Question(s) from Search: IIT

If then equals

a)

b)

c)

d)

L =  = .  .  .  .

a) – 1

b) 0

c) 1

d) 2

Let  then the value of  is

a) 3ω

b) 3ω(ω – 1)

c) 3ω²

d) 3ω(1 – ω)

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

A man walks a distance of three units from the origin towards north-east (N direction. From there he walks a distance of 4 units towards north–west (N direction to reach a point P. Then the position of P in the argand plane is

a)

b)

c)

d)

Show that, if
a, b, c, d ε ℝ

If f(x) =
then f(100) equals

a) 0

b) 1

c) 100

d) −100

Show that the area of the triangle on the argand diagram formed by the complex numbers z, iz, z + iz is   .

If the system of equations

x + ay = 0

az + y = 0

ax + z = 0

has infinite solutions then the value of a is

a) −1

b) 1

c) 0

d) No real values

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

Prove that there exists no complex number z such that  and .

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

Suppose f(x) is a function satisfying the following conditions

i) f(0) = 2, f(1) = 1

ii) f has a minimum value at x = 5/2 and

iii) for all x

 
where a, b are constants. Determine the constants a and b, and the function f(x).

a)

b)

c)

d)

Let  for n ≥ 2 and
 

Then equals

a)

b)

c)

d)

 =

a)

b)

c)

d)

Show that the integral
 =

Show that  =

Let f : ℝ → ℝ be a differentiable function and f (1) = 4. Then show that the value of   =

If  then  is equal to

a)

b)

c)

d)

The value of  where [.] represents the greatest integer function is

a)

b)

c)

d)

If  then the value of f(1) is

a)

b) 0

c) 1

d)

Given  find

Coefficient of x⁴ in  is

a)

b)

c)

d) None of these