Question(s) from Search: IIT

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

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to be on a particular side and three others on the other side. Determine the number of ways in which the seating arrangements can be made?

Tangent is drawn to the ellipse  at  where  . Then the value of θ such that the sum of intercept on the axes made by the tangents is minimum is

a)

b)

c)

d)

Let C₁ , C₂ be two circles with C₂ lying inside C₁. A circle C lying inside C₁ touches C₁ internally and C₂ externally. Identify the locus of the center of C .

The sides of a triangle are in the ratio  then the angles of the triangle are in the ratio

a) 1 : 3 : 5

b) 2 : 3 : 4

c) 3 : 2 : 1

d) 1 : 2 : 3

Subjective problem

Let y =

Find all real values of x for which y takes real values

a) for x ≥ 3, y is real

b) for 2 < x < 3, y is imaginary

c) for – 1 ≤ x < 2, y is real

d) for x < – 1,  y is imaginary

If f(x) is differentiable and strictly increasing function then the value of  is

a) 1

b) 0

c) – 1

d) 2

Let R be the set of real numbers and f : R  R such that for all x, y ε R, |f (x) – f (y)| ≤ | x – y |². Then

a)

b) f (x) is a constant

c) none of the above

If  and = and f(0) = 0. Find the value of . Given that 0 < <

a)

b)

c)

d) 1

The area bounded by the curves

y = (x + 1)²

y = (x – 1)²

and the line  is

a)

b)

c)

d)