Question(s) from Search: IIT

A circle passes through the point of intersection of the coordinate axes with the lines  and x , then λ = . . . . .

If x, y, z are real and distinct then
8u =
is always

a) Non–negative

b) Non–positive

c) Zero

d) None of these

a) 0

b) 1

c) e

d) e²

Show that   for all x ≥ 0.

For each natural number k, let C_k denote the circle with radius k centimeters and center at the origin. On the circle C_k, a particle moves k centimeters in the counterclockwise direction. After completing its motion on C_k the particle moves to C_{k + 1} in the radial direction. The motion of the particle continues in this manner. The particle starts at ( 1, 0 ). If the particle crosses the positive direction of the X–axis for the first time on the circle C_n then n = . . . . .

If  are any real numbers and n is any positive integer then

a)

b)

c)

d) none of these

If |z| = 1 and z ≠ ±1 then the value of  lie on

a) a line not passing through the origin

b)

c) the X – axis

d) the Y axis

Let a + b + c = 0, then the quadratic equation  has

a) at least one root in (0, 1)

b) one root in (2, 3) and the other in

c) imaginary roots

d) none of these

If x = a + b, y = aα + bβ, z = aβ + bα where α, β are cube roots of unity show that .

If  is a normal to  then k is

a) 3

b) 9

c) – 9

d) – 3

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)

If the expression  is real then the set of all possible values of x is .  .  .  .

a) x = 2nπ or mπ + π/4

b) x = nπ or mπ + π/4

c) x = 2nπ or 2mπ + π/4

d) x = nπ or 2mπ + π/4

(Assertion and reason)

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Statement 1 – The curve  is symmetric with respect to the line x = 1

Statement 2 – The parabola is symmetric about its axis.