Question(s) from Search: IIT

The left hand derivative of f (x) = [x] sinπx at k, k an integer is

a) (k – 1)π

b) (k – 1)π

c) kπ

d)  kπ

Determine the value of

a)

b)

c)

d)

Let f : ℝ → ℝ be such that f (1) = 3 and  then

 equals

a) 1

b)

c)

d)

For x > 0, let  find the function  and show that . Here .

a)

b)

c)

d)

Evaluate

a)

b)

c) 1

d) 2

The value of  is equal to

a)

b)

c)

d) None of these

Evaluate

a) 0

b)

c) 1

d) 2

Determine the values of a, b, c for which the function

is continuous at x = 0

a)

b)

c)

d)

If

a)

b) [2, ∞)

c)

d)

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

Find

a) 0

b) e

c) e^z

d) e³

The relatives of a man comprise 4 ladies and 3 gentlemen and his wife has 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that so that three of man’s relatives and three of wife’s relatives are included?

For a fixed value of n
D =
Then show that  is divisible by n

For any positive integers m, n (with n ≥ m), prove that
  

A =  is equal to

a) 0

b) 1

c)

d)

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

Let z₁ and z₂ be n^th roots of unity which subtend a right angle at the origin then n must be of the form

a) 4k + 1

b) 4k + 2

c) 4k + 3

d) 4k

a) 0

b) 1

c) e³

d) e⁵

If |z| = 1 and  then Re (w) is

a) 0

b)

c)

d)

If w ( ≠1 ) is cube root of unity, then
 

a) 0

b) 1

c) - 1

d) w

The determinant
 = 0 if

a) x, y, z are in arithmetic progression

b) x, y, z are in geometric progression

c) x, y, z are in harmonic progression

d) xy, yz, zx are in arithmetic progression

If  are the n roots of unity then show that .

The number of distinct roots of
 = 0
in the interval   ≤ x ≤   is

a) 0

b) 2

c) 1

d) 3

If A =  and | A³| = 125 then the value of α is

a) ± 1

b) ±2

c) ± 3

d) ± 5