Question(s) from Search: IIT

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

A man walks a distance of three units from the origin towards north-east (N direction. From there he walks a distance of 4 units towards north–west (N direction to reach a point P. Then the position of P in the argand plane is

a)

b)

c)

d)

Find the coordinates of the point on the curve  where the tangent to the curve has the greatest slope.

a) (0, 0)

b)

c)

d)

If p, q, r are any real numbers, then

a) Max ( p, q ) < max ( p, q, r )

b) Min ( p, q ) =  

c) Max ( p, q ) < min ( p, q, r )

d) none of these

Show that, if
a, b, c, d ε ℝ

The equation of the common tangent touching the circle
 and the parabola , above X–axis is

a)

b)

c)

d)

The expression  is a polynomial of degree

a) 5

b) 6

c) 7

d) 8

If f(x) =
then f(100) equals

a) 0

b) 1

c) 100

d) −100