Question(s) from Search: IIT

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

Determine the values of x for which the following function fails to be continuous or differentiable.

Justify your answer.

a) f(x) is continuous and differentiable

b) f(x) is continuous everywhere but not differentiable at
x = 1, 2

c) f(x) is continuous everywhere but not differentiable at x = 2

d) f(x) is neither continuous nor differentiable at x = 1, 2

Let  

And

where a and b are non-negative real numbers. Determine the composite function gof. If (gof)(x) is continuous for all real x, determine the values of a and b. Is gof differentiable at x = 0?

a) a = b = 0

b) a = 0, b = 1

c) a = 1, b = 0

d) a = b = 1

Find the equation of the circle touching the line 2x + 3y + 1 = 0 at the point (1, −1) and is orthogonal to the circle which has the line segment having end points (0, −1) and (−2, 3) as diameter.

Show that the value of  wherever defined

a) always lies between  and 3

b) never lies between  and 3

c) depends on the value of x

Show that f(x) is differentiable at the value of α = 1. Also,

a) b² +c² = 4

b) 4 b²  = 4 − c²  

c) 64 b² = 4 − c²

d) 64 b² = 4 + c²

The product of r consecutive natural numbers is divisible by r!

a) True

b) False

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

a) (x – 1) cos (3x + b)

b) sin (3x + 4)

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

d) none of these

The sum  where  equals

a) i

b) i – 1

c) – i

d) 0

Fill in the blank

The value of f (x) =  lies in the interval …………….

a)

b)

c)

d)

Find the area bounded by the curve x² = 4y and the straight line
x = 4y – 2.

a) 3/2

b) 3/4

c) 9/4

d) 9/8

If f(x) and g(x) are differentiable functions for 0 ≤ x ≤ 1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2 then show that there exists c satisfying 0 < c < 1 and .

a) 0 < c < 1 and

b) 0 < c < 1 and

c) 0 < c < 1 and

d) 0 < c < 1 and

Let a > 0, b > 0, c > 0 then both the roots of the equation  

a) are real and positive

b) have negative real parts

c) have positive real parts

d) none of these

If f(x) is a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x and f(2) = 10 then f(1.5) = .  .  .  .

a) 2

b) 5

c) 10

d) 20

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

a) Non – negative

b) Non – positive

c) Zero

d) None of these

Match the following
Let [x] denote the greatest integer less than or equal to x

a) (i)→ A, B, C, (ii)→ A, D, (iii)→ C, D, (iv)→ A, B

b) (i)→ A, (ii)→ A, (iii)→ C, (iv)→ B

c) (i)→ B, (ii)→ D, (iii)→ C, (iv)→ A

d) (i)→ A, B, (ii)→ A, D, (iii)→ C, D, (iv)→ B

(One or more than one correct answer)
If  are complex numbers such that  and  then the pair of complex numbers  and  satisfy

a)

b)

c)

d) None of these