Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 36

HARD

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IMPORTANT

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Let $f (x) = 1 - x - x^{3}$ . Then, the real values of $x$ satisfying the inequality, $1 - f (x) - f^{3} (x) > f (1 - 5 x)$ , are

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Important Questions on Functions

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JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 37

If a function satisfies

(x - y) f (x + y) - (x + y) f (x - y) = 2 (x^{2} y - y^{3}), \forall x, y \in R

and

f (1) = 2,

then

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 38

If the fundamental period of function

f (x) = s i n x + c o s (\sqrt{4 - a^{2}}) x

is

4 π

, then the value of

a

is/are:

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JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 39

Let

f (x)

be a real-valued function such that

f (0) = \frac{1}{2}

and

f (x + y) = f (x) f (a - y) + f (y) f (a - x) \forall x, y \in R

, then for some real

a

:

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 40

If

f (g (x))

is one-one function, then:

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 41

Which of the following functions have their range equal to

R

(the set of real numbers)?

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JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 42

Which of the following pairs of functions are identical?

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Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 43

For the following questions, choose the correct answers from the codes

(a), (b), (c)

and

(d)

defined as follows:
Statement

I

: The function

f (x) = x \sin x

and

f^{'} (x) = x \cos x + \sin x

are both non-periodic.
Statement

II

: The derivative of differentiable function (non-periodic) is non-periodic function.

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JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 3 - Functions>Proficiency in 'Functions' Exercise 1>Q 44

Statement

I

: The maximum value of

s i n \sqrt{2} x + s i n a x

cannot be

2

.
(where

a

is positive rational number)
Statement

II

:

\frac{\sqrt{2}}{a}

is irrational.