Question(s) from Search: IIT

One or more than one correct option

Let RS be a diameter of the circle x² + y² = 1 where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and the tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersect a line drawn through Q parallel to RS at a point E. Then the locus of E passes through the point(s)

a) $\left(\frac{1}{3},\frac{1}{\sqrt{3}}\right)$

b) $\left({\frac{1}{4},}^{}\frac{1}{2}\right)$

c) $\left(\frac{1}{3},-\frac{1}{3}\right)$

d) $\left(\frac{1}{4},-\frac{1}{2}\right)$

If x is not an integral multiple of 2π use mathematical induction to prove that
 

A circle passing through (1, −2) and touching the axis of X at (3, 0) also passes through the point

a) (−5, 2)

b) (2, −5)

c) (5, −2)

d) (−2, 5)

The circles  and  intersect each other in distinct points if

a) r < 2

b) r > 8

c) 2 < r < 8

d) 2 ≤ r ≤ 8

Prove by induction that
P_n = Aαⁿ + Bβⁿ for all n ≥ 1
Where α and β are roots of the quadratic equation
x² – (1 – P) x – P (1 – P) = 0,
P₁ = 1, P₂ = 1 – P², .  .  .,
P_n = (1 – P) P_{n – 1} + P (1 – P) P_{n – 2}  n ≥ 3,
and ,

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.

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