Let ℝ be the set of real numbers and f : ℝ → ℝ such that for all x and y in ℝ, . Then f (x) is a constant.

a) True

b) False

A function f : ℝ → ℝ satisfies the equation

f(x + y) = f(x) . f(y)  x, y in ℝ and f(x) ≠ 0 for any x in ℝ. Let the function be differentiable at x = 0 and . Show that. Hence determine f(x).

a) e^x

b) e^2x

c) 2e^x

d) 2e^2x

Find

a) 0

b) e

c) e^z

d) e³

Let

Determine a and b so that f is continuous at x = 0.

a)

b)

c)

d)

Let

Test whether

f(x) is continuous at x = 0

f(x) is differentiable at x = 0

a) f(x) is differentiable and continuous at x = 0

b) f(x) is continuous but not differentiable at x = 0

c) f(x) is neither continuous nor differentiable at x = 0

If a function f : is an odd function such that  for x ε [a, 2a] and the left hand derivative at

x = a is 0 then find the left hand derivative at x =  

a) 0

b) 1

c) a

d) 2a

P(x) is a polynomial function such that P(1) = 0, > P(x)

 x > 1. Then  x > 1,

a) P(x) > 0

b) P(x) = 0

c) P(x) < 1

If  and = and f(0) = 0. Find the value of . Given that 0 < <

a)

b)

c)

d) 1

f(x) is a function such that  and the tangent at any point passes through (1, 2). Find the equation of the tangent.

a) x = 2

b) y = 2

c) x + y = 2

d) x – y = 2

If  exists then both the limits  and  exist

a) True

b) False

A =  is equal to

a) 0

b) 1

c)

d)

Let f(x) =

If f is continuous for all x, then k is equal to

a) 3

b) 5

c) 7

d) 9

Identify a discontinuous function y = f(x) satisfying  

For the function

The derivative from right  .  .  .  . and the derivative from the left  .  .  .  .

a) 0, 0

b) 0, 1

c) 1, 0

d) 1, 1

 

 

Then

a) 0

b) 1

c) 2

d) 4

L =  = .  .  .  .

a) – 1

b) 0

c) 1

d) 2

If f(9) = 9,  then  equals

a) 0

b) 1

c) 2

d) 4

a) 0

b) 1

c) e³

d) e⁵

Let f(x) = x|x|. The set of points where f(x) is twice differentiable is .  .  .  .

a) ℝ

b) 0

c) ℝ − {0, 1}

a) 0

b) 1

c) e

d) e²

a) – 1

b) 0

c) 1

d) 2

Match the following

Let the function defined in column 1 has domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

The line y = x meets y =  for k ≤ 0 at

a) No point

b) One point

c) Two points

d) More than two points