Question 1: Palette 1/216

JEE Mains 2026 MCQ

Let $f(x) = \lim\limits_{\theta \to 0} \left( \frac{\cos \pi x - x^\left( \frac{2}{\theta} \right) \sin(x-1)}{1 + x^\left( \frac{2}{\theta} \right) (x-1)} \right),\ x \in \mathbb{R}$. Consider the following two statements :

(I) $f(x)$ is discontinuous at $x=1$.

(II) $f(x)$ is continuous at $x = -1$.

Then,

A

Neither (I) nor (II) is True

B

Only (II) is True

C

Only (I) is True

D

Both (I) and (II) are True

Question 2: Palette 2/216

JEE Mains 2026 MCQ

The value of

$ \lim\limits_{x \rightarrow 0} \frac{\log _e\left(\sec (e x) \cdot \sec \left(e^2 x\right) \cdot \ldots \cdot \sec \left(e^{10} x\right)\right)}{e^2-e^{2 \cos x}} $

is equal to

A

$ \frac{\left(e^{10}-1\right)}{2 e^2\left(e^2-1\right)} $

B

$ \frac{\left(e^{20}-1\right)}{2 e^2\left(e^2-1\right)} $

C

$ \frac{\left(e^{10}-1\right)}{2\left(e^2-1\right)} $

D

$ \frac{\left(e^{20}-1\right)}{2\left(e^2-1\right)} $

Question 3: Palette 3/216

JEE Mains 2026 MCQ

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $\lim\limits_{t \rightarrow x}\left(\frac{t^2 y(x)-x^2 y(t)}{x-t}\right)=3$ for each $x > 0$. Then $2 y(2)$ is equal to :

A

27

B

18

C

23

D

12

Question 4: Palette 4/216

JEE Mains 2026 MCQ

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function

$ f(x)=\left\{\begin{array}{cl} b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^3} & , x>0 \\ a & , x=0 \end{array}\right. $

is continuous at $x=0$, then $a^2+b^2$ is equal to :

A

$\frac{1}{2}$

B

$\frac{5}{8}$

C

$\frac{3}{4}$

D

$\frac{9}{16}$

Question 5: Palette 5/216

JEE Mains 2026 MCQ

Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x)= \begin{cases}2 \alpha\left(x^2-2\right)+2 \beta x & , x<1 \\ (\alpha+3) x+(\alpha-\beta) & , x \geq 1\end{cases}$ be differentiable at all $x \in \mathbb{R}$. Then $34(\alpha+\beta)$ is equal to

A

48

B

84

C

36

D

24

Question 6: Palette 6/216

JEE Mains 2026 MCQ

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log _e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

A

$\frac{2}{3}$

B

$\frac{1}{2}$

C

2

D

$\frac{3}{2}$

Question 7: Palette 7/216

JEE Mains 2026 MCQ

If $f(x)=\left\{\begin{array}{cc}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0\end{array}\right.$

is continuous at $x=0$, then $a+b$ is equal to :

A

1

B

2

C

0

D

4

Question 8: Palette 8/216

JEE Mains 2026 MCQ

Let $f(x)= \begin{cases}\frac{\mathrm{a} x^2+2 \mathrm{a} x+3}{4 x^2+4 x-3} & , x \neq-\frac{3}{2}, \frac{1}{2} \\ \mathrm{~b} & , x=-\frac{3}{2}, \frac{1}{2}\end{cases}$ be continuous at $x=-\frac{3}{2}$. If $f \circ f(x)=\frac{7}{5}$, then $x$ is equal to:

A

1.4

B

2

C

1

D

0

Question 9: Palette 9/216

JEE Mains 2026 MCQ

If $\lim\limits_{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to :

A

3

B

5

C

9

D

7

Question 10: Palette 10/216

JEE Mains 2026 MCQ

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^2\right\}$.

Let $\mathrm{S}=\left\{x \in(-2,2)\right.$ : the function $\mathrm{g}(x)=|x|\left[x^2\right]$ is discontinuous at $\left.x\right\}$.

Then $\sum\limits_{x \in \mathrm{~S}} f(x)$ equals

A

$2-\sqrt{2}$

B

$2 \sqrt{6}-3 \sqrt{2}$

C

$1-\sqrt{2}$

D

$\sqrt{6}-2 \sqrt{2}$

Question 11: Palette 11/216

JEE Mains 2026 MCQ

Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $f(3)=18, f^{\prime}(3)=0$ and $f^{\prime \prime}(3)=4$.

Then $\lim\limits _{x \rightarrow 1}\left(\log _e\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to :

A

9

B

18

C

1

D

2

Question 12: Palette 12/216

JEE Mains 2026 MCQ

For the function $f(x)=\mathrm{e}^{\sin |x|}-|x|, x \in \mathbf{R}$, consider the following statements :

Statement I : $ f$ is differentiable for all $x \in \mathbf{R}$.

Statement II : $ f$ is increasing in $\left(-\pi,-\frac{\pi}{2}\right)$.

In the light of the above statements, choose the correct answer from the options given below :

A

Both Statement I and Statement II are true

B

Both Statement I and Statement II are false

C

Statement I is true but Statement II is false

D

Statement I is false but Statement II is true

Question 13: Palette 13/216

JEE Mains 2026 MCQ

$ \text { The value of } \lim\limits_{x \rightarrow 0}\left(\frac{x^2 \sin ^2 x}{x^2-\sin ^2 x}\right) \text { is : } $

A

2

B

3

C

4

D

6

Question 14: Palette 14/216

JEE Mains 2026 MCQ

Let $f(x)=\lim \limits_{y \rightarrow 0} \frac{(1-\cos (x y)) \tan (x y)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x \in \mathbf{R}$ is :

A

0

B

2

C

3

D

1

Question 15: Palette 15/216

JEE Mains 2026 MCQ

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$ for all $x \in \mathbf{R}, f^{\prime}(1)=2 g^{\prime}(1)=4$ and $g(2)=3 f(2)=9$. Then $f(25)-g(25)$ is equal to :

A

20

B

40

C

-20

D

-40

Question 16: Palette 16/216

JEE Mains 2026 MCQ

The product of all possible values of $\alpha$, for which

$\lim \limits_{x \rightarrow 0}\left(\frac{1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x)}{\sin ^2((\alpha+1) x)}\right)=2$, is :

A

-2

B

1

C

-1

D

${\frac{5}{4}}$

Question 17: Palette 17/216

JEE Mains 2026 MCQ

If $ \lim\limits_{x\to 2} \frac{\sin\left(x^3 - 5x^2 + ax + b\right)}{\left(\sqrt{x-1}-1\right) \log_e(x-1)} = m $, then $a + b + m$ is equal to :

A

5

B

6

C

8

D

10

Question 18: Palette 18/216

JEE Mains 2025 MCQ

Given below are two statements:

Statement I: $ \lim\limits_{x \to 0} \left( \frac{\tan^{-1} x + \log_e \sqrt{\frac{1+x}{1-x}} - 2x}{x^5} \right) = \frac{2}{5} $

Statement II: $ \lim\limits_{x \to 1} \left( x^{\frac{2}{1-x}} \right) = \frac{1}{e^2} $

In the light of the above statements, choose the correct answer from the options given below:

A

Statement I is false but Statement II is true

B

Both Statement I and Statement II are false

C

Both Statement I and Statement II are true

D

Statement I is true but Statement II is false

Question 19: Palette 19/216

JEE Mains 2025 MCQ

$\lim _\limits{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^2\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is equal to

A

$\frac{5}{3}$

B

1

C

$\frac{1}{3}$

D

$\frac{1}{15}$

Question 20: Palette 20/216

JEE Mains 2025 MCQ

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f^{\prime}(2)=4$. Let $\lim \limits_{x \rightarrow 0}(f(2+x))^{3 / x}=\mathrm{e}^\alpha$. Then the number of times the curve $y=4 x^3-4 x^2-4(\alpha-7) x-\alpha$ meets $x$-axis is :

A

3

B

1

C

2

D

0

Question 21: Palette 21/216

JEE Mains 2025 MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x \in \mathbb{R}$. If $\lim _\limits{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then $\sum_\limits{r=1}^{10} G\left(r^2\right)$ is equal to

A

215

B

420

C

385

D

540

Question 22: Palette 22/216

JEE Mains 2025 MCQ

If $\lim _\limits{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^3}=-1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda+\mu$ is equal to

A

20

B

19

C

18

D

17

Question 23: Palette 23/216

JEE Mains 2025 MCQ

Let $\quad f(x)= \begin{cases}(1+a x)^{1 / x} & , x<0 \\ 1+b, & x=0 \\ \frac{(x+4)^{1 / 2}-2}{(x+c)^{1 / 3}-2}, & x>0\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to:

