Question 1: Palette 1/196

JEE Mains 2026 MCQ

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ :

(I) $f$ is differentiable at all $x>0$.

(II) $f$ is increasing in $(0,1)$.

(III) $f$ is decreasing in $(1, \infty)$.

Then.

A

Only (I) is TRUE.

B

Only (I) and (III) are TRUE.

C

Only (II) and (III) are TRUE.

D

All (I), (II) and (III) are TRUE.

Question 2: Palette 2/196

JEE Mains 2026 MCQ

The least value of $\left(\cos ^2 \theta-6 \sin \theta \cos \theta+3 \sin ^2 \theta+2\right)$ is

A

$4-\sqrt{10}$

B

-1

C

$4+\sqrt{10}$

D

1

Question 3: Palette 3/196

JEE Mains 2026 MCQ

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11 \pi+\theta)\right)-2\left(\sin ^6\left(\frac{3 \pi}{2}-\theta\right)+\sin ^6(9 \pi-\theta)\right), \theta \in \mathbf{R}$.

Then $\alpha+2 \beta$ is equal to :

A

6

B

5

C

4

D

3

Question 4: Palette 4/196

JEE Mains 2026 MCQ

Let $f(x)=x^{2025}-x^{2000}, x \in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

A

-40

B

-41

C

-80

D

-81

Question 5: Palette 5/196

JEE Mains 2026 MCQ

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$ and $f'(a-1) = 0$, where $a$ is a real number.

Let $g(x) = f(\tan^2 x - 2 \tan x + a),\ 0 < x < \frac{\pi}{2}$.

Consider the following two statements:

(I) g is increasing in $\left(0, \frac{\pi}{4}\right)$

(II) g is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Then,

A

Both (I) and (II) are True

B

Neither (I) nor (II) is True

C

Only (I) is True

D

Only (II) is True

Question 6: Palette 6/196

JEE Mains 2026 MCQ

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f^{\prime}(0)=3$. Then the minimum value of the function $g(x)=3+e^x f(x)$, is:

A

$3\left(\frac{e+1}{e}\right)$

B

$3\left(\frac{e-1}{e}\right)$

C

$\frac{3-e}{e}$

D

3e

Question 7: Palette 7/196

JEE Mains 2026 MCQ

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x = 1$ and $x = -1$. If $\lim\limits_{x \to 0} \left( \frac{f(x)}{x^3} \right) = -5$, then $f(2) - f(-2)$ is equal to:

A

0

B

50

C

92

D

112

Question 8: Palette 8/196

JEE Mains 2026 MCQ

The number of critical points of the function

$f(x) = \begin{cases} |\frac{\sin x}{x}|, & x \neq 0 \\ 1, & x = 0 \end{cases}$ in the interval $(-2\pi, 2\pi)$ is equal to :

A

1

B

3

C

5

D

7

Question 9: Palette 9/196

JEE Mains 2025 MCQ

Let the function $ f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0 $ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $ \sum\limits_{i=1}^{5} \alpha_i^2 $ is equal to

A

48

B

40

C

36

D

28

Question 10: Palette 10/196

JEE Mains 2025 MCQ

Let f : ℝ $ \to $ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $ \lim\limits_{x \to 0} \frac{f(x)}{x^2} = 5 $, then f(2) is equal to :

A

8

B

10

C

12

D

14

Question 11: Palette 11/196

JEE Mains 2025 MCQ

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+a x^2+b \log _{\mathrm{e}}|x|+1, x \neq 0$. Let $m$ and M respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\frac{1}{2}\right]$. Then $|\mathrm{M}+m|$ is equal to $\left(\right.$ Take $\left.\log _{\mathrm{e}} 2=0.7\right):$

A

21.1

B

19.8

C

22.1

D

20.9

Question 12: Palette 12/196

JEE Mains 2025 MCQ

Let $\mathrm{a}>0$. If the function $f(x)=6 x^3-45 \mathrm{a} x^2+108 \mathrm{a}^2 x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1 x_2=54$, then $\mathrm{a}+x_1+x_2$ is equal to :

A

15

B

13

C

24

D

18

Question 13: Palette 13/196

JEE Mains 2025 MCQ

The shortest distance between the curves $y^2=8 x$ and $x^2+y^2+12 y+35=0$ is:

A

$2 \sqrt{3}-1$

B

$2 \sqrt{2}-1$

C

$3 \sqrt{2}-1$

D

$\sqrt{2}$

Question 14: Palette 14/196

JEE Mains 2025 MCQ

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=||x+2|-2| x \|$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is

A

3

B

4

C

2

D

5

Question 15: Palette 15/196

JEE Mains 2025 MCQ

If the function $f(x)=2 x^3-9 a x^2+12 \mathrm{a}^2 x+1$, where $\mathrm{a}>0$, attains its local maximum and local minimum values at p and q , respectively, such that $\mathrm{p}^2=\mathrm{q}$, then $f(3)$ is equal to :

A

55

B

37

C

10

D

23

Question 16: Palette 16/196

JEE Mains 2025 MCQ

The sum of all local minimum values of the function

$\mathrm{f}(x)=\left\{\begin{array}{lr} 1-2 x, & x<-1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x>2 \end{array}\right.$

is

A

$\frac{167}{72}$

B

$\frac{157}{72}$

C

$\frac{171}{72}$

D

$\frac{131}{72}$

Question 17: Palette 17/196

JEE Mains 2025 MCQ

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2$ is strictly decreasing. Then $100(\mathrm{a}+\mathrm{b}-\mathrm{c})$ is equal to :

A

360

B

420

C

160

D

280

Question 18: Palette 18/196

JEE Mains 2025 MCQ

Consider the region $R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3} x^2, x \geq 0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:

A

$\frac{821}{123}$

B

$\frac{567}{121}$

C

$\frac{730}{119}$

D

$\frac{625}{111}$

Question 19: Palette 19/196

JEE Mains 2025 MCQ

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}$. The surface area (in $\mathrm{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is :

A

$128 \pi$

B

$196 \pi$

C

$225 \pi$

D

$256 \pi$

Question 20: Palette 20/196

JEE Mains 2025 MCQ

Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

A

3 and 2

B

2 and 2

C

2 and 3

D

1 and 3

Question 21: Palette 21/196

JEE Mains 2024 MCQ

If the function $f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a}> 0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :

A

$x^2-6 x+8=0$

B

$8 x^2-6 x+1=0$

C

$8 x^2+6 x-1=0$

D

$x^2+6 x+8=0$

Question 22: Palette 22/196

JEE Mains 2024 MCQ

Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in interval $(0,2 \pi)$ is

A

1

B

3

C

4

D

2

Question 23: Palette 23/196

JEE Mains 2024 MCQ

The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is

A

2

B

1

C

0

D

3

Question 24: Palette 24/196

JEE Mains 2024 MCQ

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements

(S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$.

(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

A

Both (S1) and (S2) are incorrect.

B

Only (S1) is correct.

C

Only (S2) is correct.

D

Both (S1) and (S2) are correct.

Question 25: Palette 25/196

JEE Mains 2024 MCQ

The interval in which the function $f(x)=x^x, x>0$, is strictly increasing is

A

$(0, \infty)$

B

$\left(0, \frac{1}{e}\right]$

C

$\left[\frac{1}{e^2}, 1\right)$

D

$\left[\frac{1}{e}, \infty\right)$

Question 26: Palette 26/196

JEE Mains 2024 MCQ

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a+b)$^2$ is equal to :

A

64

B

80

C

60

D

72

Question 27: Palette 27/196

JEE Mains 2024 MCQ

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

A

42

B

7

C

1

D

14

Question 28: Palette 28/196

JEE Mains 2024 MCQ

For the function

$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$

consider the following two statements :

(I) $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.

(II) $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.

Between the above two statements,

A

only (I) is true.

B

both (I) and (II) are true.

C

only (II) is true.

D

neither (I) nor (II) is true.

Question 29: Palette 29/196

JEE Mains 2024 MCQ

Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2+2 \beta^2$ is equal to

A

42

B

38

C

24

D

44

Question 30: Palette 30/196

JEE Mains 2024 MCQ

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :

A

217

B

182

C

201

D

195

Question 31: Palette 31/196

JEE Mains 2024 MCQ

If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :

A

$\left(0, \frac{1}{\sqrt{5}}\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$

B

$\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$

C

$\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$

D

$\left(-\infty, \frac{1}{\sqrt{5}}\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$

Question 32: Palette 32/196

JEE Mains 2024 MCQ

Let $f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of $\lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to

