Evaluate

a)

b)

c)

d) 4f(2)

Let  be a differentiable function having . Then,   is equal to

a) 18

b) 12

c) 36

d) 24

, then  equals

a)

b) 1

c)

d) 0

Suppose the cubic x³ – px + q has three distinct real roots when p > 0 and q > 0 then which one of the following is true?

a) The cubic has maximum at both  and  

b) The cubic has minimum at  and maximum at

c) The cubic has minimum at  and maximum at

d) The cubic has minimum at both and at

How many real solutions does the equation
 have
a) 5

b) 7

c) 1

d) 3

The value of c for which the conclusion of mean value theorem holds for the function f(x) = ln_ex on the interval [1, 3] is

a)  

b)  

c)  

d)  

The function f(x) = tan^{− 1} (sinx + cosx) is an increasing function in

a)

b)

c)

d)

Let f : R → R be a function defined by f(x) = min(x + 1, |x| + 1) then which of the following is true?

a) f(x) ≥ 1 for all x  R

b) f(x) is not differentiable at x = 1

c) f(x) is differentiable everywhere

d) f(x) is not differentiable at x = 0

The function f : R – {0} → R given by  can be made continuous by defining f(0) as

a) 2

b) -1

c) 0

d) 1

The function   has a local minimum at

a) x = − 2

b) x = 0

c) x = 1

d) x = 2

Angle between the tangents to the curve y = x² – 5x + 6 at the points (2, 0) and (3, 0) is

a)

b)

c)

d)

If x is real the maximum value of  is

a) 41

b) 1

c)

d)

If x^m yⁿ = (x + y)^{m + n}  then  is

a)

b) xy

c)

d)

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

Suppose f(x) is differentiable at x = 1 and
 

a) 6

b) 5

c) 4

d) 3

If f is a real differentiable function satisfying |f(x) – f(y)| ≤ (x – y)², x, y  R and f(0) = 0, then f(1) equals

a) 1

b) 2

c) 0

d) – 1

The normal to the curve x = a(cosθ + θsinθ), y = a(sinθ – θcosθ) at any point θ is such that

a) It is a constant distance from the origin

b) It passes through

c) It makes angle  with x – axis

d) It passes through the origin

Let α, β be the distinct roots of ax² + bx + c = 0 then  is equal to

a)

b)

c)

d)

A spherical iron ball 10cm in radius is coated with a layer of ice of uniform thickness that melts at the rate of 50 cm³/min. When the thickness of ice is 15 cm than the rate at which the thickness of the ice decreases, is

a)

b)

c)

d)

If  the difference between the maximum and minimum value of u² is given by

a)

b)

c)

d)

If  then the values of a, b are

a) a  R, b  R

b) a = 1, b  R

c) a  R, b = 2

d) a = 1, b = 2

Let . If f(x) is continuous in  then  is

a) 1

b)

c)

d) 1

The normal to the curve x = a (1 + cosθ), y = asinθ at θ always passes through the fixed point

a) (a, 0)

b) (0, a)

c) (0, 0)

d) (a, a)