Question(s) from Search: IIT

Let  for n ≥ 2 and
 

Then equals

a)

b)

c)

d)

India played two matches each with Australia and West indies. In any match the probability of India getting the points 0, 1, and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least seven points is

a) 0.8730

b) 0.0875

c) 0.0625

d) 0.0250

Let T_n denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If  then n equals

a) 5

b) 7

c) 6

d) 4

Three of the vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

a)

b)

c)

d)

Let a, b, c be real numbers with a ≠ 0 and let α, β be roots of the equation . Express the roots of  in terms of α, β.

Suppose a, b, c are in Arithmetic Progression and  are in Geometric Progression. If  then the value of a is

a)

b)

c)

d)

 =

a)

b)

c)

d)

If  are complementary events E and F respectively and if 0 < p(E) < 1, then

a)

b)

c)

d)

The equation  has an irrational root.

a) False

b) True

Multiple Choice

For  if
 , then

a)  

b)  

c)  

d)  

The numbers are selected from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. Probability that the minimum of the two numbers is less than 4 is

a)

b)

c)

d)

Fill in the blank

If the product of the roots of the equation
  is 7
Then the roots are real for  ………….

Show that the integral
 =

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

a)

b)

c)

d)

Fill in the blank

There are exactly two distinct linear functions ………. and ………. which map {−1, 1} onto {0, 2}.

Find three numbers a, b, c between 2 and 18 such that (i) their sum is 25 (ii) 2, a, b are consecutive terms of an Arithmetic Progression and (iii) the numbers b, c, 18 are consecutive terms of a Geometric Progression

Show that  =

Let p and q be the position vectors of P and Q respectively with respect to O and . The points R and S divide PQ internally and externally in the ratio 2:3 respectively. If OR and OS are perpendicular then

a)

b)

c)

d)

Let u, v and w be vectors such that  . If  then  is equal to

a) 47

b) –25

c) 0

d) 25

(One or more correct answers)
There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed

a)

b)

c)

d)

(Subjective problem)

Solve  
where a > 0, b = a²x.

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

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)