If g (f (x)) = |sin x| and f (g (x)) = (sin)², then

a) f (x) = sin² x, g (x) =

b) f (x) = sin x, g (x) =

c) f (x) = x², g (x) = sin

d) f and g cannot be determined

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

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

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 )

The range of the function f (x) = , x ε R is

a) ( 1, )

b)

c)

d)

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) =

Multiple choices
y = f ( x ) =  then

a) x = f (y)

b) f (1) = 3

c) y is increasing with x for x < 1

d) f is a rational function of x

Subjective problem

Let y =

Find all real values of x for which y takes real values

a) for x ≥ 3, y is real

b) for 2 < x < 3, y is imaginary

c) for – 1 ≤ x < 2, y is real

d) for x < – 1,  y is imaginary

Let R be the set of real numbers and f : R  R such that for all x, y ε R, |f (x) – f (y)| ≤ | x – y |². Then

a)

b) f (x) is a constant

c) none of the above

Let {x} and [x] denote the fractional and integral part of a real number respectively. Solve 4 {x} = x + [x]

a) x = 0

b)

c)

d)

True / False

If f (x) = ( a – xⁿ )^1/n  where a > 0 and n is a positive integer then f ( f ( x ) ) = x.

a) True

b) False

Fill in the blank

The domain of the function f (x) =  is

a) [− 2, − 1]

b) [1, 2]

c) [− 2, − 1] ⋃ [1, 2]

d) None of the above

Fill in the blank
If f (x) = sin ln  then the domain of f (x) is ………….

a) (−2, −1)

b) (−2, 1)

c) (0, 1)

d) (1, ∞)

If x, y, z are real and distinct then
8u =
is always

a) Non–negative

b) Non–positive

c) Zero

d) None of these

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

The number of real solutions of the equation | x |² – 3 | x | + 2 = 0 is

a) 4

b) 1

c) 3

d) 2

If  are any real numbers and n is any positive integer then

a)

b)

c)

d) none of these

The largest interval for which  is

a)

b)

c)

d)

Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 200 students is to be as small as possible then the school should be built at

a) Town B

b) 45 km from town A

c) Town A

d) 45 km from town B

If p, q, r are any real numbers, then

a) Max ( p, q ) < max ( p, q, r )

b) Min ( p, q ) =  

c) Max ( p, q ) < min ( p, q, r )

d) none of these

The equation  has

a) No root

b) One root

c) Two equal roots

d) Infinitely many roots

If then ab + bc + ca lies in the interval

a)  

b)  

c)  

d)  

If α and β are roots of  and  are roots of  then the equation  has always

a) Two real roots

b) Two positive roots

c) Two negative roots

d) One positive and one negative root