Limits and Derivatives

The value of x 0 ( x2 2 xx2- 2 x ) is :

1. 2
2. 3
3. 4
4. 6

Let x 2 ( (x-2) ) (rx2+(p-2)x-2p )(x-2)2=5 for some r, p R. If the set of all possible values of q, such that the roots of the equation rx2-px+q=0 lie in (0, 2), be the interval ( , ], then 4( + ) equals :

1. 11
2. 13
3. 17
4. 21

The product of all possible values of , for which x 0 ( 1- ( x) (( +1)x) (( +2)x) 2(( +1)x) )=2, is :

1. -2
2. 1
3. -1
4. 54

Let f(x)= y 0 (1- (xy)) (xy)y3. Then the number of solutions of the equation f(x)= x, x R is :

1. 0
2. 2
3. 3
4. 1

If x 2 (x3-5x2+ax+b ) ( x-1-1 ) e(x-1)=m, then a+b+m is equal to :

1. 5
2. 6
3. 8
4. 10

The value of x 0 e ( (ex) (e2x) (e10x) )e2 - e2 x is equal to

1. (e10 - 1)2e2(e2 - 1)
2. (e20 - 1)2e2(e2 - 1)
3. (e10 - 1)2(e2 - 1)
4. (e20 - 1)2(e2 - 1)

If the function f(x) = ex (e x - x - 1 ) + e( x + x) - x x - x is continuous at x = 0, then the value of f(0) is equal to

1. 23
2. 12
3. 2
4. 32

If x 0 e(a - 1)x + 2 bx + (c - 2)e-xx x - e(1 + x) = 2, then a2 + b2 + c2 is equal to :

1. 3
2. 5
3. 9
4. 7

Let f : R (0, ) be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f''(3) = 4. Then x 1 e ( f(2 + x)f(3) ) 18(x - 1)2 is equal to :

1. 9
2. 18
3. 1
4. 2

Let f(x) = x3 + x2 f'(1) + 2x f''(2) + f'''(3), x R. Then the value of f'(5) is :

1. 6575
2. 1175
3. 25
4. 625

Given below are two statements: In the light of the above statements, choose the correct answer from the options given below:

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

x 0+ (5(x) 13 ) e (1 + 3x2 ) ( -13 x )2 (e5(x) 43 - 1 ) is equal to

1. 53
2. 1
3. 13
4. 115

If x 1+ (x - 1)(6 + (x - 1)) + (1 - x)(x - 1)3 = -1, where , R, then + is equal to

1. 20
2. 19
3. 18
4. 17

Let f be a differentiable function on R such that f(2)=1, f'(2)=4. Let x 0(f(2+x))3/x=e . Then the number of times the curve y=4x3-4x2-4( -7)x- meets x-axis is :

1. 3
2. 1
3. 2
4. 0

If x 0 ( xx ) 1x2 = p, then 96 e p is equal to

For , , R, if x 0 x2 x + ( - 1)ex2 2x - x = 3, then + - is equal to :

1. -1
2. 4
3. 6
4. 7

If x 0 (2x) + a (4x) - bx4 is finite, then (a + b) is equal to:

1. 0
2. 34
3. -1
4. 12

Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which x 0+ (x ( [ 1x ] + [ 2x ] + + [ px ] ) - x2 ( [ 1x2 ] + [ 22x2 ] + + [ 92x2 ] ) ) 1 is equal to .

Let f(x) = n r=0n ( (x/2r+1 ) + 3 (x/2r+1 )1 - 2 (x/2r+1 ) ). Then x 0 ex - ef(x)(x - f(x)) is equal to .

x 0 cosecx ( 2 2x+3 x- 2x+ x+4 ) is

1. 1 15
2. 12 5
3. 0
4. - 12 5

x (2x2-3x+5 )(3x-1) x2 (3x2+5x+4 ) (3x+2)x is equal to :

1. 2e3
2. 23 e
3. 2e 3
4. 2 3e

If x ( ( e1-e ) ( 1e- x1+x ) )x= , then the value of e 1+ e equals :

1. e-2
2. e2
3. e
4. e-1