Limits and Derivatives

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

x 0 e - (1 + 2x) 12xx is equal to

1. -2e
2. e - e2
3. 0
4. e

The value of x 0 2 ( 1 - x 2x [3] 3x [10] 10xx2 ) is .

If = x 0+ ( e x - e x x - x ) and = x 0(1 + x) 12 x are the roots of the quadratic equation ax2 + bx - e = 0, then 12 e(a + b) is equal to .

Let a > 0 be a root of the equation 2x2 + x - 2 = 0. If x 1a 16 (1 - (2 + x - 2x2 ) )(1 - ax)2 = + 17, where , Z, then + is equal to .

If x 1 (5x+1)1/3-(x+5)1/3(2x+3)1/2-(x+4)1/2= m 5n(2n)2/3, where (m, n)=1, then 8m+12n is equal to .

Let f(x) = 0x (t + (1 - et ) )dt, x R. Then, x 0 f(x)x3 is equal to

1. 16
2. - 16
3. 23
4. - 23

Let \x\ denote the fractional part of x and f(x)= -1 (1-\x\2 ) -1(1-\x\)\x\-\x\3, x 0. If L and R respectively denotes the left hand limit and the right hand limit of f(x) at x=0, then 32 2 (L2+R2 ) is equal to .