Question(s) from Search: IIT

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

A solution of the differential equation  is

a)  y = 2

b)  y = 2x

c)  

d)  2

For all x,  then the interval in which a lies is

a) a <

b)

c)

d)

Let the three digit numbers A28, 3B9 and 62C where A, B, C are integers between 0 and 9, be divisible by a fixed number k. Show that the determinant
 

is divisible by k.

If a and b are real numbers between 0 and 1 such that the points  form an equilateral triangle then a is equal to .  .  .  .

a)  

b)  

c)  

d)  

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

Let a, b, c be the sides of a triangle where a ≠ c and λ ε R. If roots of the equation  are real then

a)

b)

c)

d)

Find the value of the determinant

where a, b, c are respectively p^th, q^th and r^th term of a harmonic

progression.

a) 0

b) 1

c) ½

d) None of the above

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

Let S is the set of all real x, such that  is positive, then S contains

a)

b)

c)

d)

The equation  has

a) At least one real solution

b) Exactly three real solutions

c) Has exactly one irrational solution

d) Complex roots

Show that for for any triangle with sides a, b, c
3 (ab + bc + ac) ≤ (a + b + c)² < 4 (ab + bc + ca)

The solution set of equation  = 0 is ……….

a) {0}

b) {1, 2}

c) {−1, 2}

d) {−1, −2}