Limits and Derivatives

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 .

Let f(x)= casesx-1, & x is even,\ 2x, & x is odd, cases x . If for some a , f(f(f(a)))=21, then x - \ |x|3a- [ xa ] \, where [t] denotes the greatest integer less than or equal to t, is equal to :

1. 169
2. 121
3. 225
4. 144

x 0 e2| x|-2| x|-1x2

1. is equal to 1
2. does not exist
3. is equal to -1
4. is equal to 2

Let f: R (0, ) be a strictly increasing function such that x f(7x)f(x)=1. Then, the value of x [ f(5x)f(x)-1 ] is equal to

1. 0
2. 4
3. 1
4. 7/5

If x 0 ax2ex-b e(1+x)+cxe-xx2 x=1, then 16 (a2+b2+c2 ) is equal to .

Let f: [- 2, 2 ] be a differentiable function such that f(0)= 12. If the x 0 x 0x f(t) dtex2-1= , then 8 2 is equal to :

1. 4
2. 2
3. 1
4. 16