Question(s) from Search: IIT

The function  is

a) Increasing on (0, ∞)

b) Decreasing on (0, ∞)

c) Increasing on  and decreasing on  

d) Increasing on  and decreasing on

A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of ΔPQR.

If we consider only the principal values of the inverse trigonometric functions then the value of

 is

a)

b)

c)

d)

Let g (x) = 1 + x – [ x ] and f (x) =  then for all x,
f (g (x)) is equal to

a) x

b) 1

c) f ( x )

d) g ( x )

Let P(asecθ, btanθ) and Q(asecɸ, btanɸ) where θ + ɸ =  be two points on the hyperbola . If (h, k) be the point of intersection of the normals at P and Q then k is equal to

a)

b)

c)

d)

Find the value of  at  where
.

a) 1

b)

c)

d)

Let ℝ be the set of real numbers and f : ℝ → ℝ such that for all x and y in ℝ, . Then f (x) is a constant.

a) True

b) False

Let

Then at x = 0, f has

a) A local maximum

b) No local maximum

c) A local minimum

d) No extremum

Let C be any circle with centre (0, . Prove that at the most two rational points can be there on C (A rational point is a point both of whose coordinates are rational numbers).

Find

a) 0

b) e

c) e^z

d) e³

The relatives of a man comprise 4 ladies and 3 gentlemen and his wife has 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that so that three of man’s relatives and three of wife’s relatives are included?

Let   then the real roots of the equation

 are

a) ± 1

b)

c)

d) 0 and 1

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