Question(s) from Search: IIT

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.

In a triangle ABC, let ∠ C = . If r is the inradius and R is the circumradius of the triangle then 2(r+R) = ………….

a) a+b

b) b+c

c) c+a

d) a+b+c

Let f(x), x ≥ 0 be a non-negative function and let F(x) = . For some c > 0, f(x) ≤ cF(x) for all x ≥ 0. Then for all x ≥ 0, f(x) =

a) 0

b) 1

c) 2

d) 4

Tangents are drawn from P (6, 8) to the circle  . Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.

Find the natural number a for which
  
where the function f satisfies the relation f (x + y) = f (x) . f (y)
for all natural numbers x and y and further f (1) = 2

a) 1

b) 2

c) 3

d) 4

In a certain test  students gave wrong answers to at least i questions where i = 1, 2, …, k. No student gave more than k correct answers. Total number of wrong answers given is .  .  .

Multiple choice

If

a) f(x) is increasing on [– 1, 2]

b) f(x) is continuous on [– 1, 3]

c)  does not exist

d) f(x) has maximum value at x = 2

If arg(z) < 0 then arg(−z) – arg(z) is equal to

a) π

b) –π

c) – π/2

d) π/2

Multiple choice

f(x) is a cubic polynomial with f(2) = 18 and f(1) = − 1. Also f(x) has a local maxima at x = − 1 and  has a local minima at x = 0 then

a) The distance between (− 1, 2) and (a, f(a)), where x = a is the point of local minimum, is

b) f(x) is increasing for

c) f(x) has a local minima at x = 1

d) The value of f(0) = 15

From the point A (0, 3) on the circle , a chord AB is drawn and extended to a point M such that AˆM = 2AˆB. The equation of locus of M is . . . . .

In Δ ABC the median to the side BC is of length  and divides ∠A into 30° and 45°. Then find the length of side BC.

a) 1

b) 2

c)

d)