A

0

B

4

C

1

D

7/5

Question 33: Palette 33/196

JEE Mains 2024 MCQ

If the function $f:(-\infty,-1] \rightarrow(a, b]$ defined by $f(x)=e^{x^3-3 x+1}$ is one - one and onto, then the distance of the point $P(2 b+4, a+2)$ from the line $x+e^{-3} y=4$ is :

A

$2 \sqrt{1+e^6}$

B

$\sqrt{1+e^6}$

C

$3 \sqrt{1+e^6}$

D

$4 \sqrt{1+e^6}$

Question 34: Palette 34/196

JEE Mains 2024 MCQ

$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$

A

24

B

18

C

42

D

48

Question 35: Palette 35/196

JEE Mains 2024 MCQ

Let $f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is

A

108

B

392

C

608

D

600

Question 36: Palette 36/196

JEE Mains 2024 MCQ

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2 x^2+54$ at points $(x, y)$ and $(-x, y)$, where $y>0$, is :

A

108

B

122

C

88

D

92

Question 37: Palette 37/196

JEE Mains 2024 MCQ

The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$

A

decreases in $(-\infty,-2) \cup(-2,8) \cup(8, \infty)$

B

increases in $(-\infty,-2) \cup(-2,8) \cup(8, \infty)$

C

decreases in $(-2,8)$ and increases in $(-\infty,-2) \cup(8, \infty)$

D

decreases in $(-\infty,-2)$ and increases in $(8, \infty)$

Question 38: Palette 38/196

JEE Mains 2024 MCQ

The function $f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$, has

A

exactly one point of local minima and no point of local maxima

B

exactly one point of local maxima and exactly one point of local minima

C

exactly two points of local maxima and exactly one point of local minima

D

exactly one point of local maxima and no point of local minima

Question 39: Palette 39/196

JEE Mains 2024 MCQ

Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements

(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.

(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi}{12}$.

Then

A

Both (I) and (II) are correct.

B

Only (I) is correct.

C

Both (I) and (II) are incorrect.

D

Only (II) is correct.

Question 40: Palette 40/196

JEE Mains 2024 MCQ

Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)>0$ for all $x \in(0,3)$. If $g$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8 \alpha$ is :

A

0

B

24

C

18

D

20

Question 41: Palette 41/196

JEE Mains 2023 MCQ

$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$

A

$\frac{5 \pi+2+3 \sqrt{3}}{6}$

B

0

C

$\frac{\pi+2-3 \sqrt{3}}{6}$

D

$\pi$

Question 42: Palette 42/196

JEE Mains 2023 MCQ

If the local maximum value of the function $f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$ , is $\frac{k}{e}$, then $\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}$ is equal to

A

$e^{3}+e^{6}+e^{10}$

B

$e^{3}+e^{5}+e^{11}$

C

$e^{3}+e^{6}+e^{11}$

D

$e^{5}+e^{6}+e^{11}$

Question 43: Palette 43/196

JEE Mains 2023 MCQ

Let $f:[2,4] \rightarrow \mathbb{R}$ be a differentiable function such that $\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{4}$.

Consider the following two statements :

(A) : $f(x) \leq 1$, for all $x \in[2,4]$

(B) : $f(x) \geq \frac{1}{8}$, for all $x \in[2,4]$

Then,

A

Neither statement (A) nor statement (B) is true

B

Only statement (A) is true

C

Only statement (B) is true

D

Both the statements $(\mathrm{A})$ and (B) are true

Question 44: Palette 44/196

JEE Mains 2023 MCQ

Let $\mathrm{g}(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x) > 0, x \in(0,1)$. If $\mathrm{g}$ is decreasing in the interval $(0, a)$ and increasing in the interval $(\alpha, 1)$, then $\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)$ is equal to :

A

$\frac{3 \pi}{4}$

B

$\pi$

C

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

D

$\frac{3 \pi}{2}$

Question 45: Palette 45/196

JEE Mains 2023 MCQ

The slope of tangent at any point (x, y) on a curve $y=y(x)$ is ${{{x^2} + {y^2}} \over {2xy}},x > 0$. If $y(2) = 0$, then a value of $y(8)$ is :

A

$ - 4\sqrt 2 $

B

$2\sqrt 3 $

C

$4\sqrt 3 $

D

$ - 2\sqrt 3 $

Question 46: Palette 46/196

JEE Mains 2023 MCQ

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to :

A

1025

B

900

C

800

D

675

Question 47: Palette 47/196

JEE Mains 2023 MCQ

The sum of the absolute maximum and minimum values of the function $f(x)=\left|x^{2}-5 x+6\right|-3 x+2$ in the interval $[-1,3]$ is equal to :

