Question(s) from Search: IIT

Suppose p(x) =
If  prove that
 

The sum of the first 2n terms of the Arithmetic Progression 2, 5, 8, . . . .  is equal to the sum of the first n terms of the Arithmetic Progression 57, 59, 61, . . . . then n equals

a) 100

b) 12

c) 11

d) 13

Show that  =

Seven white balls and three black balls are randomly placed in a row. The possibility that no two black balls are placed adjacently equals

a)

b)

c)

d)

 where a, b ε R then find the value of a for which equation has unequal roots for all values of b.

If α, β are roots of  and  are in Geometric Progression and  then

a)

b)

c)

d)

 =

a)

b)

c)

d)

A fair coin is tossed repeatedly. If the tail appears on first four times, then the probability of the head appearing on in the fifth toss equals

a)

b)

c)

d)

If x and y are positive real numbers and m and n are any positive integers then

a) True

b) False

If x, y, z are in Harmonic Progression then show that
 

 =

a)

b)

c)

d)

The points with position vectors  are collinear if

a)

b)

c)

d)

If P (B) =  and  then

P (B ∩ C) is

a)

b)

c)

d)

Fill in the blank

The solution of the equation
 is …………..

If  are in Arithmetic Progression where  for all i, show that
 

Show that
 =

Let a, b, c be three non co–planar vectors and p, q, r are vectors defined by the relations
  
Then the value of the expression
 is equal to

a) 0

b) 1

c) 2

d) 3

Fill in the blank

If x < 0, y < 0,  and  then x  ……….. and y  ………..

The sum of the squares of three distinct real numbers which are in Geometric Progression is . If their sum is , show that
 

Show that  =

Let  If d is a unit vector such that  then d equals

a)

b)

c)

d)

(One or more correct answers)
Let E and F are two independent events. The probability that both E and F happen is  and the probability that neither E nor F happens is  , then

a)   

b)    

c)        

d)     

If then x lies in the interval

a) (2, ∞)

b) (1, 2)

c) (−2, −1)

d) None of these

The value of the integral
 is

a)

b)

c) π

d) None of these

Six boys and six girls sit in a row at random. Find the probability that the six girls sit together,