8/2/2019 AIEEE Maths ShortcutFormulae

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AIEEE MATHEMATICS SHORTCUT FORMULAE

If

If

(for a > 0, a 1)

If lx + my + n = 0 is a tangent to the circle x2+y2+ 2gx + 2fy + c = 0

then (lg+mf-n)2=

(l2+m2) (g2+f2-c)

If the circle x2+y2+ 2gx + 2fy + c = 0 touches X - axis then g2= c.

If the circle x2+y2+ 2gx + 2fy + c = 0 touches Y-axis then f2=c.

If the circle x2+y2+ 2gx + 2fy + c = 0 touches both the coordinate axes

the g2= f2=c.

Let 'C' is centre, r is radius of the circle

S = 0, the points P and Q are said to be inverse points with respect S =

0 if (i) C, P, Q are collinear, (ii) P, Q lies on the same side of C (iii) CP.

CQ =r2.

General solution of Sinis n(-1)n if principle value is

General solution of tanis n+ if principle value is

General solution of Cosis 2n if principle value is General solution of if Sin= 0 is =n General solution of if Cos=0 is (2n+1)/2

Number of terms in the expansion(x+a)n is n+1

Number of terms in the expansion (x1+x2+...+xr)n is n+r-1Cr-1

In

For independent te

In above , the term containing xs is

(1 + x)n - 1 is divisible by x and

(1 + x)n - nx -1 is divisible by x2.

Coefficient of xn in (x+1)(x+2)... (x+n)=n

Number of ways of arranging 'r' things out of 'n' different things =nPr Number of ways of arranging 'n' different things in circular manner =

(n-1)!

For a number 2a3b5c

1) Number of divisors = (a+1) (b +1) (c +1)

2) Number of divisors excluding unity =

(a +1) (b +1) (c +1) -1

3) Excluding number & unity =

(a+1) (b +1) (c +1) - 2

i.e proper divisors or non-trival solutions

4) Odd divisors = (b +1) (c +1)

5) Even divisors=

total divisors - Odd divisors

6) Sum of divisors =

7) Sum of even divisors = sum of total divisors - sum of odd divisors

Number of points of intersections of circles out of 'n' circles =nC2x2

Number of points of intersections of lines out of 'n' lines =nC2x1

Number of points of intersections with 'n' circles and 'm' straight lines

= nC1xmC1x2 Number of lines formed out of n points in a plane out of which 'm' are

collinear isnC

2- mC

2+1

Number of triangles formed out of 'n' points out of which 'm' points are

collinear isnC3-

mC3 Number of diagnols in a polygon of 'n' sides

is =

Number of triangles whose vertices are vertices of an polygon, but

none of whose sides happen to be sides of polygon =

If 'm' parallel lines intersected by 'n' other lines, the number of paral-

lelograms formed = mC2xnC

2

Number of rectangles in a chess board =9c

2x 9c

2= 1296

Number of Squares in a chess board =

Number of rectangles which are not Squares = 1296 - 204 = 1092

The Number of ways of selecting the things among 'a' bananas, 'b'

apples 'c' mangoes is (a+1)(b+1)(c+1)

28 204=

1( 4)( 5)

6 n n n

( 3)

2

n n

1 1 12 1 3 1 5 1

2 1 3 1 5 1

+ + +

a b c

1

1

= +r

r

n

n

P

P

n r

1

++

np s

p q

1++np

p q

+p

q

bax

x

( ) 1 1, + ++ =n rr

T n rx a

T r

1 1 1 1tan

1

+ + + + =

x y z xyzTan x Tan y Tan z

xy yz zx

2( )( ) ( )

8 =

b

a

x a b x dx b a

4

0

log(1 cot ) log 28

+ = d

42

0

1[tan tan ]

1

+ =

n nx x dx

n

( ) ( ) ( ) ( )2

+= + =

b b

a a

a bxf x dx f x dx if f a b x f x

2

00

2 ( ) if f(a-x)=f(x)( )

0 if f(a-x)= f(x)

=

a

af x dx

f x dx

1 1log

( 1) 1= +

+ +n

n n

xdx c

x x n x

1 1

2 2

1 1 1tan tan

+ x x

cb a a a b b

2 2 2 2

1

( )( )=

+ + dxx a x b

{ } ( )1

1 { ( )}( ) . ( ) 1

1

+

= + +

nn f x

f x f x dx C nn

1( )log ( )

( )= +

f xdx f x C

f x

( ) ( )1 ++ = +f ax b

f ax b dx Ca

log= +

xx a

a dx C a

. = +x x

e dx e C

1log= + dx x C

x

1

1,1

+

= ++n

n xn x dx C n

, = +K R Kdx Kx C