A

64

B

48

C

36

D

72

Question 24: Palette 24/216

JEE Mains 2025 MCQ

$If\,\mathop {\lim }\limits_{x \to 0} {{\cos (2x) + a\cos (4x) - b} \over {{x^4}}}is\,finite,\,then\,(a + b)\,is\,equal\,to:$

A

0

B

$\frac{3}{4}$

C

-1

D

$\frac{1}{2}$

Question 25: Palette 25/216

JEE Mains 2025 MCQ

For $\alpha, \beta, \gamma \in \mathbf{R}$, if $\lim _\limits{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) \mathrm{e}^{x^2}}{\sin 2 x-\beta x}=3$, then $\beta+\gamma-\alpha$ is equal to :

A

$-$1

B

4

C

6

D

7

Question 26: Palette 26/216

JEE Mains 2025 MCQ

Let the function $f(x)=\left(x^2-1\right)\left|x^2-a x+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12 x+5 y+10=0$ is equal to :

A

5

B

2

C

4

D

3

Question 27: Palette 27/216

JEE Mains 2025 MCQ

The value of $\lim \limits_{n \rightarrow \infty}\left(\sum\limits_{k=1}^n \frac{k^3+6 k^2+11 k+5}{(k+3)!}\right)$ is :

A

5/3

B

2

C

4/3

D

7/3

Question 28: Palette 28/216

JEE Mains 2025 MCQ

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2< x<3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

A

6

B

9

C

8

D

7

Question 29: Palette 29/216

JEE Mains 2025 MCQ

$\lim _\limits{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)$ is:

A

$\frac{1}{\sqrt{15}}$

B

$\frac{1}{2 \sqrt{5}}$

C

$0$

D

$-\frac{1}{2 \sqrt{5}}$

Question 30: Palette 30/216

JEE Mains 2025 MCQ

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the $\lim\limits _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in \mathbb{R}$, then $\alpha+2 \beta$ is equal to

A

6

B

5

C

3

D

4

Question 31: Palette 31/216

JEE Mains 2025 MCQ

$\lim \limits_{x \rightarrow \infty} \frac{\left(2 x^2-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^2+5 x+4\right) \sqrt{(3 x+2)^x}}$ is equal to :

A

$\frac{2 e}{3}$

B

$\frac{2}{3 \sqrt{\mathrm{e}}}$

C

$\frac{2 \mathrm{e}}{\sqrt{3}}$

D

$\frac{2}{\sqrt{3 \mathrm{e}}}$

Question 32: Palette 32/216

JEE Mains 2025 MCQ

If the function

$ f(x)=\left\{\begin{array}{l} \frac{2}{x}\left\{\sin \left(k_1+1\right) x+\sin \left(k_2-1\right) x\right\}, \quad x<0 \\ 4, \quad x=0 \\ \frac{2}{x} \log _e\left(\frac{2+k_1 x}{2+k_2 x}\right), \quad x>0 \end{array}\right. $

is continuous at $x=0$, then $k_1^2+k_2^2$ is equal to :

A

5

B

10

C

20

D

8

Question 33: Palette 33/216

JEE Mains 2025 MCQ

If $\lim _\limits{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :

A

$e^{-2}$

B

$\mathrm{e}^2$

C

$e$

D

$e^{-1}$

Question 34: Palette 34/216

JEE Mains 2025 MCQ

If $\sum_\limits{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}$, then $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=1}^n\left(\frac{1}{T_r}\right)$ is equal to :

A

$\frac{2}{3}$

B

$\frac{1}{3}$

C

1

D

0

Question 35: Palette 35/216

JEE Mains 2024 MCQ

$\lim _\limits{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to

A

$\frac{-2}{e}$

B

$e-e^2$

C

0

D

$e$

Question 36: Palette 36/216

JEE Mains 2024 MCQ

For $\mathrm{a}, \mathrm{b}>0$, let $f(x)= \begin{cases}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x< 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}}, & x> 0\end{cases}$ be a continuous function at $x=0$. Then $\frac{\mathrm{b}}{\mathrm{a}}$ is equal to :

A

4

B

5

C

8

D

6

Question 37: Palette 37/216

JEE Mains 2024 MCQ

$\lim _\limits{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}$ is equal to :

A

$\frac{2}{3}$

B

$\frac{1}{2}$

C

$\frac{3}{4}$

D

$\frac{1}{3}$

Question 38: Palette 38/216

JEE Mains 2024 MCQ

Let ,$f:[-1,2] \rightarrow \mathbf{R}$ be given by $f(x)=2 x^2+x+\left[x^2\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

A

5

B

6

C

4

D

3

Question 39: Palette 39/216

JEE Mains 2024 MCQ

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}$, is continuous at $x=0$, then $f(0)$ is equal to :

A

4

B

$-$2

C

$-$4

D

2

Question 40: Palette 40/216

JEE Mains 2024 MCQ

If the function

$f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$

is continuous at $x=0$, then the value of $a^2$ is equal to

A

968

B

1250

C

1152

D

746

Question 41: Palette 41/216

JEE Mains 2024 MCQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by

$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0\end{cases}$

where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to :

A

48

B

6

C

3

D

12

Question 42: Palette 42/216

JEE Mains 2024 MCQ

Let $f(x)=\left|2 x^2+5\right| x|-3|, x \in \mathbf{R}$. If $\mathrm{m}$ and $\mathrm{n}$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $\mathrm{m}+\mathrm{n}$ is equal to :

A

5

B

3

C

2

D

0

Question 43: Palette 43/216

JEE Mains 2024 MCQ

Let $f(x)=\left\{\begin{array}{l}x-1, x \text { is even, } \\ 2 x, \quad x \text { is odd, }\end{array} x \in \mathbf{N}\right.$.

If for some $\mathrm{a} \in \mathbf{N}, f(f(f(\mathrm{a})))=21$, then $\lim\limits_{x \rightarrow \mathrm{a}^{-}}\left\{\frac{|x|^3}{\mathrm{a}}-\left[\frac{x}{\mathrm{a}}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :

A

169

B

121

C

225

D

144

Question 44: Palette 44/216

JEE Mains 2024 MCQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$ f(x)= \begin{cases}\frac{a-b \cos 2 x}{x^2} ; & x<0 \\\\ x^2+c x+2 ; & 0 \leq x \leq 1 \\\\ 2 x+1 ; & x>1\end{cases} $

If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ is the number of points where $f$ is NOT differential then $\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}$ equals :

A

1

B

4

C

3

D

2

Question 45: Palette 45/216

JEE Mains 2024 MCQ

Consider the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=e^{-\left|\log _e x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

A

0

B

1

C

2

D

3

Question 46: Palette 46/216

JEE Mains 2024 MCQ

$\lim _\limits{x \rightarrow 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

A

is equal to 1

B

does not exist

C

is equal to $-1$

D

is equal to 2

Question 47: Palette 47/216

JEE Mains 2024 MCQ

Let $g(x)$ be a linear function and $f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$, is continuous at $x=0$. If $f^{\prime}(1)=f(-1)$, then the value $g(3)$ is

A

$\log _e\left(\frac{4}{9}\right)-1$

B

$\frac{1}{3} \log _e\left(\frac{4}{9 e^{1 / 3}}\right)$

C

$\log _e\left(\frac{4}{9 e^{1 / 3}}\right)$

D

$\frac{1}{3} \log _e\left(\frac{4}{9}\right)+1$

Question 48: Palette 48/216

JEE Mains 2024 MCQ

Consider the function $f:(0,2) \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by

$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x < 2 \end{array} .\right. \text { Then, }$

A

$g$ is continuous but not differentiable at $x=1$

B

$g$ is continuous and differentiable for all $x \in(0,2)$

C

$g$ is not continuous for all $x \in(0,2)$

D

$g$ is neither continuous nor differentiable at $x=1$

Question 49: Palette 49/216

JEE Mains 2024 MCQ

$\text { If } \lim _\limits{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3} \text {, then } 2 \alpha-\beta \text { is equal to : }$

A

2

B

1

C

5

D

7

Question 50: Palette 50/216

JEE Mains 2024 MCQ

Consider the function.