A

13

B

24

C

10

D

12

Question 48: Palette 48/196

JEE Mains 2023 MCQ

A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$. If $2 A_{1}+3 A_{2}$ is minimum then $\left(\pi l_{1}\right): l_{2}$ is equal to :

A

6 : 1

B

1 : 6

C

4 : 1

D

3 : 1

Question 49: Palette 49/196

JEE Mains 2023 MCQ

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$

and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$

have a common extreme point, then $a+2 b+7$ is equal to :

A

6

B

$\frac{3}{2}$

C

3

D

4

Question 50: Palette 50/196

JEE Mains 2023 MCQ

The number of points on the curve $y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$ at which the normal lines are parallel to $x+90 y+2=0$ is :

A

2

B

3

C

4

D

0

Question 51: Palette 51/196

JEE Mains 2023 MCQ

Let the function $f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of p is

A

$\left( { - {9 \over 2},{9 \over 2}} \right)$

B

$\left( {{9 \over 2},\infty } \right)$

C

$\left( {0,{9 \over 2}} \right)$

D

$\left( { - \infty ,{9 \over 2}} \right)$

Question 52: Palette 52/196

JEE Mains 2023 MCQ

Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12,x\in(-4,4)$. If M is local maximum value of the function $f$ in ($-4,4)$, then M =

A

$18\sqrt6-\frac{33}{2}$

B

$18\sqrt6-\frac{31}{2}$

C

$12\sqrt6-\frac{33}{2}$

D

$12\sqrt6-\frac{31}{2}$

Question 53: Palette 53/196

JEE Mains 2023 MCQ

Let $f:(0,1)\to\mathbb{R}$ be a function defined $f(x) = {1 \over {1 - {e^{ - x}}}}$, and $g(x) = \left( {f( - x) - f(x)} \right)$. Consider two statements

(I) g is an increasing function in (0, 1)

(II) g is one-one in (0, 1)

Then,

A

Both (I) and (II) are true

B

Neither (I) nor (II) is true

C

Only (II) is true

D

Only (I) is true

Question 54: Palette 54/196

JEE Mains 2022 MCQ

Let $f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathrm{R}$. Then which of the following statements are true?

$\mathrm{P}: x=0$ is a point of local minima of $f$

$\mathrm{Q}: x=\sqrt{2}$ is a point of inflection of $f$

$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

A

Only P and Q

B

Only P and R

C

Only Q and R

D

All P, Q and R

Question 55: Palette 55/196

JEE Mains 2022 MCQ

The function $f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$, is :

A

increasing in $\left(-\frac{1}{2}, 1\right)$

B

decreasing in $\left(\frac{1}{2}, 2\right)$

C

increasing in $\left(-1,-\frac{1}{2}\right)$

D

decreasing in $\left(-\frac{1}{2}, \frac{1}{2}\right)$

Question 56: Palette 56/196

JEE Mains 2022 MCQ

If the minimum value of $f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0$, is 14 , then the value of $\alpha$ is equal to :

A

32

B

64

C

128

D

256

Question 57: Palette 57/196

JEE Mains 2022 MCQ

If the maximum value of $a$, for which the function $f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$, is $\bar{a}$, then $f_{\bar{a}}\left(\frac{\pi}{8}\right)$ is equal to :

A

$ 8-\frac{9 \pi}{4\left(9+\pi^{2}\right)} $

B

$8-\frac{4 \pi}{9\left(4+\pi^{2}\right)}$

C

$8\left(\frac{1+\pi^{2}}{9+\pi^{2}}\right)$

D

$8-\frac{\pi}{4}$

Question 58: Palette 58/196

JEE Mains 2022 MCQ

If the absolute maximum value of the function $f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$ in the interval $[-3,0]$ is $f(\alpha)$, then :

A

$\alpha=0$

B

$ \alpha=-3$

C

$\alpha \in(-1,0)$

D

$\alpha \in(-3,-1]$

Question 59: Palette 59/196

JEE Mains 2022 MCQ

The curve $y(x)=a x^{3}+b x^{2}+c x+5$ touches the $x$-axis at the point $\mathrm{P}(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y^{\prime}$ is equal to 3 . Then the local maximum value of $y(x)$ is:

