Question(s) from Search: IIT

A circle is given by , another circle C touches it externally and also the X-axis, then the locus of the centre of C is

a)

b)

c)

d)

Find all solutions of
 in

a)

b)

c)

d)

Let f (x) = sin x and g (x) = ln|x|. If the range of the composition

functions fog and gof are R₁ and R₂ respectively, then

a) R₁ = [ u : −1 ≤ u < 1], R₂ = [ v : − < v < 0 ]

b) R₁ = [ u : − < u < 0 ], R₂ = [ v : −1 ≤ v ≤ 0]

c) R₁ = [ u : −1 < u < 1], R₂ = [ v : − < v < 0 ]

d) R₁ = [ u : −1 ≤ u ≤ 1], R₂ = [ v : − < v ≤ 0 ]

Let f (x) =   then for all x

a)

b) f is differentiable

c)  is continuous

d) f is continuous

If for non-zero x, where a ≠ b thenis equal to

a)

b)

c)

d)

Find the equation of the circle which passes through the point (2, 0) and whose centre is the limit of the point of intersection of the lines .

Multiple choices

If f (x) = then

a) f (x) is continuous ∀ x  ℝ

b) f (x) > 0 ∀ x > 1

c) f (x) is continuous but not differentiable ∀ x  ℝ

d) f (x) is not differentiable at two points

Eight chairs are numbered one to eight. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 then the men select the chairs from amongst the remaining. The number of possible arrangements is

a)

b)

c)

d) None of these

AB is a diameter of a circle and C is any point on the circumference of the circle. Then

a) The area of △ABC is maximum if it is isosceles

b) The area of △ABC is minimum if it is isosceles

c) The perimeter of △ABC is minimum when it is isosceles

d) None of these

The abscissas of two points A and B are the roots of the equation  and their ordinates are the roots of the equation . Find the equation of the circle on AB as diameter.

Find  if f(x) =

a) 0

b)

c)

d)

An n digit number is a positive number with exactly n–digits. Nine hundred distinct n–digit numbers are to be formed with only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

a) 6

b) 7

c) 8

d) 9

Let  be a given circle. Find the locus of the foot of perpendicular drawn from the origin upon any chord of S which subtends a right angle at the origin.

Let f be a twice differentiable function such that  and , . Find h (10) if

h (5) = 1.

a) 0

b) 1

c) 2

d) 4

The number of arrangements of two letters of the word BANANA in which two N’s do not appear adjacently is

a) 40

b) 60

c) 80

d) 100

The circles each of radius 5 units touch each other at (1, 2). If the equation of the common tangent is , find the equation of the circles.

If , then find the values of n and r

The function  increases if

a)

b)

c)

d)

a) True

b) False

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many different ways can we place the balls so that no box is empty?

If  then f(x) is

a) Increasing on

b) Decreasing on ℝ

c) Increasing on ℝ

d) Decreasing on

In a triangle ABC, ∠ B = , ∠ C = . Let D divides BC internally in the ratio 1:3 then  is equal to

a)

b)

c)

d)

Let

Test whether

f(x) is continuous at x = 0

f(x) is differentiable at x = 0

a) f(x) is differentiable and continuous at x = 0

b) f(x) is continuous but not differentiable at x = 0

c) f(x) is neither continuous nor differentiable at x = 0

A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which he can select at least one book is 63, find the value of n?

Let  be the equation of pair of tangents from the origin O to a circle of radius 3 with centre in the first quadrant. If A is a point of contact, find the length of OA.