$ f(x)=\left\{\begin{array}{cc} \frac{\mathrm{a}\left(7 x-12-x^2\right)}{\mathrm{b}\left|x^2-7 x+12\right|} & , x<3 \\\\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\\\ \mathrm{~b} & , x=3, \end{array}\right. $

where $[x]$ denotes the greatest integer less than or equal to $x$. If $\mathrm{S}$ denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x=3$, then the number of elements in $\mathrm{S}$ is :

A

Infinitely many

B

4

C

2

D

1

Question 51: Palette 51/216

JEE Mains 2024 MCQ

If $\mathrm{a}=\lim\limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\lim\limits _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a b^3$ is :

A

36

B

25

C

32

D

30

Question 52: Palette 52/216

JEE Mains 2023 MCQ

Let $[x]$ denote the greatest integer function and

$f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be the number of

points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in

$(0,2)$, where $f$ is not differentiable. Then $(m+n)^{2}+2$ is equal to :

A

3

B

6

C

2

D

11

Question 53: Palette 53/216

JEE Mains 2023 MCQ

If $\lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17$, then $5 a^{2}+b^{2}$ is equal to

A

64

B

68

C

72

D

76

Question 54: Palette 54/216

JEE Mains 2023 MCQ

Let $f$ and $g$ be two functions defined by

$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ |x-1|, & x \geq 0\end{array}\right.$ and $\mathrm{g}(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ 1, & x \geq 0\end{array}\right.$

Then $(g \circ f)(x)$ is :

A

continuous everywhere but not differentiable at $x=1$

B

differentiable everywhere

C

not continuous at $x=-1$

D

continuous everywhere but not differentiable exactly at one point

Question 55: Palette 55/216

JEE Mains 2023 MCQ

Let $f(x)=\left[x^{2}-x\right]+|-x+[x]|$, where $x \in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is :

A

continuous at $x=0$, but not continuous at $x=1$

B

continuous at $x=0$ and $x=1$

C

continuous at $x=1$, but not continuous at $x=0$

D

not continuous at $x=0$ and $x=1$

Question 56: Palette 56/216

JEE Mains 2023 MCQ

If $\alpha > \beta > 0$ are the roots of the equation $a x^{2}+b x+1=0$, and $\lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^{2}+b x+a\right)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right), \text { then } \mathrm{k} \text { is equal to }$ :

A

$2 \beta$

B

$\beta$

C

$\alpha$

D

$2 \alpha$

Question 57: Palette 57/216

JEE Mains 2023 MCQ

$\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+1)\right)^{5}}\right)\right)$ is equal to _____________.

A

15

B

18

C

9

D

24

Question 58: Palette 58/216

JEE Mains 2023 MCQ

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}$ be $\mathrm{n}$ positive consecutive terms of an arithmetic progression. If $\mathrm{d} > 0$ is its common difference, then

$\lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right)$ is

A

$\frac{1}{\sqrt{d}}$

B

1

C

0

D

$\sqrt{d}$

Question 59: Palette 59/216

JEE Mains 2023 MCQ

$ \lim\limits_{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3 $

A

is equal to 9

B

is equal to $\frac{27}{2}$

C

does not exist

D

is equal to 27

Question 60: Palette 60/216

JEE Mains 2023 MCQ

Let $f, g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as

$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.$

$g(x)=\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-1\end{array}\right.$

and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the

value of $\lim\limits_{x \rightarrow 1} g(h(x-1))$ is :

A

1

B

$-1$

C

$\sin (1)$

D

0

Question 61: Palette 61/216

JEE Mains 2023 MCQ

Suppose $f: \mathbb{R} \rightarrow(0, \infty)$ be a differentiable function such that $5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}$. If $f(3)=320$, then $\sum_\limits{n=0}^{5} f(n)$ is equal to :

A

6875

B

6525

C

6575

D

6825

Question 62: Palette 62/216

JEE Mains 2023 MCQ

Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.$

Then $\mathop {\lim }\limits_{x \to 2{p^ + }} [f(x)]$, where $\left[ . \right]$ denotes greatest integer function, is

A

2

B

1

C

0

D

$-1$

Question 63: Palette 63/216

JEE Mains 2023 MCQ

If the function $f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$

is continuous at $x = {\pi \over 2}$, then $9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$ is equal to

A

11

B

10

C

8

D

2e$^4$ + 8

Question 64: Palette 64/216

JEE Mains 2023 MCQ

The value of $\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + 3} - \sqrt {{n^4} + 5n + 4} }}$ is :

A

${3 \over {2\sqrt 2 }}$

B

${3 \over 2}(\sqrt 2 + 1)$

C

$3(\sqrt 2 + 1)$

D

${{\sqrt 2 + 1} \over 2}$

Question 65: Palette 65/216

JEE Mains 2023 MCQ

The set of all values of $a$ for which $\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0$, where [$\alpha$] denotes the greatest integer less than or equal to $\alpha$ is equal to

A

$[-7.5,-6.5]$

B

$(-7.5,-6.5]$

C

$[-7.5,-6.5)$

D

$(-7.5,-6.5)$

Question 66: Palette 66/216

JEE Mains 2023 MCQ

$\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t}}}}} \right)^{{{\sin }^2}t}}$ is equal to

A

${{n(n + 1)} \over 2}$

B

n

C

n$^2$ + n

D

n$^2$

Question 67: Palette 67/216

JEE Mains 2023 MCQ

Let $f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$

Then at $x=0$

A

$f$ is continuous but $f'$ is not continuous

B

$f$ and $f'$ both are continuous

C

$f$ is continuous but not differentiable

D

$f'$ is continuous but not differentiable

Question 68: Palette 68/216

JEE Mains 2022 MCQ

$ \text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text { if } x=0 \end{array} \text { be continuous at } x=0 .\right. $

Then $\alpha$ is equal to

A

10

B

$-$10

C

5

D

$-$5

Question 69: Palette 69/216

JEE Mains 2022 MCQ

If $\lim\limits_{x \rightarrow 0} \frac{\alpha \mathrm{e}^{x}+\beta \mathrm{e}^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}$, where $\alpha, \beta, \gamma \in \mathbf{R}$, then which of the following is NOT correct?

A

$\alpha^{2}+\beta^{2}+\gamma^{2}=6$

B

$\alpha \beta+\beta \gamma+\gamma \alpha+1=0$

C

$\alpha\beta^{2}+\beta \gamma^{2}+\gamma \alpha^{2}+3=0$

D

$\alpha^{2}-\beta^{2}+\gamma^{2}=4$

Question 70: Palette 70/216

JEE Mains 2022 MCQ

The number of points, where the function $f: \mathbf{R} \rightarrow \mathbf{R}$,

$f(x)=|x-1| \cos |x-2| \sin |x-1|+(x-3)\left|x^{2}-5 x+4\right|$, is NOT differentiable, is :

A

1

B

2

C

3

D

4

Question 71: Palette 71/216

JEE Mains 2022 MCQ

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x)=\lim\limits_{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}$ is continuous for all x in :

A

$R-\{-1\}$

B

$ \mathbb{R}-\{-1,1\}$

C

$R-\{1\}$

D

$R-\{0\}$

Question 72: Palette 72/216

JEE Mains 2022 MCQ

If for $\mathrm{p} \neq \mathrm{q} \neq 0$, the function $f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}$ is continuous at $x=0$, then :

A

$7 p q \,f(0)-1=0$

B

$63 q \,f(0)-\mathrm{p}^{2}=0$

C

$21 q \,f(0)-\mathrm{p}^{2}=0$

D

$7 p q \,f(0)-9=0$

Question 73: Palette 73/216

JEE Mains 2022 MCQ

Let $\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}$ for some $\alpha \in \mathbb{R}$. Then the value of $\alpha+\beta$ is :

A

$\frac{14}{5}$

B

$\frac{3}{2}$

C

$\frac{5}{2}$

D

$\frac{7}{2}$

Question 74: Palette 74/216

JEE Mains 2022 MCQ

Let f : R $\to$ R be a continuous function such that $f(3x) - f(x) = x$. If $f(8) = 7$, then $f(14)$ is equal to :

A

4

B

10

C

11

D

16

Question 75: Palette 75/216

JEE Mains 2022 MCQ

If the function $f(x) = \left\{ {\matrix{ {{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr k & , & {x = 0} \cr } } \right.$ is continuous at x = 0, then k is equal to:

A

1

B

$-$1

C

e

D

0

Question 76: Palette 76/216

JEE Mains 2022 MCQ

If $f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {x + 1} & , & {x < 0} \cr {{{(x - 4)}^2} + b} & , & {x \ge 0} \cr } } \right.$ are continuous on R, then $(gof)(2) + (fog)( - 2)$ is equal to :

A

$-$10

B

10

C

8

D

$-$8

Question 77: Palette 77/216

JEE Mains 2022 MCQ

Let $f(x) = \left\{ {\matrix{ {{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr { - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr } } \right.$.

