Question(s) from Search: IIT

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

Show that the area of the triangle on the argand diagram formed by the complex numbers z, iz, z + iz is   .

The angle between the tangents drawn from the point (1, 4) to the parabola  is

a)

b)

c)

d)

The equation  has

a) No solution

b) One solution

c) Two solutions

d) More than two solutions

If the system of equations

x + ay = 0

az + y = 0

ax + z = 0

has infinite solutions then the value of a is

a) −1

b) 1

c) 0

d) No real values

Let z and ω be two complex numbers such that |z| ≤ 1 and |w| ≤ 1 then show that .

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

If a, b, c, d are positive real numbers such that a + b + c + d = 2 then M = ( a + b ) ( c + d ) satisfies

a) 0 ≤ M ≤ 1

b) 1 ≤ M ≤ 2

c) 2 ≤ M ≤ 3

d) 3 ≤ M ≤ 4

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

Let α, β be the roots of  and γ, δ roots of . If α, β, γ, δ are in geometric progression then the integral values of p and q respectively are

a) −2, −32

b) −2, 3

c) −6, 3

d) −6, −32

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

Prove that there exists no complex number z such that  and .

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

If  are positive real numbers whose product is a fixed number c then the minimum value of  is

a)

b)

c)

d)

If three complex numbers are in arithmetic progression then they lie on a circle in the complex plane.

a) True

b) False

The cube roots of unity when represented on argand diagram form the vertices of an equilateral triangle.

a) True

b) False

If  is a solution of  and  then  is equal to

a)

b)

c) 1

d)

If one root is square of the other root of the equation  then the relation between p and q is

a)

b)

c)

d)