Question(s) from Search: IIT

Ten different letters of an alphabet are given. Words with five letters are formed from the given letters. Then the number of words which have at least one letter repeated is

a) 69760

b) 30240

c) 99748

d) None of these

Let a, b, c be non-zero real numbers such that
 
 
Then the quadratic function  has

a) no root in (0, 2)

b) at least one root in (1, 2)

c) a double root in (0, 2)

d) two imaginary roots

Prove that the value of the function  do not lie between  and 3 for any real x.

a) True

b) False

If g (f (x)) = |sin x| and f (g (x)) = (sin)², then

a) f (x) = sin² x, g (x) =

b) f (x) = sin x, g (x) =

c) f (x) = x², g (x) = sin

d) f and g cannot be determined

Evaluate

a) 0

b)

c)

d) 1

If   then  equals

a)

b)

c)

d) None of these

Let  be a polynomial in a real variable x with 0< then the function p(x) has

a) neither maximum nor minimum

b) only one maximum

c) only one minimum

d) only one maximum and only one minimum

e) none of these

Let a given line L₁ intersect the X-axis and Y-axis at P and Q respectively. Let another line L₂ perpendicular to L₁ cut the X and Y axis at R and S respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.

Fill in the blank
General values of θ satisfying the equation  are

a) θ = nπ

b)

c)

d) θ = nπ or θ =

If f (x + y) = f (x) + f (y) for all x and y. If the function f is continuous at x = 0 then f is continuous for all x.

a) True

b) False

How many different 9 digit numbers can be formed from the numbers 223355888 by rearranging its digits so that the odd digits occupy even positions

a) 16

b) 36

c) 60

d) 180

The function defined by  is

a) Decreasing for all x

b) Decreasing in  and increasing in

c) Increasing for all x

d) Decreasing in  and increasing in  

The principal value of is

a)

b)

c)

d)

e) None of these

Let f(x) =

Discuss the continuity of  on [0, 2]

a)  is continuous for all x  ℝ

b)  is continuous for all x  ℝ except at x = 1

c)  is continuous for all x  ℝ except at x = 1 and x = 2

d)  is continuous for all x  ℝ except at x = 0, x = 1 and x = 2

Let a circle be given by . Find the condition on a and b if two chords each bisected by the X–axis can be drawn from .

The value of x for which  is

a)

b) 1

c) 0

d)

Consider the following Statement (S) and Reason (R)

S: Both sinx, cosx are decreasing functions in the interval

R: If a differentiable function decreases in an interval (a, b) then the derivative also decreases in (a, b)

Which of the following is true?

a) Both S and R are wrong

b) Both S and R are correct but R is not the correct explanation of S

c) S is correct and R is the correct explanation of S

d) S is correct and R is wrong

The numerical value of  is

a)

b)

c)

d)

The range of the function f (x) = , x ε R is

a) ( 1, )

b)

c)

d)

A function f : ℝ → ℝ satisfies the equation

f(x + y) = f(x) . f(y)  x, y in ℝ and f(x) ≠ 0 for any x in ℝ. Let the function be differentiable at x = 0 and . Show that. Hence determine f(x).

a) e^x

b) e^2x

c) 2e^x

d) 2e^2x

m men and n women are to be seated in a row so that no two women sit together. If m > n, then find the number of ways in which they can be seated.

In a triangle ABC, is equal to

a)

b)

c)

d)

If F (x) =

where  =  and  and given that F (5) = 5
then F (10) is equal to

a) 5

b) 10

c) 0

d) 15

For a fixed value of n
D =
Then show that  is divisible by n

The area bounded by the curves  

and the X–axis in the first quadrant is

a) 9

b)

c) 36

d) 18