Question(s) from Search: IIT

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

Let  then the value of  is

a) 3ω

b) 3ω(ω – 1)

c) 3ω²

d) 3ω(1 – ω)

The area of triangle formed by the positive X–axis and the normal and tangent to the circle  at  is . . . . . .

Intercepts on the line y = x by the circle  is AB. Equation of the circle with AB as diameter is  . . . . .

The number of real solutions of the equation | x |² – 3 | x | + 2 = 0 is

a) 4

b) 1

c) 3

d) 2

Match the following

Let the function defined in column 1 has domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C