Question(s) from Search: IIT

If  are in Arithmetic Progression
then

a) a, b, c are in Arithmetic Progression

b)  are in  Arithmetic Progression

c) a, b, c are in Geometric Progression

d) a, b, c are in Harmonic Progression

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, ∞)

The harmonic means of the roots of the equation
 is

a) 2

b) 4

c) 6

d) 8

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

Consider the two curves  then

a)  touch each other at only one point

b)  touch each other exactly at two points

c)  intersect(but not touch) at exactly two points

d)  neither intersect nor touch each other

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  =