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

HARD

JEE Main

IMPORTANT

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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 ______.

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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 - 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

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 28 July 2022 Shift 2>Q 74

Let

f (x) = a x^{2} + b x + c

be such that

f (1) = 3, f (- 2) = λ

and

f (3) = 4

. If

f (0) + f (1) + f (- 2) + f (3) = 14

, then

λ

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 24 January 2023 Shift 1>Q 75

The equation

x^{2} - 4 x + [x] + 3 = x [x]

, where

[x]

denotes the greatest integer function, has:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 24 January 2023 Shift 2>Q 76

Let

f (x)

be a function such that

f (x + y) = f (x) \cdot f (y)

for all

x, y \in N

, If

f (1) = 3

and

\sum_{k = 1}^{n} f (k) = 3279

, then the value of

n

is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 23 - Functions>JEE Main - 24 January 2023 Shift 2>Q 77

If

f (x) = \frac{2^{2 x}}{2^{2 x} + 2}

,

x \in R

, then

f (\frac{1}{2023}) + f (\frac{2}{2023}) + f (\frac{3}{2023}) . . . . . . . . . f (\frac{2022}{2023})

is equal to