Limits and Derivatives

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