Question(s) from Search: IIT

Consider a family of circles . If in the first quadrant, the common tangent to a circle of the family and the ellipse  meet the coordinate axes at A and B, then find the locus of the mid-point of AB.

Multiple choices

Let g (x) be a function defined on [−1, 1]. If the area of the equilateral triangle with the area of its vertices at ( 0, 0) and ( x, g (x)) is  then the function g (x) is

a) g (x) =

b) g (x) =

c) g (x) =

d) g (x) =

For a fixed value of n
D =
Then show that  is divisible by n

The area bounded by the curves  

and the X–axis in the first quadrant is

a) 9

b)

c) 36

d) 18

Find the point on   which is nearest to the line

Which one of the following is true in a triangle ABC?

a)

b)

c)

d)

Given A =  and f (x) = cosx – x (x + 1). Find the range of f (A).

a)

b)

c)

d)

For any positive integers m, n (with n ≥ m), prove that
  

If f(x) = x^a lnx and f(0) = 0 then the value of a for which Rolle’s theorem can be applied in [0, 1] is

a) – 2

b) – 1

c) 0

d)

The points of intersection of the line  and the circle  is . . . . . 

Let the angles A, B, C of Δ ABC be in arithmetic progression and
b : c = . Find the angle A.

a)

b)

c)

d)

A =  is equal to

a) 0

b) 1

c)

d)

Multiple choice

For which value of m, is the area of the region bounded by the curve y = x –x² and the line y = mx equal to

a) – 4

b) – 2

c) 2

d) 4

The equation of the line passing through the points of intersection of the circles
 and
 is . . . . .

For the function

The derivative from right  .  .  .  . and the derivative from the left  .  .  .  .

a) 0, 0

b) 0, 1

c) 1, 0

d) 1, 1

Let z₁ and z₂ be n^th roots of unity which subtend a right angle at the origin then n must be of the form

a) 4k + 1

b) 4k + 2

c) 4k + 3

d) 4k

If the triangle  another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and a slope equal to  , then the coordinates of the centre of C₂ are . . . . .

Let ABC be a triangle such that
 

If A, B, C are in arithmetic progression, determine the values of A, B, C.

a) 30°, 60°, 90°

b) 30°, 75°, 75°

c) 45°, 60°, 75°

d) 60°, 60°, 60°

If f (x) =  and  

then (gof)(x) = …………

a) 0

b) 1

c) 2

d) 3

a) 0

b) 1

c) e³

d) e⁵

Let f(x) = x|x|. The set of points where f(x) is twice differentiable is .  .  .  .

a) ℝ

b) 0

c) ℝ − {0, 1}

Find the shortest distance of the point (0, c) from the parabola
y = x², where 0 ≤ c ≤ 5.

a)

b)

c)

d)

Both roots of the equation

( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always

a) positive

b) negative

c) real

d) none of these

a) – 1

b) 0

c) 1

d) 2

If  is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is

a)  

b)  

c)  

d)