Then the set of all values of b, for which f(x) has maximum value at x = 1, is :

A

($-$6, $-$2)

B

(2, 6)

C

$[ - 6, - 2) \cup (2,6]$

D

$\left[ {-\sqrt 6 , - 2} \right) \cup \left( {2,\sqrt 6 } \right]$

Question 78: Palette 78/216

JEE Mains 2022 MCQ

$\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}$ is equal to

A

14

B

7

C

14$\sqrt2$

D

7$\sqrt2$

Question 79: Palette 79/216

JEE Mains 2022 MCQ

If $\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$, then $8(\alpha+\beta)$ is equal to :

A

4

B

$-$8

C

$-$4

D

8

Question 80: Palette 80/216

JEE Mains 2022 MCQ

The value of $\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}$ is equal to:

A

${{{\pi ^2}} \over 6}$

B

${{{\pi ^2}} \over 3}$

C

${{{\pi ^2}} \over 2}$

D

$\pi$²

Question 81: Palette 81/216

JEE Mains 2022 MCQ

Let f, g : R $\to$ R be functions defined by

$f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {{e^x} - x} & , & {x < 0} \cr {{{(x - 1)}^2} - 1} & , & {x \ge 0} \cr } } \right.$ where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :

A

one point

B

two points

C

three points

D

four points

Question 82: Palette 82/216

JEE Mains 2022 MCQ

The value of

$\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}$ is equal to :

A

1

B

2

C

3

D

6

Question 83: Palette 83/216

JEE Mains 2022 MCQ

Let f : R $\to$ R be defined as

$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$

where a, b, c $\in$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

A

There exists a, b, c $\in$ R such that f is continuous on R.

B

If f is discontinuous at exactly one point, then a + b + c = 1

C

If f is discontinuous at exactly one point, then a + b + c $\ne$ 1

D

f is discontinuous at at least two points, for any values of a, b and c

Question 84: Palette 84/216

JEE Mains 2022 MCQ

Let a be an integer such that $\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}$ exists, where [t] is greatest integer $\le$ t. Then a is equal to :

A

$-$6

B

$-$2

C

2

D

6

Question 85: Palette 85/216

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}$ is equal to :

A

${1 \over 3}$

B

${1 \over 4}$

C

${1 \over 6}$

D

${1 \over 12}$

Question 86: Palette 86/216

JEE Mains 2022 MCQ

Let f(x) = min {1, 1 + x sin x}, 0 $\le$ x $\le$ 2$\pi $. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

A

(2, 0)

B

(1, 0)

C

(1, 1)

D

(2, 1)

Question 87: Palette 87/216

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$ is equal to :

A

$\sqrt 2 $

B

$ - \sqrt 2 $

C

${1 \over {\sqrt 2 }}$

D

$ - {1 \over {\sqrt 2 }}$

Question 88: Palette 88/216

JEE Mains 2022 MCQ

Let f, g : R $\to$ R be two real valued functions defined as $f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x < 0} \cr {{e^x}} & , & {x \ge 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {{x^2} + {k_1}x} & , & {x < 0} \cr {4x + {k_2}} & , & {x \ge 0} \cr } } \right.$, where k₁ and k₂ are real constants. If (gof) is differentiable at x = 0, then (gof) ($-$ 4) + (gof) (4) is equal to :

A

$4({e^4} + 1)$

B

$2(2{e^4} + 1)$

C

$4{e^4}$

D

$2(2{e^4} - 1)$

Question 89: Palette 89/216

JEE Mains 2022 MCQ

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)$ is equal to

A

${1 \over {12}}$

B

$-$${1 \over {18}}$

C

$-$${1 \over {12}}$

D

${1 \over {6}}$

Question 90: Palette 90/216

JEE Mains 2022 MCQ

Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$. Then, the value of $\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$ is equal to:

A

$-$15

B

$-$60

C

60

D

15

Question 91: Palette 91/216

JEE Mains 2022 MCQ

Let $f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.$

where [t] denotes greatest integer $\le$ t. If m is the number of points where $f$ is not continuous and n is the number of points where $f$ is not differentiable, then the ordered pair (m, n) is :

A

(3, 3)

B

(2, 4)

C

(2, 3)

D

(3, 4)

Question 92: Palette 92/216

JEE Mains 2021 MCQ

If $\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$ and $\beta = \mathop {\lim }\limits_{x \to 0 } {(\cos x)^{\cot x}}$ are the roots of the equation, ax² + bx $-$ 4 = 0, then the ordered pair (a, b) is :

A

(1, $-$3)

B

($-$1, 3)

C

($-$1, $-$3)

D

(1, 3)

Question 93: Palette 93/216

JEE Mains 2021 MCQ

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

A

f''(x) = 0 for all x $\in$ (0, 2)

B

f''(x) = 0 for some x $\in$ (0, 2)

C

f'(x) = 0 for some x $\in$ [0, 2]

D

f''(x) > 0 for all x $\in$ (0, 2)

Question 94: Palette 94/216

JEE Mains 2021 MCQ

The function

$f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}$ is not differentiable at exactly :

A

four points

B

three points

C

two points

D

one point

Question 95: Palette 95/216

JEE Mains 2021 MCQ

If the function
$f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x < 0} \cr k & , & {x = 0} \cr {{{{{\cos }^2}x - {{\sin }^2}x - 1} \over {\sqrt {{x^2} + 1} - 1}}} & , & {x > 0} \cr } } \right.$ is continuous

at x = 0, then ${1 \over a} + {1 \over b} + {4 \over k}$ is equal to :

A

$-$5

B

5

C

$-$4

D

4

Question 96: Palette 96/216

JEE Mains 2021 MCQ

$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}$ is equal to :

A

${\pi ^2}$

B

$2{\pi ^2}$

C

$4{\pi ^2}$

D

$4\pi $

Question 97: Palette 97/216

JEE Mains 2021 MCQ

If $\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b$, then the ordered pair (a, b) is :

A

$\left( {1,{1 \over 2}} \right)$

B

$\left( {1, - {1 \over 2}} \right)$

C

$\left( { - 1,{1 \over 2}} \right)$

D

$\left( { - 1, - {1 \over 2}} \right)$

Question 98: Palette 98/216

JEE Mains 2021 MCQ

If $\alpha$, $\beta$ are the distinct roots of x² + bx + c = 0, then

$\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}}$ is equal to :

A

b² + 4c

B

2(b² + 4c)

C

2(b² $-$ 4c)

D

b² $-$ 4c

Question 99: Palette 99/216

JEE Mains 2021 MCQ

Let [t] denote the greatest integer less than or equal to t. Let
f(x) = x $-$ [x], g(x) = 1 $-$ x + [x], and h(x) = min{f(x), g(x)}, x $\in$ [$-$2, 2]. Then h is :

A

continuous in [$-$2, 2] but not differentiable at more than
four points in ($-$2, 2)

B

not continuous at exactly three points in [$-$2, 2]

C

continuous in [$-$2, 2] but not differentiable at exactly
three points in ($-$2, 2)

D

not continuous at exactly four points in [$-$2, 2]

Question 100: Palette 100/216

JEE Mains 2021 MCQ

$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$ is equal to :

A

${9 \over {44}}$

B

${5 \over {24}}$

C

${1 \over 5}$

D

${7 \over {36}}$

Question 101: Palette 101/216

JEE Mains 2021 MCQ

The value of

$\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right)$ is equal to :

A

0

B

4

C

$-$4

D

$-$1

Question 102: Palette 102/216

JEE Mains 2021 MCQ

Let $f:[0,\infty ) \to [0,3]$ be a function defined by

$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$

Then which of the following is true?

A

f is continuous everywhere but not differentiable exactly at one point in (0, $\infty$)

B

f is differentiable everywhere in (0, $\infty$)

C

f is not continuous exactly at two points in (0, $\infty$)

D

f is continuous everywhere but not differentiable exactly at two points in (0, $\infty$)

Question 103: Palette 103/216

JEE Mains 2021 MCQ

Let $f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$ be defined as $f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right.$

If f is continuous at x = 0, then the value of 6a + b² is equal to :

A

1 $-$ e

B

e $-$ 1

C

1 + e

D

e

Question 104: Palette 104/216

JEE Mains 2021 MCQ

Let f : R $\to$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}$ is equal to :

A

4

B

8

C

16

D

12

Question 105: Palette 105/216

JEE Mains 2021 MCQ

Let f : R $\to$ R be defined as

$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x < 2} \cr {{e^{{{\tan (x - 2)} \over {x - [x]}}}},} & {x > 2} \cr {\mu ,} & {x = 2} \cr } } \right.$

where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $\lambda$ + $\mu$ is equal to :

A

e($-$e + 1)

B

e(e $-$ 2)

C

1

D

2e $-$ 1

Question 106: Palette 106/216

JEE Mains 2021 MCQ

Let f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$

If f is continuous at x = 0, then $\alpha$ is equal to :

A

1

B

3

C

0

D

2

Question 107: Palette 107/216

JEE Mains 2021 MCQ

If $f:R \to R$ is given by $f(x) = x + 1$, then the value of $\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right]$ is :

A

${3 \over 2}$

B

${5 \over 2}$

C

${1 \over 2}$

D

${7 \over 2}$

Question 108: Palette 108/216

JEE Mains 2021 MCQ

Let a function f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right.$

where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:

A

4

B

3

C

2

D

5

Question 109: Palette 109/216

JEE Mains 2021 MCQ

Let f : R $ \to $ R be a function defined as

$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\, = 0 \hfill \cr {{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill \cr} \right.$

