ellipse , if 2a > |z1 - z2|

(ii) hyperbola , if 2a < |z1 - z2|

4)in qs like..

|z - z1| + |z - z2| = constant

and coeff of z on both mods is 1 then for sure its locus is an ellipse

and if coeff are different , then it will be parabola for sure