A

$\frac{27}{4}$

B

$\frac{29}{4}$

C

$\frac{37}{4}$

D

$\frac{9}{2}$

Question 60: Palette 60/196

JEE Mains 2022 MCQ

If xy⁴ attains maximum value at the point (x, y) on the line passing through the points (50 + $\alpha$, 0) and (0, 50 + $\alpha$), $\alpha$ > 0, then (x, y) also lies on the line :

A

y = 4x

B

x = 4y

C

y = 4x + $\alpha$

D

x = 4y $-$ $\alpha$

Question 61: Palette 61/196

JEE Mains 2022 MCQ

Let $f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R$. Then f :

A

has a local minina at $x = {1 \over 2}$

B

has a local minima at $x = {3 \over 4}$

C

is increasing in $\left( {{1 \over 2},{3 \over 4}} \right)$

D

is decreasing in $\left( {{1 \over 2},{4 \over 3}} \right)$

Question 62: Palette 62/196

JEE Mains 2022 MCQ

Let f : R $\to$ R be a function defined by f(x) = (x $-$ 3)^n₁ (x $-$ 5)^n₂, n₁, n₂ $\in$ N. Then, which of the following is NOT true?

A

For n₁ = 3, n₂ = 4, there exists $\alpha$ $\in$ (3, 5) where f attains local maxima.

B

For n₁ = 4, n₂ = 3, there exists $\alpha$ $\in$ (3, 5) where f attains local minima.

C

For n₁ = 3, n₂ = 5, there exists $\alpha$ $\in$ (3, 5) where f attains local maxima.

D

For n₁ = 4, n₂ = 6, there exists $\alpha$ $\in$ (3, 5) where f attains local maxima.

Question 63: Palette 63/196

JEE Mains 2022 MCQ

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

A

${{22} \over {9 + 4\sqrt 3 }}$

B

${{66} \over {9 + 4\sqrt 3 }}$

C

${{22} \over {4 + 9\sqrt 3 }}$

D

${{66} \over {4 + 9\sqrt 3 }}$

Question 64: Palette 64/196

JEE Mains 2022 MCQ

The number of real solutions of

${x^7} + 5{x^3} + 3x + 1 = 0$ is equal to ____________.

A

0

B

1

C

3

D

5

Question 65: Palette 65/196

JEE Mains 2022 MCQ

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :

A

2 : 5

B

19 : 45

C

3 : 8

D

19 : 15

Question 66: Palette 66/196

JEE Mains 2022 MCQ

The sum of the absolute minimum and the absolute maximum values of the

function f(x) = |3x $-$ x² + 2| $-$ x in the interval [$-$1, 2] is :

A

${{\sqrt {17} + 3} \over 2}$

B

${{\sqrt {17} + 5} \over 2}$

C

5

D

${{9 - \sqrt {17} } \over 2}$

Question 67: Palette 67/196

JEE Mains 2022 MCQ

Let S be the set of all the natural numbers, for which the line ${x \over a} + {y \over b} = 2$ is a tangent to the curve ${\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2$ at the point (a, b), ab $\ne$ 0. Then :

A

S = $\phi$

B

n(S) = 1

C

S = {2k : k $\in$ N}

D

S = N

Question 68: Palette 68/196

JEE Mains 2022 MCQ

Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :

A

11

B

11 $-$ $\pi$

C

11 + $\pi$

D

15 $-$ $\pi$

Question 69: Palette 69/196

JEE Mains 2022 MCQ

Water is being filled at the rate of 1 cm³ / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm² / sec) at which the wet conical surface area of the vessel increases is

A

5

B

${{\sqrt {21} } \over 5}$

C

${{\sqrt {26} } \over 5}$

D

${{\sqrt {26} } \over {10}}$

Question 70: Palette 70/196

JEE Mains 2022 MCQ

If the angle made by the tangent at the point (x₀, y₀) on the curve $x = 12(t + \sin t\cos t)$, $y = 12{(1 + \sin t)^2}$, $0 < t < {\pi \over 2}$, with the positive x-axis is ${\pi \over 3}$, then y₀ is equal to:

A

$6\left( {3 + 2\sqrt 2 } \right)$

B

$3\left( {7 + 4\sqrt 3 } \right)$

C

27

D

48

Question 71: Palette 71/196

JEE Mains 2022 MCQ

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by ${{{x^2}} \over {xy - {x^2}{y^2} - 1}}$. If the curve passes through the point (1, 1), then e . y(e) is equal to

