Question(s) from Search: IIT

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   =

A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and kept aside from the remaining balls in the box. Another ball is drawn at random and kept besides the first. This process is repeated till all the balls are drawn from the box. Find the probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red.

The expression
 
 is equal to

a) 0

b) 1

c) 3

d) sin4α + cosα

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.