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 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)

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 blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Let  then one of the possible value of k is

a) 1

b) 2

c) 4

d) 16

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 f(x) = ∫e^x (x – 1) (x − 2) dx, then f(x) decreases in the interval

a) (−∞, −2)

b) (−2, −1)

c) (1, 2)

d) (2, ∞)

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)

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

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.

Show that  =

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].

 =

a)

b)

c)

d)

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

a)  

b)  3

c)  

d)  

 =

a)

b)

c)

d)

Suppose the probability for A winning a game against B is 0.4. If A has an option of playing either a best of 3 games or best of 5 games match against B, which option should he choose so that the probability of his winning the match is higher.

Show that
 =

Let  be unit vectors such that . Which one of the following is correct

a)

b)

c)

d)  are mutually perpendicular

The complex number z = x + iy which satisfies the equation  lies on

a) The real axis

b) The straight line y = 5

c) Circle passing through origin

d) None of these

Show that  =

Multiple choice

Let a and b be two non-collinear unit vectors. If  and  then  is

a) ||

b)

c)

d)

The value of the integral
 is

a)

b)

c) π

d) None of these

If c be a given non-zero scalar and A and B be given non-zero vectors such that A is perpendicular to B, find the vector X which satisfies the equations