If f is continuous at x = 0, then the value of a + b is equal to :

A

$-$3

B

$-$2

C

$ - {5 \over 2}$

D

$ - {3 \over 2}$

Question 110: Palette 110/216

JEE Mains 2021 MCQ

If $\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}}$ is equal to L, then the value of (6L + 1) is

A

${1 \over 6}$

B

${1 \over 2}$

C

6

D

2

Question 111: Palette 111/216

JEE Mains 2021 MCQ

If $f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right.$ is differentiable at every point of the domain, then the values of a and b are respectively :

A

${1 \over 2},{1 \over 2}$

B

${1 \over 2}, - {3 \over 2}$

C

${5 \over 2}, - {3 \over 2}$

D

$ - {1 \over 2},{3 \over 2}$

Question 112: Palette 112/216

JEE Mains 2021 MCQ

The value of the limit

$\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}$ is equal to :

A

0

B

$-$${1 \over 2}$

C

${1 \over 4}$

D

$-$${1 \over 4}$

Question 113: Palette 113/216

JEE Mains 2021 MCQ

The value of $\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

A

r

B

${r \over 2}$

C

0

D

2r

Question 114: Palette 114/216

JEE Mains 2021 MCQ

The value of
$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$, where [ x ] denotes the greatest integer $ \le $ x is :

A

$\pi$

B

${\pi \over 4}$

C

${\pi \over 2}$

D

0

Question 115: Palette 115/216

JEE Mains 2021 MCQ

Let f : S $ \to $ S where S = (0, $\infty $) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $ \to $ R be defined as g(x) = log_e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

A

1

B

${{187} \over {144}}$

C

${{197} \over {144}}$

D

${{205} \over {144}}$

Question 116: Palette 116/216

JEE Mains 2021 MCQ

Let $\alpha$ $\in$ R be such that the function $f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$ is continuous at x = 0, where {x} = x $-$ [ x ] is the greatest integer less than or equal to x. Then :

A

no such $\alpha$ exists

B

$\alpha$ = 0

C

$\alpha$ = ${\pi \over 4}$

D

$\alpha$ = ${\pi \over {\sqrt 2 }}$

Question 117: Palette 117/216

JEE Mains 2021 MCQ

Let ${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $. Then $\mathop {\lim }\limits_{k \to \infty } {S_k}$ is equal to :

A

${\cot ^{ - 1}}\left( {{3 \over 2}} \right)$

B

${\pi \over 2}$

C

tan^$-$1 (3)

D

${\tan ^{ - 1}}\left( {{3 \over 2}} \right)$

Question 118: Palette 118/216

JEE Mains 2021 MCQ

Let the functions f : R $ \to $ R and g : R $ \to $ R be defined as :

$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$ and

$g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$

Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :

A

0

B

3

C

1

D

2

Question 119: Palette 119/216

JEE Mains 2021 MCQ

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4.

Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :

A

4 $-$ 2a

B

2a + 4

C

a + 4

D

2a $-$ 4

Question 120: Palette 120/216

JEE Mains 2021 MCQ

Let $f(x) = {\sin ^{ - 1}}x$ and $g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$. If $g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$, then the domain of the function fog is :

A

$( - \infty , - 2] \cup \left[ { - {4 \over 3},\infty } \right)$

B

$( - \infty , - 2] \cup [ - 1,\infty )$

C

$( - \infty , - 2] \cup \left[ { - {3 \over 2},\infty } \right)$

D

$( - \infty , - 1] \cup [2,\infty )$

Question 121: Palette 121/216

JEE Mains 2021 MCQ

Let f : R $ \to $ R be defined as

$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$ If f(x) is continuous on R, then a + b equals :

A

$-$3

B

3

C

$-$1

D

1

Question 122: Palette 122/216

JEE Mains 2021 MCQ

The value of $\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\}$ is :

A

${4 \over 3}$

B

${2 \over 3}$

C

${3 \over 4}$

D

${2 \over {\sqrt 3 }}$

Question 123: Palette 123/216

JEE Mains 2021 MCQ

$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$ is equal to :

A

${{1 \over 2}}$

B

1

C

0

D

${{1 \over e}}$

Question 124: Palette 124/216

JEE Mains 2021 MCQ

If f : R $ \to $ R is a function defined by f(x)= [x - 1] $\cos \left( {{{2x - 1} \over 2}} \right)\pi $, where [.] denotes the greatest integer function, then f is :

A

continuous for every real x

B

discontinuous at all integral values of x except at x = 1

C

discontinuous only at x = 1

D

continuous only at x = 1

Question 125: Palette 125/216

JEE Mains 2020 MCQ

Let f : R $ \to $ R be a function defined by
f(x) = max {x, x²}. Let S denote the set of all points in R, where f is not differentiable. Then :

A

{0, 1}

B

{0}

C

$\phi $(an empty set)

D

{1}

Question 126: Palette 126/216

JEE Mains 2020 MCQ

For all twice differentiable functions f : R $ \to $ R,
with f(0) = f(1) = f'(0) = 0

A

f''(x) $ \ne $ 0, at every point x $ \in $ (0, 1)

B

f''(x) = 0, for some x $ \in $ (0, 1)

C

f''(0) = 0

D

f''(x) = 0, at every point x $ \in $ (0, 1)

Question 127: Palette 127/216

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2} + {x^4}} - 1}}$

A

is equal to 0.

B

is equal to $\sqrt e $.

C

is equal to 1.

D

does not exist.

Question 128: Palette 128/216

JEE Mains 2020 MCQ

If the function
$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr {{k_2}\cos x,} & {x > \pi } \cr } } \right.$ is
twice differentiable, then the ordered pair (k₁, k₂) is equal to :

A

$\left( {{1 \over 2},-1} \right)$

B

(1, 1)

C

(1, 0)

D

$\left( {{1 \over 2},1} \right)$

Question 129: Palette 129/216

JEE Mains 2020 MCQ

If $\alpha $ is positive root of the equation, p(x) = x² - x - 2 = 0, then

$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}$ is equal to :

A

${1 \over \sqrt2}$

B

${1 \over 2}$

C

${3 \over \sqrt2}$

D

${3 \over 2}$

Question 130: Palette 130/216

JEE Mains 2020 MCQ

Let $f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$ be a differentiable function such that f(1) = e and
$\mathop {\lim }\limits_{t \to x} {{{t^2}{f^2}(x) - {x^2}{f^2}(t)} \over {t - x}} = 0$. If f(x) = 1, then x is equal to :

A

${1 \over e}$

B

e

C

${1 \over 2e}$

D

2e

Question 131: Palette 131/216

JEE Mains 2020 MCQ

The function
$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr {{1 \over 2}\left( {\left| x \right| - 1} \right),} & {\left| x \right| > 1} \cr } } \right.$ is :

A

continuous on R–{–1} and differentiable on R–{–1, 1}

B

both continuous and differentiable on R–{1}

C

both continuous and differentiable on R–{–1}

D

continuous on R–{1} and differentiable on R–{–1, 1}

Question 132: Palette 132/216

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}$ ($a$ $ \ne $ 0) is equal to :

A

$\left( {{2 \over 9}} \right){\left( {{2 \over 3}} \right)^{{1 \over 3}}}$

B

$\left( {{2 \over 3}} \right){\left( {{2 \over 9}} \right)^{{1 \over 3}}}$

C

${\left( {{2 \over 3}} \right)^{{4 \over 3}}}$

D

${\left( {{2 \over 9}} \right)^{{4 \over 3}}}$

Question 133: Palette 133/216

JEE Mains 2020 MCQ

Let [t] denote the greatest integer $ \le $ t. If for some
$\lambda $ $ \in $ R - {1, 0}, $\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda - x + \left[ x \right]}}} \right|$ = L, then L is equal to :

A

1

B

2

C

0

D

${1 \over 2}$

Question 134: Palette 134/216

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$ is equal to :

A

2

B

1

C

$e$

D

$e$²

Question 135: Palette 135/216

JEE Mains 2020 MCQ

If a function f(x) defined by

$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1 \le x \le 3} \cr {a{x^2} + 2cx,} & {3 < x \le 4} \cr } } \right.$

be continuous for some $a$, b, c $ \in $ R and f'(0) + f'(2) = e, then the value of of $a$ is :

A

${e \over {{e^2} - 3e - 13}}$

B

${1 \over {{e^2} - 3e + 13}}$

C

${e \over {{e^2} - 3e + 13}}$

D

${e \over {{e^2} + 3e + 13}}$

Question 136: Palette 136/216

JEE Mains 2020 MCQ

Let [t] denote the greatest integer $ \le $ t and $\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A$.
Then the function, f(x) = [x²]sin($\pi $x) is discontinuous, when x is equal to :

