Question(s) from Search: IIT

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

a)

b)

c)

d)

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

The radius of the circle passing through the focii of the ellipse  and having centre at (0, 3) is

a) 4

b) 3

c)

d)

The number of solutions of the pair of equations

in the interval [ 0, 2π ] is

a) 0

b) 1

c) 2

d) 4

Multiple choice question

On the ellipse  the points at which the tangents are parallel to the line  are

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}

An ellipse has eccentricity  and one of the focus at the point  It’s one directrix is the common tangent near to the point P to the circle  and the hyperbola . Then the equation of the ellipse in the statement form is . . . . .

The equation  has

a) no real solutions

b) one real solution

c) two real solutions

d) infinite real solutions

For positive numbers x, y and z the numerical value of the determinant
 is ………..

a) 1

b) –1

c) ±1

d) 0

The equation  represents

a) No locus if k > 0

b) An ellipse if k < 0

c) A point if k = 0

d) A hyperbola if k > 0