If f(x) = xⁿ then the value of
  is

a) 2ⁿ

b) 2^{n – 1}

c) 0

d) 1

 is

a)

b) 0

c)

d) ∞

If  then the value of k is

a) 0

b)

c)

d)

Let f(a) = g(a) = k and their n^th derivatives f⁽ⁿ⁾(a), g⁽ⁿ⁾(a) exists and are not equal for some n. Further if  then k =

a) 4

b) 2

c) 1

d) 0

If   then f(x) is

a) Continuous as well as differentiable for all x

b) Continuous for all x but not differentiable at x = 0

c) Neither continuous nor differentiable at x = 0

d) Discontinuous everywhere

The function f(x) = 2x³ – 9ax² + 12a²x + 1 where a > 0 attains its maximum and minimum at p and q respectively such that p² = q then a equals

a) 3

b) 1

c) 2

d)

If  then  is

a) n²y

b) – n²y

c) – y

d) 2x²y

 is equal to

a) e⁴

b) e²

c) e³

d) e

If siny = xsin(a + y), then  is

a)

b)

c) sina sin²(a + y)

d)

If x^y = e^{x – y}, then  is

a)

b)

c) Not defined

d)  

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ – 2 = 0

a) Cut at right angles

b) Touch each other

c) Cut at an angle of

d) Cut at an angle of

The function f(x) = cot⁻¹x + x increases in the interval

a) (1, ∞)

b) (−1, ∞)

c) (−∞, ∞)

d) (0, ∞)

The greatest value of f(x) = (x + 1)^1/3 – (x – 1)^1/3 on [0, 1] is

a) 1

b) 2

c) 3

d) 1/3

For x ε R,  is equal to

a) e

b) e⁻¹

c) e⁻⁵

d) e⁵

Let f(2) = 4 and  then  is given by

a) 2

b) – 2

c) – 4

d) 3

If α, β be the roots of the equation ax² + bx + c = 0 then  is

a) ln|a(α – β)|

b) e^{a(β – α)}

c) e^{a(α – β)}

d) None of these

If  then the value of  is

a)

b)

c)

d)

If  then  is equal to

a) – 1

b) 1

c) ln2

d) – ln2

If  then  is

a) – 2

b) – 1

c) 2

d) 1

Statement 1 :  is decreasing in x  (−∞, 5) U (5, ∞)

Statement 2 :  for all x ≠ 5

a) Statement 1 is false, Statement 2 is true

b) Statement 1 is true, Statement 2 is true. Statement 2 is a correct explanation of Statement 1.

c) Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation of Statement 1.

d) Statement 1 is true, Statement 2 is false

The values of constants a, b so that  is

a) a = 0, b = 0

b) a = 2, b = −1

c) a = −1, b = 1

d) a = 1, b = −1

Let f(x) = (x+|x|) |x| then for all x

a) f  is not continuous

b)  is differentiable for all x

c)  is continuous

d) None ofthe above

If  then the value of    is

a) 0

b) 1

c) −1

d) 2