A

$\sqrt {A + 1} $

B

$\sqrt {A + 5} $

C

$\sqrt {A + 21} $

D

$\sqrt {A} $

Question 137: Palette 137/216

JEE Mains 2020 MCQ

If $f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr {{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \over 3}}}} \over {{x^{{4 \over 3}}}}};} & {x > 0} \cr } } \right.$
is continuous at x = 0, then a + 2b is equal to :

A

0

B

-1

C

-2

D

1

Question 138: Palette 138/216

JEE Mains 2020 MCQ

Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $ \in $ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $ \in $ (a, b), ${{f(c) - f(a)} \over {f(b) - f(c)}}$ is greater than :

A

1

B

${{b - c} \over {c - a}}$

C

${{b + a} \over {b - a}}$

D

${{c - a} \over {b - c}}$

Question 139: Palette 139/216

JEE Mains 2020 MCQ

Let S be the set of all functions ƒ : [0,1] $ \to $ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $ \in $ (0,1), depending on ƒ, such that

A

$\left| {f(c) - f(1)} \right| < \left| {f'(c)} \right|$

B

$\left| {f(c) + f(1)} \right| < \left( {1 + c} \right)\left| {f'(c)} \right|$

C

$\left| {f(c) - f(1)} \right| < \left( {1 - c} \right)\left| {f'(c)} \right|$

D

None

Question 140: Palette 140/216

JEE Mains 2020 MCQ

$\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}$ is equal to

A

e

B

e²

C

${1 \over {{e^2}}}$

D

${1 \over e}$

Question 141: Palette 141/216

JEE Mains 2019 MCQ

Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x $ \in $ R. If f(x) attains maximum value at $\alpha $ and g(x) attains minimum value at $\beta $, then $\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)} \over {{x^2} - 6x + 8}}$ is equal to :

A

${1 \over 2}$

B

$-{1 \over 2}$

C

${3 \over 2}$

D

$-{3 \over 2}$

Question 142: Palette 142/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$ is :

A

6

B

1

C

3

D

2

Question 143: Palette 143/216

JEE Mains 2019 MCQ

If $\alpha $ and $\beta $ are the roots of the equation 375x² – 25x – 2 = 0, then $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\alpha ^r}} + \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\beta ^r}} $ is equal to :

A

${7 \over {116}}$

B

${{29} \over {348}}$

C

${1 \over {12}}$

D

${{21} \over {346}}$

Question 144: Palette 144/216

JEE Mains 2019 MCQ

If $\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5$, then a + b is equal to :

A

1

B

- 4

C

- 7

D

5

Question 145: Palette 145/216

JEE Mains 2019 MCQ

If$f(x) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}}}} & {,x > 0} \cr } } \right.$
is continuous at x = 0, then the ordered pair (p, q) is equal to

A

$\left( { - {3 \over 2}, - {1 \over 2}} \right)$

B

$\left( { - {1 \over 2},{3 \over 2}} \right)$

C

$\left( { - {3 \over 2}, {1 \over 2}} \right)$

D

$\left( { {5 \over 2}, {1 \over 2}} \right)$

Question 146: Palette 146/216

JEE Mains 2019 MCQ

Let f : R $ \to $ R be differentiable at c $ \in $ R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :

A

differentiable if f '(c) = 0

B

differentiable if f '(c) $ \ne $ 0

C

not differentiable

D

not differentiable if f '(c) = 0

Question 147: Palette 147/216

JEE Mains 2019 MCQ

If $\mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \over {{x^2} - {k^2}}}$, then k is :

A

${3 \over 2}$

B

${8 \over 3}$

C

${4 \over 3}$

D

${3 \over 8}$

Question 148: Palette 148/216

JEE Mains 2019 MCQ

If $f(x) = [x] - \left[ {{x \over 4}} \right]$ ,x $ \in $ 4 , where [x] denotes the greatest integer function, then

A

Both $\mathop {\lim }\limits_{x \to 4 - } f(x)$ and $\mathop {\lim }\limits_{x \to 4 + } f(x)$ exist but are not equal

B

f is continuous at x = 4

C

$\mathop {\lim }\limits_{x \to 4 + } f(x)$ exists but $\mathop {\lim }\limits_{x \to 4 - } f(x)$ does not exist

D

$\mathop {\lim }\limits_{x \to 4 - } f(x)$ exists but $\mathop {\lim }\limits_{x \to 4 + } f(x)$ does not exist

Question 149: Palette 149/216

JEE Mains 2019 MCQ

If the function $f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } \right.$
is continuous at x = 5, then the value of a – b is :-

A

${2 \over {\pi - 5 }}$

B

${2 \over {5 - \pi }}$

C

${-2 \over {\pi + 5 }}$

D

${2 \over {\pi + 5 }}$

Question 150: Palette 150/216

JEE Mains 2019 MCQ

Let ƒ(x) = 15 – |x – 10|; x $ \in $ R. Then the set of all values of x, at which the function, g(x) = ƒ(ƒ(x)) is not differentiable, is :

A

{10,15}

B

{5,10,15,20}

C

{10}

D

{5,10,15}

Question 151: Palette 151/216

JEE Mains 2019 MCQ

If the function ƒ defined on , $\left( {{\pi \over 6},{\pi \over 3}} \right)$ by $$f(x) = \left\{ {\matrix{ {{{\sqrt 2 {\mathop{\rm cosx}\nolimits} - 1} \over {\cot x - 1}},} & {x \ne {\pi \over 4}} \cr {k,} & {x = {\pi \over 4}} \cr } } \right.$$ is continuous, then k is equal to

A

1

B

1 / $\sqrt 2$

C

${1 \over 2}$

D

2

Question 152: Palette 152/216

JEE Mains 2019 MCQ

Let ƒ : R $ \to $ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0.
Then $\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3)} \over {1 + f(2 - x) - f(2)}}} \right)^{{1 \over x}}}$ is equal to

A

e

B

e²

C

e^–1

D

1

Question 153: Palette 153/216

JEE Mains 2019 MCQ

Let ƒ : [–1,3] $ \to $ R be defined as

$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr {x + \left| x \right|} & , & {1 \le x < 2} \cr {x + \left[ x \right]} & , & {2 \le x \le 3} \cr } } \right.$

where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:

A

only three points

B

four or more points

C

only two points

D

only one point

Question 154: Palette 154/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}$ equals:

A

$ \sqrt 2$

B

$2 \sqrt 2$

C

4

D

$4 \sqrt 2$

Question 155: Palette 155/216

JEE Mains 2019 MCQ

Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x $ \in $ R R. If h(x) = f(f(x)), then h'(1) is equal to :

A

4e

B

2e²

C

4e²

D

2e

Question 156: Palette 156/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to {1^ - }} {{\sqrt \pi - \sqrt {2{{\sin }^{ - 1}}x} } \over {\sqrt {1 - x} }}$ is equal to :

A

$\sqrt {{2 \over \pi }} $

B

${1 \over {\sqrt {2\pi } }}$

C

$\sqrt {{\pi \over 2}} $

D

$\sqrt \pi $

Question 157: Palette 157/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$ is :

A

$8\sqrt 2 $

B

4

C

$4\sqrt 2 $

D

8

Question 158: Palette 158/216

JEE Mains 2019 MCQ

Let S be the set of all points in (–$\pi $, $\pi $) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of which of the following ?

A

$\left\{ { - {\pi \over 2}, - {\pi \over 4},{\pi \over 4},{\pi \over 2}} \right\}$

B

$\left\{ { - {{3\pi } \over 4}, - {\pi \over 2},{\pi \over 2},{{3\pi } \over 4}} \right\}$

C

$\left\{ { - {\pi \over 4},0,{\pi \over 4}} \right\}$

D

$\left\{ { - {{3\pi } \over 4}, - {\pi \over 4},{{3\pi } \over 4},{\pi \over 4}} \right\}$

Question 159: Palette 159/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{x \to 0} {{x\cot \left( {4x} \right)} \over {{{\sin }^2}x{{\cot }^2}\left( {2x} \right)}}$ is equal to :

A

0

B

4

C

1

D

2

Question 160: Palette 160/216

JEE Mains 2019 MCQ

Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x – $\pi $) cos |x| is not differentiable. Then the set K is equal to :

A

{0, $\pi $}

B

$\phi $ (an empty set)

C

{ r }

D

{0}

Question 161: Palette 161/216

JEE Mains 2019 MCQ

Let [x] denote the greatest integer less than or equal to x. Then $\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}}$

A

equals $\pi $ + 1

B

equals 0

C

does not exist

D

equals $\pi $

Question 162: Palette 162/216

JEE Mains 2019 MCQ

Let $f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$ and

$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$

Then, in the interval (–2, 2), g is :

A

non continuous

B

differentiable at all points

C

not differentiable at two points

D

not differentiable at one point

Question 163: Palette 163/216

JEE Mains 2019 MCQ

Let f : ($-$1, 1) $ \to $ R be a function defined by f(x) = max $\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.$ If K be the set of all points at which f is not differentiable, then K has exactly -

A

one element

B

three elements

C

five elements

D

two elements

Question 164: Palette 164/216

JEE Mains 2019 MCQ

For each t $ \in $ R , let [t] be the greatest integer less than or equal to t

Then  $\mathop {\lim }\limits_{x \to 1^ + } {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \right]} \right)} \over {\left| {1 - x} \right|.\left[ {1 - x} \right]}}$

A

equals $-$ 1

B

equals 1

C

equals 0

D

does not exist

Question 165: Palette 165/216

JEE Mains 2019 MCQ

Let  $f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 - 2\left| x \right|} & {2 < \left| x \right| \le 4} \cr } } \right.$

Let S be the set of points in the interval (– 4, 4) at which f is not differentiable. Then S

A

equals $\left\{ { - 2, - 1,1,2} \right\}$

B

equals $\left\{ { - 2, - 1,0,1,2} \right\}$

C

equals $\left\{ { - 2,2} \right\}$

D

is an empty set

Question 166: Palette 166/216

JEE Mains 2019 MCQ

For each x$ \in $R, let [x] be the greatest integer less than or equal to x.

