Question(s) from Search: IIT

Let f (x) be defined for all x > 0 and be continuous. If f (x) satisfies
f  = f (x) – f (y) for all x and y and f (e) = 1 then

a) f (x) is bounded

b) f  → 0 as x → 0

c) x f  → 0 as x → 0

d) f (x) = lnx

The value of  is equal to

a)

b)

c)

d) None of these

The area bounded by the curve y = f(x), the X–axis and the ordinate x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + 4)

b) sin(3x + 4)

c) sin(3x + 4) + 3(x – 1) cos (3x + 4)

d) none of these

Through a fixed point (h, k) secants are drawn to the circle  . Show that the locus of the mid points of the secant intercepted by the circle is

There exists a solution of θ between 0 and 2π that satisfies the equation .

a) True

b) False

The number of values of x where the function
f (x) = cos x + cos () attains the maximum is

a) 0

b) 1

c) 2

d) Infinite

Evaluate

a) 0

b)

c) 1

d) 2

The circle  is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circum centre of the triangle is  find k.

The domain of definition of the function f (x) given by the equation

2^x + 2^y = 2 is

a) 0 < x ≤ 1

b) 0 ≤ x ≤ 1

c)  < x ≤ 0

d)  < x ≤ 1

Determine the values of a, b, c for which the function

is continuous at x = 0

a)

b)

c)

d)

If

a)

b) [2, ∞)

c)

d)

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) =