Question(s) from Search: IIT

If f is a continuous function with  as |x| → ∞ then show that every line y = mx intersects the curve .

Let f : ℝ → ℝ be any function. Define g : ℝ → ℝ by g (x) = |f (x)| for all x. Then g is

a) Onto if f is onto

b) One to one if f is one to one

c) Continuous if f is continuous

d) Differentiable if f is differentiable

Evaluate

a)

b)

c)

d)

Let f : ℝ → ℝ be a function defined by f (x) =  . The set of points where f (x) is not differentiable is

a) }

b)

c) {0, 1}

d)

If f is a differentiable function satisfying  for all n ≥ 1,

n  I then

a)

b)

c)

d)  is not necessarily zero

Evaluate

Multiple choices

The function f (x) = 1 + |sinx| is

a) continuous nowhere

b) continuous everywhere

c) differentiable nowhere

d) not differentiable at x = 0

e) not differentiable at infinite number of points

Let f (x) be continuous and g (x) be a discontinuous function. Prove that f (x) + g (x) is a discontinuous function.

a) True

b) False

c) Could be continuous or discontinuous

Draw the graph of the function y = [x] + |1 – x|, – 1 ≤ x ≤ 3. Determine the points, if any, where the function is not differentiable.

a) y is differentiable everywhere

b) y is not differentiable at x = 0

c) y is not differentiable at x = 0, 1, 2

d) y is not differentiable at x = 0, 1, 2 and 3

In how many ways can a pack of 52 cards be divided in 4 sets, three of them having 17 cards each and fourth just one card.

Let f(x), x ≥ 0 be a non-negative function and let F(x) = . For some c > 0, f(x) ≤ cF(x) for all x ≥ 0. Then for all x ≥ 0, f(x) =

a) 0

b) 1

c) 2

d) 4

In a certain test  students gave wrong answers to at least i questions where i = 1, 2, …, k. No student gave more than k correct answers. Total number of wrong answers given is .  .  .

If arg(z) < 0 then arg(−z) – arg(z) is equal to

a) π

b) –π

c) – π/2

d) π/2

If ω(≠1) be a cube root of unity and  then the least positive value of n is

a) 2

b) 3

c) 5

d) 6

If A = , 6A^-1 = A² + cA + dI

then (c, d ) is

a) (−11, 6)

b) (−6, 11)

c)  (6, 11 )

d)  (11, 6 )

Prove that

The function f(x) = |px – q|+ r|x|, x  when p > 0, q > 0, r > 0 assumes minimum value only on one point if

a)  p ≠ q

b)  r ≠ q

c)  r ≠ p

d)  p = q = r

Let f(θ) = sinθ (sinθ + sin3θ) then f(θ)

a) ≥ 0 only when θ ≥ 0

b)  ≤ 0 for all real θ

c)  ≥ 0 for all real θ

d) ≤ θ only when θ ≤ 0

Let y = f(x) is a cubic polynomial having maximum at x = − 1 and  has a minimum at x = 1 and f(−1) = 10, f(1) = − 6. Find the cubic polynomial and also find the distance between the points which are maxima or minima.

a)

b)

c)

d)

The domain of definition of the function           is

a)  

b)  

c)  

d)  

The set of values of x which ln(1 + x) ≤ x is equal to .  .  .  .

a) (−∞, −1)

b) (−1, 0)

c) (0, 1)

d) (1, ∞)

If,  then g(f(x)) is invertible in the domain

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

Show that the integral of sinxsin2xsin3x + sec²xcos²2x + sin⁴xcos⁴x is

 =

a) +c

b) +c

c) +c

d)