EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 June 2022 Shift 2>Q 77

HARD

JEE Main

IMPORTANT

Earn 100

Let $f, g : R \to R$ be functions defined by
$f (x) = \{\begin{cases} [x] & , x < 0 \\ |1 - x| & , x \geq 0 \end{cases}$ and
$g (x) = \{\begin{cases} e^{x} - x, & x < 0 \\ {(x - 1)}^{2} - 1, & x \geq 0 \end{cases}$
where $[x]$ denote the greatest integer less than or equal to $x$ . Then, the function fog is discontinuous at exactly

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Important Questions on Continuity and Differentiability

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 25 July 2022 Shift 1>Q 78

Let

f (x) = \{\begin{matrix} |4 x^{2} - 8 x + 5|, & if 8 x^{2} - 6 x + 1 \geq 0 \\ [4 x^{2} - 8 x + 5], & if 8 x^{2} - 6 x + 1 < 0 \end{matrix}

, where

[α]

denotes the greatest integer less than or equal to

α

. Then the number of points in

R

where

f

is not differentiable is _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 July 2022 Shift 1>Q 79

If

f (x) = \{\begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}

and

g (x) = \{\begin{cases} x + 1, & x < 0 \\ {(x - 4)}^{2} + b, & x \geq 0 \end{cases}

are continuous on

R

, then

(g o f) (2) + (f o g) (- 2)

is equal to:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 27 July 2022 Shift 2>Q 80

If for

p \neq q \neq 0

, then function

f (x) = \frac{\sqrt[7]{p (729 + x)} - 3}{\sqrt[3]{729 + q x} - 9}

is continuous at

x = 0

, then

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 July 2022 Shift 1>Q 81

Let

f : [0, 1] \to R

be a twice differentiable function in

(0, 1)

such that

f (0) = 3

and

f (1) = 5

. If the line

y = 2 x + 3

intersects the graph of

f

at only two distinct points in

(0, 1)

, then the least number of points

x \in (0, 1)

, at which

f^{"} (x) = 0

, is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 July 2022 Shift 2>Q 82

The function

f : R \to R

defined by

f (x) = \lim_{n \to \infty} \frac{\cos (2 π x) - x^{2 n} \sin (x - 1)}{1 + x^{2 n + 1} - x^{2 n}}

is continuous for all

x

in

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 29 July 2022 Shift 1>Q 83

The number of points, where the function

f : R \to R, f (x) = |x - 1| \cos |x - 2| \sin |x - 1| + (x - 3) |x^{2} - 5 x + 4|

, is NOT differentiable, is

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 29 July 2022 Shift 2>Q 84

Let the function $f (x) = \{\begin{matrix} \frac{\log_{e} (1 + 5 x) - \log_{e} (1 + α x)}{x} & if x \neq 0 \\ 10 & if x = 0 \end{matrix}$ be continuous at $x = 0$ . Then $α$ is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 29 July 2022 Shift 2>Q 85

If

[t]

denotes the greatest integer

\leq t

, then number of points, at which the function

f (x) = 4 |2 x + 3| +

9 [x + \frac{1}{2}] - 12 [x + 20]

is not differentiable in the open interval

(- 20, 20)

, is ______.