A

${{1 - \tan (1)} \over {1 + \tan (1)}}$

B

tan(1)

C

1

D

${{1 + \tan (1)} \over {1 - \tan (1)}}$

Question 72: Palette 72/196

JEE Mains 2022 MCQ

Let $\lambda$$^ * $ be the largest value of $\lambda$ for which the function ${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$ is increasing for all x $\in$ R. Then ${f_{{\lambda ^ * }}}(1) + {f_{{\lambda ^ * }}}( - 1)$ is equal to :

A

36

B

48

C

64

D

72

Question 73: Palette 73/196

JEE Mains 2022 MCQ

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

A

9

B

10

C

11

D

12

Question 74: Palette 74/196

JEE Mains 2022 MCQ

For the function

$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$, which one of the following is NOT correct?

A

f is increasing in (1, 2) and decreasing in (2, $\infty$)

B

f(x) = $-$1 has exactly two solutions

C

$f'(e) - f''(2) < 0$

D

f(x) = 0 has a root in the interval (e, e + 1)

Question 75: Palette 75/196

JEE Mains 2022 MCQ

If the tangent at the point (x₁, y₁) on the curve $y = {x^3} + 3{x^2} + 5$ passes through the origin, then (x₁, y₁) does NOT lie on the curve :

A

${x^2} + {{{y^2}} \over {81}} = 2$

B

${{{y^2}} \over 9} - {x^2} = 8$

C

$y = 4{x^2} + 5$

D

${x \over 3} - {y^2} = 2$

Question 76: Palette 76/196

JEE Mains 2022 MCQ

The sum of absolute maximum and absolute minimum values of the function $f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$ in the interval [0, 1] is :

A

$3 + {{\sin (1){{\cos }^2}\left( {{1 \over 2}} \right)} \over 2}$

B

$3 + {1 \over 2}(1 + 2\cos (1))\sin (1)$

C

$5 + {1 \over 2}(\sin (1) + \sin (2))$

D

$2 + \sin \left( {{1 \over 2}} \right)\cos \left( {{1 \over 2}} \right)$

Question 77: Palette 77/196

JEE Mains 2022 MCQ

Let $\lambda x - 2y = \mu $ be a tangent to the hyperbola ${a^2}{x^2} - {y^2} = {b^2}$. Then ${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$ is equal to :

A

$-$2

B

$-$4

C

2

D

4

Question 78: Palette 78/196

JEE Mains 2021 MCQ

The function $f(x) = {x^3} - 6{x^2} + ax + b$ is such that $f(2) = f(4) = 0$. Consider two statements :

Statement 1 : there exists x₁, x₂ $\in$(2, 4), x₁ < x₂, such that f'(x₁) = $-$1 and f'(x₂) = 0.

Statement 2 : there exists x₃, x₄ $\in$ (2, 4), x₃ < x₄, such that f is decreasing in (2, x₄), increasing in (x₄, 4) and $2f'({x_3}) = \sqrt 3 f({x_4})$.

Then

A

both Statement 1 and Statement 2 are true

B

Statement 1 is false and Statement 2 is true

C

both Statement 1 and Statement 2 are false

D

Statement 1 is true and Statement 2 is false

Question 79: Palette 79/196

JEE Mains 2021 MCQ

The number of real roots of the equation

${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$ is :

A

2

B

4

C

1

D

0

Question 80: Palette 80/196

JEE Mains 2021 MCQ

A box open from top is made from a rectangular sheet of dimension a $\times$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

A

${{a + b - \sqrt {{a^2} + {b^2} - ab} } \over {12}}$

B

${{a + b - \sqrt {{a^2} + {b^2} + ab} } \over 6}$

C

${{a + b - \sqrt {{a^2} + {b^2} - ab} } \over 6}$

D

${{a + b + \sqrt {{a^2} + {b^2} + ab} } \over 6}$

Question 81: Palette 81/196

JEE Mains 2021 MCQ

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is :

A

${5 \over {2 + \sqrt 3 }}$

B

${{10} \over {2 + 3\sqrt 3 }}$

C

${5 \over {3 + \sqrt 3 }}$

D

${{10} \over {3 + 2\sqrt 3 }}$

Question 82: Palette 82/196

JEE Mains 2021 MCQ

The local maximum value of the function $f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$, x > 0, is

