Question(s) from Search: IIT

If one of the diameters of the circle, given by the equation x² + y² – 4x + 6y – 12 = 0 is a chord of a circle S whose centre is at (−3, 2), then the radius of S is

a) $5\sqrt{2}$

b) $5\sqrt{3}$

c) $5$

d) $10$

If  for all k ≥ n then show that

The function  (where [y] is the greatest integer less than or equal to y) is discontinuous at

a) All integers

b) All integers except 0 and 1

c) All integers except 0

d) All integers except 1

If  are three non-coplanar unit vectors and α, β, γ are the angles between  , v and w, w and u respectively and x, y and z are unit vectors along the bisector of the angles α, β, γ respectively. Prove that
  

For how many values of p, the circlex² + y² + 2x + 4y – p = 0 and the coordinate axis have exactly three common points

a) 0

b) 1

c) 2

d) 3

A tangent PT is drawn to the circle x² + y² = 4 at the point $P\left(\sqrt{3},1\right)$

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)² + y² = 1A common tangent to the circles is

a) x = 4

b) y = 2

c) $x+\sqrt{3}y=4$

d) $x+2\sqrt{2}y=6$

The integer n, for which  is a finite

non–zero number is

a) 1

b) 2

c) 3

d) 4

The locus of the middle points of the chord of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x² + y² = 9 is

a) 20(x² + y²) – 36x + 45y = 0

b) 20(x² + y²) + 36x − 45y = 0

c) 36(x² + y²) – 20x + 45y = 0

d) 36(x² + y²) + 20x − 45y = 0

Let P be a point on the parabola y² = 8x which is at a minimum distance from the centre C of the circle x² + (y + 6)² = 1. Then the equation of the circle passing through C and having its centre at P is

a) x² + y² – 4x + 8y + 12 = 0

b) x² + y² –x + 4y − 12 = 0

c) x² + y² –x + 2y − 24 = 0

d) x² + y² – 4x + 9y + 18 = 0

Let  then points where f (x) is not differentiable is (are)

a) 0

b) 1

c) ± 1

d) 0, ± 1

The slope of the line touching both parabolas y² = 4x and x² = −32y is

a) $\frac{1}{2}$

b) $\frac{3}{2}$

c) $\frac{1}{8}$

d) $\frac{2}{3}$

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and QR intersect at a point x on the circumference of the circle, then 2r equals

a)

b)

c)

d)

Multiple choices

Let [x] denote the greatest integer less than or equal to x. If

f (x) = [xsinπx] then f(x) is

a) Continuous at x = 0

b) Continuous in  

c) f (x) is differentiable at x = 1

d) differentiable in

e) None of these

Let  then

a)

b)

c)

d)

Let a, r, s, t be non-zero real numbers. Let P(at², 2at), Q, R(ar², 2ar and S(as², 2as) be distinct points on the parabola y² = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)

The value of r is

a) $\frac{-1}{t}$

b) $\frac{t^2+1}{t}$

c) $\frac{1}{t}$

d) $\frac{t^2-1}{t}$

Find all solutions of

a)

b)

c)

d)

Multiple choices

Which of the following functions are continuous on (0, π)

a) tanx

b)

c)

d)

One or more than one correct option

If the normals of the parabola y² = 4x drawn at the end points of the latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² then the value of r² is

a) 4

b) 1

c) 2

d) 0

Multiple choices

Let  for every real number x then

a) h (x) is continuous for all x

b) h is differentiable for all x

c)  for all x > 1

d) h is not differentiable for two values of x

Number of divisors of the form 4n + 2(n ≥ 0) of integer 240 is

a) 4

b) 8

c) 10

d) 3

The smallest positive root of the equation tan x – x = 0 lies in

a)

b)

c)

d)

e) None of these

Let f (x) be defined on the interval  such that

g (x) = f (|x|) + |f(x)|

Test the differentiability of g (x) in

a) g(x) is differentiable at all x  ℝ

b) g(x) is differentiable at all x  ℝ except at x = 1

c) g(x) is differentiable at all x  ℝ except at x = 0, 1

d) g(x) is differentiable at all x  ℝ except at x = 0, 1, 2

If the LCM of p, q is  where r, s, t are prime numbers and p, q are positive integers then the number of ordered pairs (p, q) is

a) 252

b) 254

c) 225

d) 224

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords in which the circle  cuts the members of the family are concurrent at a point. Find the coordinates of this point.

In how many ways can a pack of 52 cards be divided into four groups of 13 cards each.