Question(s) from Search: IIT

If α and β are roots of  and  are roots of  then the equation  has always

a) Two real roots

b) Two positive roots

c) Two negative roots

d) One positive and one negative root

The number of points of intersection of the two curves y = 2sinx and y =  is

a) 0

b) 1

c) 2

d)

If the system of equations

x – ky – z = 0

kx – y –z = 0

x + y –z = 0

has a non zero solution then possible values of k are

a) −1, 2

b) 1, 2

c) 0, 1

d) −1, 1

The axis of the parabola is along the line  and the distance of the vertex and focus from origin are  and  respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

a)

b)

c)

d)

The roots of the equation  are real and less than 3, then

a) a < 2

b) 2 < a < 3

c) 3 ≤ a ≤ 4

d) a > 4

Given 2x – y – z = 2, x – 2y + z = − 4, x + y + λz = 4 then the value of λ such that the given system of equations has no solution is

a) 3

b) −2

c) 0

d) −3

Find all non zero complex numbers satisfying .

Sketch the region bounded by the curves y = x² and  . Find the area.

a)

b)

c)

d)

Find the equation of the normal to the curve  which passes through the point (1, 2).

(Multiple choices)
The determinant
  is equal to zero if

a) a, b, c are in arithmetic progression

b) a, b, c are in geometric progression

c) a, b, c are in harmonic progression

d) α is a root of the equation ax² + bx + c = 0

e) x – α is a factor of ax² + 2bx + c

Let f(x) =  and m(b) be the minimum value of f(x). As b varies, range of m(b) is

a)

b) [ 0,

c) [

d)

At any point P on the parabola  , a tangent is drawn which meets the directrix at Q. Find the locus of the point R which divides QP externally in the ratio  .

The set of all real numbers x for which  is

a)

b)

c)

d)

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)

If a ≠ p, b ≠ q, c ≠ r and
 = 0

Then find the value of
  +  +

a) 0

b) 1

c) 2

d) 3

For all A, B, C, P, Q, R show that
 = 0

Let f(x) = |x – 1|, then

a)

b)

c)

d) None of these

The differential equation representing the family of curves  where c is a positive parameter, is of

a) Order 1

b) Order 2

c) Degree 3

d) Degree 4

Let a, b, c be real numbers with a² + b² + c² = 1. Show that the equation represents a straight line
 = 0

Let , then the set  is

a)  

b)  

c)  

d)  ϕ

A normal is drawn at a point  of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is  

If f(x) = 3x – 5 then  

a) is given by

b) is given by

c) does not exist because f is not one-one

d) does not exist because f is not onto

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4