Question(s) from Search: IIT

For any natural number m, show that
 

Let α, β, γ be distinct real numbers. The points with position vectors
 

a) Are collinear

b) Form an equilateral triangle

c) Form a scalene triangle

d) Form a right angled triangle

The least value of the expression  for x > 1 is

a) 10

b) 2

c) −0.01

d) none of these

The probability that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has 75% chances of passing in at least one, a 50% chance of passing in at least two and 40% chance of passing in exactly two. Which of the following relations is true?

a)

b)

c)

d)

If  satisfies ,
find the value of

Let f : ℝ → ℝ and g : ℝ → ℝ be continuous functions. Then the value of integral  is

a) π

b) 1

c) – 1

d) 0

Let  If c is a vector such that  and the angle between  and c is 30° then  is equal to

a)

b)

c) 2

d) 3

An anti–aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probability of hitting the plane at the first shot, 2^nd, 3^rd and 4^th shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

The value of the expression  is equal to

a) 2

b)

c) 4

d)

Let f(x) =  where p is a constant

Then  at x = 0 is

a) p

b)

c)

d) Independent of p

a)

b)

c)

d)

A and B are independent events. The probability that both A and B occur is  and probability that neither of them occur is . Find the probability of the occurrence of A.

Let  then  equals

a) tan

b) tan

c) tan

d) tan²

 equals

a)

b)

c)

d)

If  f(x) = x – [x] for every real number x, where [x] is the integral part of x, then  is

a) 1

b) 2

c) 0

d)

If a and b are two unit vectors such that  are perpendicular to each other then the angle between a and b is

a) 45°

b) 60°

c)

d)

A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point.

The length of longest interval in which the function  is increasing is

a)

b)

c)

d)

Let p, q be the roots of the equation , and r and s are roots of the equation . If  are in arithmetic progression then A = .  .  .  .  . , B = .  .  .  .  .

Let y =  Find

a)

b)

c)

d) 0

If

Then  =

a) 0

b) 1

c) 2

d) 3

If  are non-coplanar vectors and
  then a.b₁ and a.are orthogonal.

Let A be a set containing n elements. A subset P of A is constructed at random. The set A is reconstructed by replacing the elements of P. A subset of Q of A is again chosen at random. Find the probability that P and Q have no elements in common.

The derivative of an even function is always an odd function.

a) False

b) True

If  then

a) Re(z) = 0

b) Im(z) = 0

c) Re(z) = 0, Im(z) > 0

d) Re(z) > 0, Im(z) < 0