A

${\left( {2\sqrt e } \right)^{{1 \over e}}}$

B

${\left( {{4 \over {\sqrt e }}} \right)^{{e \over 4}}}$

C

${(e)^{{2 \over e}}}$

D

1

Question 83: Palette 83/196

JEE Mains 2021 MCQ

Let $f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$, $x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$. Then, f is :

A

increasing in $\left( { - {\pi \over 6},{\pi \over 2}} \right)$

B

decreasing in $\left( {0,{\pi \over 2}} \right)$

C

increasing in $\left( { - {\pi \over 6},0} \right)$

D

decreasing in $\left( { - {\pi \over 6},0} \right)$

Question 84: Palette 84/196

JEE Mains 2021 MCQ

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

$f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le 0} \cr } } \right.$. Then f is increasing function in the interval

A

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

B

(0,2)

C

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

D

($-$3, $-$1)

Question 85: Palette 85/196

JEE Mains 2021 MCQ

The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :

A

$-$22

B

5

C

$-$27

D

0

Question 86: Palette 86/196

JEE Mains 2021 MCQ

Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where ${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.$

Let a function f : R $\to$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

A

$ - {{20} \over {27}}$

B

${{88} \over {27}}$

C

${{20} \over {27}}$

D

$ - {{88} \over {27}}$

Question 87: Palette 87/196

JEE Mains 2021 MCQ

Let 'a' be a real number such that the function f(x) = ax² + 6x $-$ 15, x $\in$ R is increasing in $\left( { - \infty ,{3 \over 4}} \right)$ and decreasing in $\left( {{3 \over 4},\infty } \right)$. Then the function g(x) = ax² $-$ 6x + 15, x$\in$R has a :

A

local maximum at x = $-$ ${{3 \over 4}}$

B

local minimum at x = $-$${{3 \over 4}}$

C

local maximum at x = ${{3 \over 4}}$

D

local minimum at x = ${{3 \over 4}}$

Question 88: Palette 88/196

JEE Mains 2021 MCQ

Consider the function f : R $ \to $ R defined by

$f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr 0,\,\,x = 0 \hfill \cr} \right.$. Then f is :

A

not monotonic on ($-$$\infty $, 0) and (0, $\infty $)

B

monotonic on (0, $\infty $) only

C

monotonic on ($-$$\infty $, 0) only

D

monotonic on ($-$$\infty $, 0) $\cup$ (0, $\infty $)

Question 89: Palette 89/196

JEE Mains 2021 MCQ

Let f be a real valued function, defined on R $-$ {$-$1, 1} and given by

f(x) = 3 log_e $\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}$.

Then in which of the following intervals, function f(x) is increasing?

A

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

B

($-$$\infty $, $\infty $) $-$ {$-$1, 1)

C

($-$$\infty $, ${{1 \over 2}}$] $-$ {$-$1}

D

($-$1, ${{1 \over 2}}$]

Question 90: Palette 90/196

JEE Mains 2021 MCQ

The maximum value of

$f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R$ is :

A

$\sqrt 5 $

B

${3 \over 4}$

C

5

D

$\sqrt 7 $

Question 91: Palette 91/196

JEE Mains 2021 MCQ

Let slope of the tangent line to a curve at any point P(x, y) be given by ${{x{y^2} + y} \over x}$. If the curve intersects the line x + 2y = 4 at x = $-$2, then the value of y, for which the point (3, y) lies on the curve, is :

A

$ - {{18} \over {19}}$

B

$ - {{4} \over {3}}$

C

${{18} \over {35}}$

D

$ - {{18} \over {11}}$

Question 92: Palette 92/196

JEE Mains 2021 MCQ

The maximum slope of the curve $y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$ occurs at the point :

A

$\left( {3,{{21} \over 2}} \right)$

B

(0, 0)

C

(2, 9)

D

(2, 2)

Question 93: Palette 93/196

JEE Mains 2021 MCQ

Let f be any function defined on R and let it satisfy the condition : $|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$

If f(0) = 1, then :

A

f(x) can take any value in R

B

$f(x) < 0,\forall x \in R$

C

$f(x) > 0,\forall x \in R$

D

$f(x) = 0,\forall x \in R$

Question 94: Palette 94/196

JEE Mains 2021 MCQ

If the curves, ${{{x^2}} \over a} + {{{y^2}} \over b} = 1$ and ${{{x^2}} \over c} + {{{y^2}} \over d} = 1$ intersect each other at an angle of 90$^\circ$, then which of the following relations is TRUE?

