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

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)

If f is an even function defined on (−5, 5) then the real values of x satisfying the equation f (x) =  are ……………

a)

b)

c)

d)

If ω(≠1) be a cube root of unity and  then the least positive value of n is

a) 2

b) 3

c) 5

d) 6

A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is . . . . .

The number of all possible triplets  such that
 for all x is

a) Zero

b) One

c) Three

d) Infinite

e) None

A swimmer S is in the sea at a distance d km. from the closest point A on a straight shore. The house of the swimmer is on the shore at a distance L km. from A. He can swim at a speed of
u km/hour and walk at a speed of v km/hr (v > u). At what point on the shore should he land so that he reaches his house in the shortest possible time.

a)

b)

c)

d)