Then $\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}$ is equal to :

A

$-$ sin 1

B

1

C

sin 1

D

0

Question 167: Palette 167/216

JEE Mains 2019 MCQ

$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$

A

exists and equals ${1 \over {2\sqrt 2 }}$

B

exists and equals ${1 \over {4\sqrt 2 }}$

C

exists and equals ${1 \over {2\sqrt 2 (1 + \sqrt {2)} }}$

D

does not exists

Question 168: Palette 168/216

JEE Mains 2019 MCQ

Let f : R $ \to $ R be a function defined as
$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.$

Then, f is

A

continuous if a = 0 and b = 5

B

continuous if a = –5 and b = 10

C

continuous if a = 5 and b = 5

D

not continuous for any values of a and b

Question 169: Palette 169/216

JEE Mains 2018 MCQ

$\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \right)}^{{2 \over 3}}}}}$ equals.

A

${1 \over 3}$

B

$-$ ${1 \over 3}$

C

$-$ ${1 \over 6}$

D

${1 \over 6}$

Question 170: Palette 170/216

JEE Mains 2018 MCQ

If the function f defined as

$f\left( x \right) = {1 \over x} - {{k - 1} \over {{e^{2x}} - 1}},x \ne 0,$ is continuous at

x = 0, then the ordered pair (k, f(0)) is equal to :

A

(3, 2)

B

(3, 1)

C

(2, 1)

D

$\left( {{1 \over 3},\,2} \right)$

Question 171: Palette 171/216

JEE Mains 2018 MCQ

For each t $ \in R$, let [t] be the greatest integer less than or equal to t.

Then $\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)$

A

does not exist in R

B

is equal to 0

C

is equal to 15

D

is equal to 120

Question 172: Palette 172/216

JEE Mains 2018 MCQ

Let S = { t $ \in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)$$\sin \left| x \right|$ is not differentiable at t}, then the set S is equal to

A

{0, $\pi $}

B

$\phi $ (an empty set)

C

{0}

D

{$\pi $}

Question 173: Palette 173/216

JEE Mains 2018 MCQ

Let f(x) be a polynomial of degree $4$ having extreme values at $x = 1$ and $x = 2.$

If   $\mathop {lim}\limits_{x \to 0} \left( {{{f\left( x \right)} \over {{x^2}}} + 1} \right) = 3$   then f($-$1) is equal to :

A

${9 \over 2}$

B

${5 \over 2}$

C

${3 \over 2}$

D

${1 \over 2}$

Question 174: Palette 174/216

JEE Mains 2018 MCQ

Let f(x) = $\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {,x = 2} \cr } } \right.$

Thevaue of k for which f s continuous at x = 2 is :

A

1

B

e

C

e^-1

D

e^-2

Question 175: Palette 175/216

JEE Mains 2018 MCQ

$\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}$ equals :

A

${1 \over 4}$

B

1

C

${1 \over 2}$

D

$-$ ${1 \over 2}$

Question 176: Palette 176/216

JEE Mains 2018 MCQ

Let S = {($\lambda $, $\mu $) $ \in $ R $ \times $ R : f(t) = (|$\lambda $| e^|t| $-$ $\mu $). sin (2|t|), t $ \in $ R, is a differentiable function}. Then S is a subset of :

A

R $ \times $ [0, $\infty $)

B

[0, $\infty $) $ \times $ R

C

R $ \times $ ($-$ $\infty $, 0)

D

($-$ $\infty $, 0) $ \times $ R

Question 177: Palette 177/216

JEE Mains 2017 MCQ

The value of k for which the function

$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$

is continuous at x = ${\pi \over 2},$ is :

A

${{17} \over {20}}$

B

${{2} \over {5}}$

C

${{3} \over {5}}$

D

$-$ ${{2} \over {5}}$

Question 178: Palette 178/216

JEE Mains 2017 MCQ

$\mathop {\lim }\limits_{x \to 3} $ ${{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$ is equal to :

A

$\sqrt 3 $

B

${1 \over {\sqrt 2 }}$

C

${{\sqrt 3 } \over 2}$

D

${1 \over {2\sqrt 2 }}$

Question 179: Palette 179/216

JEE Mains 2017 MCQ

$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}$ equals

A

${1 \over {16}}$

B

${1 \over 8}$

C

${1 \over {4}}$

D

${1 \over {24}}$

Question 180: Palette 180/216

JEE Mains 2016 MCQ

$\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}$ is :

A

$-$ 2

B

$-$ ${1 \over 2}$

C

${1 \over 2}$

D

2

Question 181: Palette 181/216

JEE Mains 2016 MCQ

Let a, b $ \in $ R, (a $ \ne $ 0). If the function f defined as

$f\left( x \right) = \left\{ {\matrix{ {{{2{x^2}} \over a}\,\,,} & {0 \le x < 1} \cr {a\,\,\,,} & {1 \le x < \sqrt 2 } \cr {{{2{b^2} - 4b} \over {{x^3}}},} & {\sqrt 2 \le x < \infty } \cr } } \right.$

is continuous in the interval [0, $\infty $), then an ordered pair ( a, b) is :

A

$\left( {\sqrt 2 ,1 - \sqrt 3 } \right)$

B

$\left( { - \sqrt 2 ,1 + \sqrt 3 } \right)$

C

$\left( {\sqrt 2 , - 1 + \sqrt 3 } \right)$

D

$\left( { - \sqrt 2 ,1 - \sqrt 3 } \right)$

Question 182: Palette 182/216

JEE Mains 2016 MCQ

If the function

f(x) = $\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \right),} & {1 \le x \le 2} \cr } } \right.$

is differentiable at x = 1, then ${a \over b}$ is equal to :

A

${{\pi - 2} \over 2}$

B

${{ - \pi - 2} \over 2}$

C

${{\pi + 2} \over 2}$

D

$ - 1 - {\cos ^{ - 1}}\left( 2 \right)$

Question 183: Palette 183/216

JEE Mains 2016 MCQ

If    $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$ then 'a' is equal to :

A

2

B

${3 \over 2}$

C

${2 \over 3}$

D

${1 \over 2}$

Question 184: Palette 184/216

JEE Mains 2016 MCQ

Let $p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}$ then $log$ $p$ is equal to :

A

${1 \over 2}$

B

${1 \over 4}$

C

$2$

D

$1$

Question 185: Palette 185/216

JEE Mains 2016 MCQ

For $x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,$

and $\,\,g\left( x \right) = f\left( {f\left( x \right)} \right),\,\,$ then :

A

$g$ is not differentiable at $x=0$

B

$g'\left( 0 \right) = \cos \left( {\log 2} \right)$

C

$g'\left( 0 \right) = - \cos \left( {\log 2} \right)$

D

$g$ is differentiable at $x=0$ and $g'\left( 0 \right) = - \sin \left( {\log 2} \right)$

Question 186: Palette 186/216

JEE Mains 2015 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$ is equal to

A

2

B

${1 \over 2}$

C

4

D

3

Question 187: Palette 187/216

JEE Mains 2015 MCQ

If the function.

