EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 28 June 2022 Shift 1>Q 65

MEDIUM

JEE Main

IMPORTANT

Earn 100

Let a function $f : ℕ \to ℕ$ be defined by $f (n) = [\begin{array}{l} 2 n, & n = 2, 4, 6, 8, \dots . . \\ n - 1, & n = 3, 7, 11, 15, \dots . . \\ \frac{n + 1}{2}, & n = 1, 5, 9, 13, \dots . . \end{array}$
then, $f$ is

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Important Questions on Functions

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 28 June 2022 Shift 2>Q 66

Let

S = \{1, 2, 3, 4\}

. Then the number of elements in the set {

f : S \times S \to S : f

is onto and

f (a, b) = f (b, a)

\geq a \forall (a, b) \in S \times S

} is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 29 June 2022 Shift 1>Q 67

The domain of the function

\cos^{- 1} (\frac{2 \sin^{- 1} (\frac{1}{4 x^{2} - 1})}{π})

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 29 June 2022 Shift 2>Q 68

Let

f (x)

and

g (x)

be two real polynomials of degree

2

and

1

respectively. If

f (g (x)) = 8 x^{2} - 2 x

, and

g (f (x)) = 4 x^{2} + 6 x + 1

, then the value of

f (2) + g (2)

is ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 25 July 2022 Shift 2>Q 69

Let

f (x)

be a quadratic polynomial with leading coefficient

1

such that

f (0) = p, p \neq 0

, and

f (1) = \frac{1}{3}

. If the equations

f (x) = 0

and

f o f o f o f (x) = 0

have a common real root, then

f (- 3)

is equal to ______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 26 July 2022 Shift 1>Q 70

Let

f : R \to R

be a continuous function such that

f (3 x) - f (x) = x

. If

f (8) = 7

, then

f (14)

is equal to:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 27 July 2022 Shift 1>Q 71

Let

f, g : ℕ - \{1\} \to ℕ

be functions defined by

f (a) = α

, where

α

is the maximum of the powers of those primes

p

such that

p^{α}

divides

a

, and

g (a) = a + 1

, for all

a \in ℕ - \{1\}

. Then, the function

f + g

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 27 July 2022 Shift 2>Q 72

The domain of the function

f (x) = \sin^{- 1} [2 x^{2} - 3] + \log_{2} (\log_{\frac{1}{2}} (x^{2} - 5 x + 5))

, where

[t]

is the greatest integer function, is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 28 July 2022 Shift 1>Q 73

Let

α, β

and

γ

be three positive real numbers. Let

f (x) = α x^{5} + β x^{3} + γ x, x \in R

and

g : R \to R

be such that

g (f (x)) = x

for all

x \in R

. If

a_{1}, a_{2}, a_{3}, \dots, a_{n}

be in arithmetic progression with mean zero, then the value of

f (g (\frac{1}{n} \sum_{i = 1}^{n} f (a_{i})))

is equal to