Question(s) from Search: IIT

A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which he can select at least one book is 63, find the value of n?

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

Let  be a circle. A pair of tangents from (4, 5) and a pair of radii form a quadrilateral of area . . . . .

Identify a discontinuous function y = f(x) satisfying  

If  are complex numbers such that  then  is

a) Equal to 1

b) Less than 1

c) Greater than 3

d) Equal to 3

A polygon of nine sides, each of length 2, is inscribed in a circle. The radius of the circle is . . . . .

Fill in the blank
If f (x) = sin ln  then the domain of f (x) is ………….

a) (−2, −1)

b) (−2, 1)

c) (0, 1)

d) (1, ∞)

If f(9) = 9,  then  equals

a) 0

b) 1

c) 2

d) 4

A circle passes through the point of intersection of the coordinate axes with the lines  and x , then λ = . . . . .

If x, y, z are real and distinct then
8u =
is always

a) Non–negative

b) Non–positive

c) Zero

d) None of these

a) 0

b) 1

c) e

d) e²

Show that   for all x ≥ 0.

For each natural number k, let C_k denote the circle with radius k centimeters and center at the origin. On the circle C_k, a particle moves k centimeters in the counterclockwise direction. After completing its motion on C_k the particle moves to C_{k + 1} in the radial direction. The motion of the particle continues in this manner. The particle starts at ( 1, 0 ). If the particle crosses the positive direction of the X–axis for the first time on the circle C_n then n = . . . . .

If  are any real numbers and n is any positive integer then

a)

b)

c)

d) none of these