$g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr } } \right.$

is differentiable, then the value of $k+m$ is :

A

${{10} \over 3}$

B

$4$

C

$2$

D

${{16} \over 5}$

Question 188: Palette 188/216

JEE Mains 2014 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}$ is equal to :

A

$ - \pi $

B

$ \pi $

C

${\pi \over 2}$

D

1

Question 189: Palette 189/216

JEE Mains 2013 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$ is equal to

A

$ - {1 \over 4}$

B

${1 \over 2}$

C

1

D

2

Question 190: Palette 190/216

JEE Mains 2012 MCQ

Consider the function, $f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R$

Statement - 1 : $f'\left( 4 \right) = 0$

Statement - 2 : $f$ is continuous in [2, 5], differentiable in (2, 5) and $f$(2) = $f$(5)

A

Statement - 1 is false, statement - 2 is true

B

Statement - 1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1

C

Statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1

D

Statement - 1 is true, statement - 2 is false

Question 191: Palette 191/216

JEE Mains 2012 MCQ

If $f:R \to R$ is a function defined by

$f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi $,

where [x] denotes the greatest integer function, then $f$ is

A

continuous for every real $x$

B

discontinuous only at $x=0$

C

discontinuous only at non-zero integral values of $x$

D

continuous only at $x=0$

Question 192: Palette 192/216

JEE Mains 2011 MCQ

The value of $p$ and $q$ for which the function

$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} \cr } } \right.$

is continuous for all $x$ in R, are

A

$p =$ ${5 \over 2}$, $q = $ ${1 \over 2}$

B

$p =$ $-{3 \over 2}$, $q = $ ${1 \over 2}$

C

$p =$ ${1 \over 2}$, $q = $ ${3 \over 2}$

D

$p =$ ${1 \over 2}$, $q = $ $-{3 \over 2}$

Question 193: Palette 193/216

JEE Mains 2011 MCQ

$\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)$

A

Equals $\sqrt 2 $

B

Equals $-\sqrt 2 $

C

Equals ${1 \over {\sqrt 2 }}$

D

does not exist

Question 194: Palette 194/216

JEE Mains 2010 MCQ

Let $f:R \to R$ be a positive increasing function with

$\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1$. Then $\mathop {\lim }\limits_{x \to \infty } {{f(2x)} \over {f(x)}} = $

A

${2 \over 3}$

B

${3 \over 2}$

C

3

D

1

Question 195: Palette 195/216

JEE Mains 2009 MCQ

Let $f\left( x \right) = x\left| x \right|$ and $g\left( x \right) = \sin x.$
Statement-1: gof is differentiable at $x=0$ and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at $x=0$.

A

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

B

Statement-1 is true, Statement-2 is false

C

Statement-1 is false, Statement-2 is true

D

Statement-1 is true, Statement-2 is true Statement-2 is a correct explanation for Statement-1

Question 196: Palette 196/216

JEE Mains 2008 MCQ

Let $f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right.$

Then which one of the following is true?

A

$f$ is neither differentiable at x = 0 nor at x = 1

B

$f$ is differentiable at x = 0 and at x = 1

C

$f$ is differentiable at x = 0 but not at x = 1

D

$f$ is differentiable at x = 1 but not at x = 0

Question 197: Palette 197/216

JEE Mains 2007 MCQ

Let $f:R \to R$ be a function defined by

$f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}$, then which of the following is true?

A

$f(x)$ is differentiale everywhere

B

$f(x)$ is not differentiable at x = 0

C

$f(x) > 1$ for all $x \in R$

D

$f(x)$ is not differentiable at x = 1

Question 198: Palette 198/216

JEE Mains 2007 MCQ

The function $f:R/\left\{ 0 \right\} \to R$ given by

$f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}$

can be made continuous at $x$ = 0 by defining $f$(0) as

A

0

B

1

C

2

D

$-1$

Question 199: Palette 199/216

JEE Mains 2006 MCQ

The set of points where $f\left( x \right) = {x \over {1 + \left| x \right|}}$ is differentiable is

A

$\left( { - \infty ,0} \right) \cup \left( {0,\infty } \right)$

B

$\left( { - \infty ,1} \right) \cup \left( { - 1,\infty } \right)$

C

$\left( { - \infty ,\infty } \right)$

D

$\left( {0,\infty } \right)$

Question 200: Palette 200/216

JEE Mains 2005 MCQ

If $f$ is a real valued differentiable function satisfying

$\left| {f\left( x \right) - f\left( y \right)} \right|$ $ \le {\left( {x - y} \right)^2}$, $x, y$ $ \in R$
and $f(0)$ = 0, then $f(1)$ equals

A

-1

B

0

C

2

D

1

Question 201: Palette 201/216

JEE Mains 2005 MCQ

Suppose $f(x)$ is differentiable at x = 1 and

$\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$, then $f'\left( 1 \right)$ equals

A

3

B

4

C

5

D

6

Question 202: Palette 202/216

JEE Mains 2005 MCQ

Let $\alpha$ and $\beta$ be the distinct roots of $a{x^2} + bx + c = 0$, then

$\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}$ is equal to

A

${{{a^2}{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

B

0

C

$ - {{{a^2}{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

D

${{{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

Question 203: Palette 203/216

JEE Mains 2004 MCQ

If $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$, then the value of $a$ and $b$, are

A

$a$ = 1 and $b$ = 2

B

$a$ = 1 and $b$ $ \in R$

C

$a$ $ \in R$ and $b$ = 2

D

$a$ $ \in R$ and $b$ $ \in R$

Question 204: Palette 204/216

JEE Mains 2004 MCQ

Let $f(x) = {{1 - \tan x} \over {4x - \pi }}$, $x \ne {\pi \over 4}$, $x \in \left[ {0,{\pi \over 2}} \right]$.

If $f(x)$ is continuous in $\left[ {0,{\pi \over 2}} \right]$, then $f\left( {{\pi \over 4}} \right)$ is

A

$-1$

B

${1 \over 2}$

C

$-{1 \over 2}$

D

$1$

Question 205: Palette 205/216

JEE Mains 2003 MCQ

If $f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \cr } } \right.$

then $f(x)$ is

A

discontinuous everywhere

B

continuous as well as differentiable for all x

C

continuous for all x but not differentiable at x = 0

D

neither differentiable nor continuous at x = 0

Question 206: Palette 206/216

JEE Mains 2003 MCQ

If $\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}$ = k, the value of k is

A

$ - {2 \over 3}$

B

0

C

$ - {1 \over 3}$

D

${2 \over 3}$

Question 207: Palette 207/216

JEE Mains 2003 MCQ

$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin x} \right]} \over {\left[ {1 + \tan \left( {{x \over 2}} \right)} \right]{{\left[ {\pi - 2x} \right]}^3}}}$ is

A

$\infty $

B

${1 \over 8}$

C

0

D

${1 \over 32}$

Question 208: Palette 208/216

JEE Mains 2003 MCQ

Let $f(a) = g(a) = k$ and their n^th derivatives
${f^n}(a)$, ${g^n}(a)$ exist and are not equal for some n. Further if

$\mathop {\lim }\limits_{x \to a} {{f(a)g(x) - f(a) - g(a)f(x) + f(a)} \over {g(x) - f(x)}} = 4$

then the value of k is

A

0

B

4

C

2

D

1

Question 209: Palette 209/216

JEE Mains 2002 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}$ is

A

$1$

B

$-1$

C

zero

D

does not exist

Question 210: Palette 210/216

JEE Mains 2002 MCQ

$f$ is defined in $\left[ { - 5,5} \right]$ as

$f\left( x \right) = x$ if $x$ is rational

$\,\,\,\,\,\,\,\,\,\,\,\,\,$ $ = - x$ if $x$ is irrational. Then

A

$f(x)$ is continuous at every x, except $x = 0$

B

$f(x)$ is discontinuous at every $x,$ except $x = 0$

C

$f(x)$ is continuous everywhere

D

$f(x)$ is discontinuous everywhere

Question 211: Palette 211/216

JEE Mains 2002 MCQ

If f(x + y) = f(x).f(y) $\forall $ x, y and f(5) = 2, f'(0) = 3, then
f'(5) is

A

0

B

1

C

6

D

2

Question 212: Palette 212/216

JEE Mains 2002 MCQ

$\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}$

A

${e^4}$

B

${e^2}$

C

${e^3}$

D

$1$

Question 213: Palette 213/216

JEE Mains 2002 MCQ

$\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$, $n \in N$, ( [x] denotes the greatest integer less than or equal to x )

A

has value $ -1$

B

has value $0$

C

has value $1$

D

does not exist

Question 214: Palette 214/216

JEE Mains 2002 MCQ

If $f\left( 1 \right) = 1,{f'}\left( 1 \right) = 2,$ then
$\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}}$ is

A

$2$

B

$4$

C

$1$

D

${1 \over 2}$

Question 215: Palette 215/216

JEE Mains 2002 MCQ

f(x) and g(x) are two differentiable functions on [0, 2] such that

f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9

then f(x) - g(x) at x = ${3 \over 2}$ is

A

0

B

2

C

10

D

-5

Question 216: Palette 216/216

JEE Mains 2002 MCQ

Let $f(2) = 4$ and $f'(x) = 4.$

Then $\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \over {x - 2}}$ is given by

A

$2$

B

$- 2$

C

$- 4$

D

$3$