Question(s) from Search: IIT

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

If  then  is equal to

a)

b)

c)

d)

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

a)

b)

c)

d)

Let the vectors  be such that  . Let P₁ and P₂ be the planes determined by the pairs of vectors a, b and c, d respectively. Then the angle between P₁ and P₂ is

a) 0

b)

c)

d)

If A, B, C be events such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(AB) = 0.08, P(AC) = 0.28, P(ABC) = 0.09 and P(A ∪ B ∪ C) ≥ 0.75, then show that P(BC) lies in the interval [0.23, 0.48].

If ω be the cube root of unity then the value of
 is

a)

b)

c)

d)

If  then the value of f(1) is

a)

b) 0

c) 1

d)

The value of a so that the volume of parallelopiped formed by  becomes minimum is

a)  

b)  3

c)  

d)  

 is true if

a) x + y= 0

b) x = y, x ≠ 0

c) x = y

d) x ≠ 0, y ≠ 0