Question(s) from Search: IIT

Three circles touch each other externally. The tangents at their points of contact meet at a point whose distance from a point of contact is 4. Find the ratio of the product of the radii to the sum of the radii of the circles.

The number of solutions of  is

a) 0

b) One

c) Two

d) Infinite

Let f (x) be a continuous function satisfying  If  exists, find its value.

a) 0

b) 1

c) 2

d) 4

The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order as in the English dictionary. The number of words that appear before the word COCHIN is

a) 360

b) 192

c) 96

d) 48

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many different ways can we place the balls so that no box is empty?

If  then f(x) is

a) Increasing on

b) Decreasing on ℝ

c) Increasing on ℝ

d) Decreasing on

In a triangle ABC, ∠ B = , ∠ C = . Let D divides BC internally in the ratio 1:3 then  is equal to

a)

b)

c)

d)

Let

Test whether

f(x) is continuous at x = 0

f(x) is differentiable at x = 0

a) f(x) is differentiable and continuous at x = 0

b) f(x) is continuous but not differentiable at x = 0

c) f(x) is neither continuous nor differentiable at x = 0

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?

Let  be the equation of pair of tangents from the origin O to a circle of radius 3 with centre in the first quadrant. If A is a point of contact, find the length of OA.

If the angles of a triangle are in the ratio 4:1:1 then the ratio of the longest side to the perimeter is

a)

b) 1 : 6

c)

d) 2 : 3

 If f (x) = cos [π²] x + cos [-π²] x where [x] stands of the greatest integer function then

a) f  = −1

b)

c) f (−π) = 0

d) f  = 1

Let p be a prime and m be a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides

Let I_n represents area of n sided regular polygon inscribed in a unit circle and O_n the area of n–sided regular polygon circumscribing it. Prove that

P(x) is a polynomial function such that P(1) = 0, > P(x)

 x > 1. Then  x > 1,

a) P(x) > 0

b) P(x) = 0

c) P(x) < 1

Prove that

Minimum area of the triangle formed by the tangent to the ellipse

 with co-ordinate axes is

a)

b)

c)

d) ab

If A and B are points in the plane such that (constant) for all P on a given circle then the value of k cannot be equal to - -  - - -.

Let {x} and [x] denote the fractional and integral part of a real number respectively. Solve 4 {x} = x + [x]

a) x = 0

b)

c)

d)

The sides AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .  .  .  .

Multiple choice

Let h(x) = f(x) – (f(x))² + (f(x))³ for every real number x, then

a) h increases whenever f is increasing

b) h is increasing whenever f is decreasing

c) h is decreasing whenever f is decreasing

d) nothing can be said in general

From the origin chords are drawn to the circle . The equation of the locus of the mid points of these chords is . . . . .

If then equals

a)

b)

c)

d)

The area of the triangle formed by the tangents from the point (4, 3) to the circle  and the line joining their point of contact is .

L =  = .  .  .  .

a) – 1

b) 0

c) 1

d) 2