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)

A unit vector perpendicular to the plane determined by the points P (1, -1, 2), Q (2, 0, -1) and R (0, 2, 1) is .  .  .  .  .

If one of the diameters of the circle  is a chord to the circle with centre (2, 1) then the radius of the circle is

a)

b)

c) 3

d) 2

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

Let a, b and c be three vectors having magnitudes 1, 1 and 2 respectively. If  then the acute angle between a and c is  .  .  .  .  .

The equation of the tangents drawn from the origin to the circle  are

a) x= 6

b) y = 0

c)

d)

The area bounded by the curve y = f(x), the X–axis and the ordinate x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + 4)

b) sin(3x + 4)

c) sin(3x + 4) + 3(x – 1) cos (3x + 4)

d) none of these

Through a fixed point (h, k) secants are drawn to the circle  . Show that the locus of the mid points of the secant intercepted by the circle is

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)

The circle  is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circum centre of the triangle is  find k.

The set of all real numbers x for which  is

a)

b)

c)

d)

The function  is

a) Increasing on (0, ∞)

b) Decreasing on (0, ∞)

c) Increasing on  and decreasing on  

d) Increasing on  and decreasing on

A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of ΔPQR.

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

Let P(asecθ, btanθ) and Q(asecɸ, btanɸ) where θ + ɸ =  be two points on the hyperbola . If (h, k) be the point of intersection of the normals at P and Q then k is equal to

a)

b)

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

Let

Then at x = 0, f has

a) A local maximum

b) No local maximum

c) A local minimum

d) No extremum

Let C be any circle with centre (0, . Prove that at the most two rational points can be there on C (A rational point is a point both of whose coordinates are rational numbers).

The equation  has

a) At least one real solution

b) Exactly three real solutions

c) Has exactly one irrational solution

d) Complex roots

Let   then the real roots of the equation

 are

a) ± 1

b)

c)

d) 0 and 1

Consider a family of circles . If in the first quadrant, the common tangent to a circle of the family and the ellipse  meet the coordinate axes at A and B, then find the locus of the mid-point of AB.

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}