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*Multiple options can be correct

QUESTION: 1

Solution:

Let us check each option one by one.

(b) f (1) ≠ 3 as function is not defined for x = 1

∴ (b) is not correct.

*Multiple options can be correct

QUESTION: 2

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

Solution:

As (0, 0) and (x, g (x)) are two vertices of an equilateral triangle; therefore, length of the side of D is

∴ (b), (c) are the correct answers as (a) is not a function

(∴ image of x is not unique)

*Multiple options can be correct

QUESTION: 3

stands for the greatest integer function, then

Solution:

f(x) = cos [π^{2}] x + cos [– π^{2}] x

We know 9 < π^{2} < 10 and – 10 < – π^{2} < – 9

⇒ [π^{2}] = 9 and [–π^{2}] = – 10

⇒ ∴ f (x) = cos 9x + cos (–10x)

f (x) = cos 9x + cos 10 x

Let us check each option one by one.

Thus (a) and (c) are the correct options.

*Multiple options can be correct

QUESTION: 4

If f(x) = 3x – 5, then f^{–1}(x)

Solution:

f (x) = 3x – 5 (given), which is strictly increasing on R,

so f –1 (x) exists.

Let y = f (x) = 3x – 5

...(1)

and y = f (x) ⇒ x = f^{ –1}(y) ...(2)

*Multiple options can be correct

QUESTION: 5

If g (f(x)) = | sin x | and f (g(x)) = (sin √x)^{2}, then

Solution:

Let us check each option one by one.

*Multiple options can be correct

QUESTION: 6

Let f : (0, 1) → R be defined by where b is a constant such that 0 < b < 1. Then

Solution:

∴ f is neither onto nor invertible

Hence a and b are the correct options.

*Multiple options can be correct

QUESTION: 7

Let f : (–1, 1) ⇒ IR be such that Then the value (s) of

Solution:

*Multiple options can be correct

QUESTION: 8

The function f(x) = 2|x| + |x + 2| – | |x + 2| – 2 |x| has a local minimum or a local maximum at x =

Solution:

the critical points can be obtained by solving |x| = 0

The graph of y = f(x) is as follows

From graph f(x) has local minimum at –2 and 0 and

*Multiple options can be correct

QUESTION: 9

R be given by f (x) = (log(sec x + tan x))^{3}.

Then

Solution:

∴ f is an odd function.

(a) is correct and (d) is not correct.

Also

We know that strictly increasing function is one one.

∴ f is one one

∴ (b) is correct

∴ Range of f = (–∞, ∞) = R

∴ f is an onto function.

∴ (c) is correct.

*Multiple options can be correct

QUESTION: 10

Let a ∈ R and let f : R → R be given by f (x) = x^{5} – 5x + a. Then

Solution:

f (x) = x^{5} - 5 x+a

f (x) = 0 ⇒ x^{5} - 5 x +a = 0

⇒ a = 5x – x^{5} = g(x)

⇒ g(x) = 0 when x = 0, 5^{1/4} - 5^{1/4} and g' (x ) = 0 ⇒ x = 1, – 1

Also g (– 1) = – 4 and g(1) = 4

∴ graph of g(x) will be as shown below.

From graph

if a ∈ ( -4,-4)

then g(x) = a or f (x) = 0 has 3 real roots If a > 4 or a < – 4

then f(x) = 0 has only one real root.

∴ (b) and (d) are the correct options.

*Multiple options can be correct

QUESTION: 11

Let sin x for all x ∈ R. Let (fog)(x) denote f(g(x)) and (gof)(x) denote g(f(x)). Then which of the following is (are) true?

Solution:

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