Question(s) from Search: IIT

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)

If  exists then both the limits  and  exist

a) True

b) False

Total number of ways in which six ‘+’ and four ‘’ signs can be arranged in a line so that no two ‘’signs occur together is …..

Multiple choice

The function  

has local minimum at x =

a) 0

b) 1

c) 2

d) 3

Multiple choice

For which value of m, is the area of the region bounded by the curve y = x –x² and the line y = mx equal to

a) – 4

b) – 2

c) 2

d) 4

The equation of the line passing through the points of intersection of the circles
 and
 is . . . . .

For the function

The derivative from right  .  .  .  . and the derivative from the left  .  .  .  .

a) 0, 0

b) 0, 1

c) 1, 0

d) 1, 1

Let z₁ and z₂ be n^th roots of unity which subtend a right angle at the origin then n must be of the form

a) 4k + 1

b) 4k + 2

c) 4k + 3

d) 4k

If the triangle  another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and a slope equal to  , then the coordinates of the centre of C₂ are . . . . .

Let ABC be a triangle such that
 

If A, B, C are in arithmetic progression, determine the values of A, B, C.

a) 30°, 60°, 90°

b) 30°, 75°, 75°

c) 45°, 60°, 75°

d) 60°, 60°, 60°