Question(s) from Search: IIT

Let the Harmonic Mean and Geometric Mean of two positive numbers be in the ratio of 4:5. Then the two numbers are in the ratio .  .  .  .  .

If f (x) = , find  from first principle.

a)

b)

c)

d)

If for real number y, [y] is the greatest integer less than or equal to y then the value of the integral   is

a)

b)

c)

d)

If  and  then b is equal to

a)

b)

c)

d)

A box contains two 50 paise coins, 5 twenty five paise coins and a certain number N(≥ 2) of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these coins is less than one rupee and 50 paise.

 and
where α, β ε [π, π]. Values of α, β which satisfy both the equations is/are

a) 0

b) 1

c) 2

d) 4

Given positive integers r > 1, n > 2 and the coefficients of (3r)^th term and (r + 2)^th terms in the binomial expansion of (1 + x)²ⁿ are equal then

a) n = 2r

b) n = 2r + 1

c) n = 3r

d) none of these

The number of distinct real values of λ for which   are coplanar is

a) Zero

b) One

c) Two

d) three

If , then  

a) True

b) False

If in the expansion of (1 + x)^m (1 – x)ⁿ, the coefficients of x and x² are 3 and –6 respectively. then m is

a) 6

b) 9

c) 12

d) 24

If  then  at x = e is .  .  .

a) 0

b)

c) e

d) 1

If  then the expression for  in terms of  is

a)

b)

c)

d)

Multiple choice

Which of the following expressions are meaningful

a)

b)

c)

d)

If  then at x = 0,  is equal to

a) 0

b) 1

c) 2

d) 4

is equal to

a) 0

b) 4

c) 6

d) −4

Find all values of λ such that  and

 where  are unit vectors along the coordinate vectors.

The complex numbers sinx + icos2x and cosx – isin2x are conjugate to each other for

a) a = nπ

b) x = 0

c) x =  

d) None of these

Suppose  is an identity in x where  are constants and . Then the value of n = ……….

a) 4

b) 5

c) 6

d) 7

Prove that  is divisible by 25 for any natural number n.

Evaluate   where n is a positive integer and t is a parameter independent of x.

a)

b)

c)

d)

Let OABC be a parallelogram with O as the origin and OC a diagonal. Let D be the midpoint of OA. Using vector method, prove that BD and CO intersect in the same ratio.

For positive integers n₁ and n₂ the value of the expression  where  is real if and only if

a)

b)

c)

d)

 is equal to

a) 0

b)

c)

d) None of these

Find the area bounded by the X - axis, part of the curve  and the ordinate at x = 2 and x = 4. If the ordinate at x = a divide the area into two equal parts, find a,

a)

b)

c)

d)

Determine the value of c so that for all real x the vector cx and  make an obtuse angle with each other.