A

a $-$ c = b + d

B

a + b = c + d

C

$ab = {{c + d} \over {a + b}}$

D

a $-$ b = c $-$ d

Question 95: Palette 95/196

JEE Mains 2021 MCQ

If Rolle's theorem holds for the function $f(x) = {x^3} - a{x^2} + bx - 4$, $x \in [1,2]$ with $f'\left( {{4 \over 3}} \right) = 0$, then ordered pair (a, b) is equal to :

A

($-$5, $-$8)

B

(5, $-$8)

C

($-$5, 8)

D

(5, 8)

Question 96: Palette 96/196

JEE Mains 2021 MCQ

For which of the following curves, the line $x + \sqrt 3 y = 2\sqrt 3 $ is the tangent at the point $\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)$?

A

$2{x^2} - 18{y^2} = 9$

B

${y^2} = {1 \over {6\sqrt 3 }}x$

C

${x^2} + 9{y^2} = 9$

D

${x^2} + {y^2} = 7$

Question 97: Palette 97/196

JEE Mains 2021 MCQ

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

$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr } } \right.$

Let A = {x $ \in $ R : f is increasing}. Then A is equal to :

A

$( - 5,\infty )$

B

$( - \infty , - 5) \cup (4,\infty )$

C

$( - 5, - 4) \cup (4,\infty )$

D

$( - \infty , - 5) \cup ( - 4,\infty )$

Question 98: Palette 98/196

JEE Mains 2021 MCQ

If the curve y = ax² + bx + c, x$ \in $R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :

A

a = $-$ 1, b = 1, c = 1

B

a = 1, b = 1, c = 0

C

a = ${1 \over 2}$, b = ${1 \over 2}$, c = 1

D

a = 1, b = 0, c = 1

Question 99: Palette 99/196

JEE Mains 2021 MCQ

The function
f(x) = ${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$ :

A

increases in $\left( { - \infty ,{1 \over 2}} \right]$

B

decreases in $\left( { - \infty ,{1 \over 2}} \right]$

C

increases in $\left[ {{1 \over 2},\infty } \right)$

D

decreases in $\left[ {{1 \over 2},\infty } \right)$

Question 100: Palette 100/196

JEE Mains 2021 MCQ

If the tangent to the curve y = x³ at the point P(t, t³) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

A

0

B

2t³

C

-2t³

D

-t³

Question 101: Palette 101/196

JEE Mains 2020 MCQ

If the tangent to the curve, y = f (x) = xlog_e x,
(x > 0) at a point (c, f(c)) is parallel to the line-segment
joining the points (1, 0) and (e, e), then c is equal to :

A

${{e - 1} \over e}$

B

${e^{\left( {{1 \over {1 - e}}} \right)}}$

C

${e^{\left( {{1 \over {e - 1}}} \right)}}$

D

${1 \over {e - 1}}$

Question 102: Palette 102/196

JEE Mains 2020 MCQ

The set of all real values of $\lambda $ for which the function

$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$

has exactly one maxima and exactly one minima, is :

A

$\left( { - {3 \over 2},{3 \over 2}} \right) - \left\{ 0 \right\}$

B

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

C

$\left( { - {1 \over 2},{1 \over 2}} \right) - \left\{ 0 \right\}$

D

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

Question 103: Palette 103/196

JEE Mains 2020 MCQ

The position of a moving car at time t is
given by f(t) = at² + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t₁ , t₂ ] is attained at the point :

A

${{\left( {{t_1} + {t_2}} \right)} \over 2}$

B

${{\left( {{t_2} - {t_1}} \right)} \over 2}$

C

2a(t₁ + t₂) + b

D

a(t₂ – t₁) + b

Question 104: Palette 104/196

JEE Mains 2020 MCQ

Which of the following points lies on the tangent to the curve

x⁴e^y + 2$\sqrt {y + 1} $ = 3 at the point (1, 0)?

A

(2, 2)

B

(–2, 4)

C

(2, 6)

D

(–2, 6)

Question 105: Palette 105/196

JEE Mains 2020 MCQ

If x = 1 is a critical point of the function
f(x) = (3x² + ax – 2 – a)e^x , then :

A

x = 1 is a local maxima and x = $ - {2 \over 3}$ is a local minima of f.

B

x = 1 and x = $ - {2 \over 3}$ are local maxima of f.

C

x = 1 and x = $ - {2 \over 3}$ are local minima of f.

D

x = 1 is a local minima and x = $ - {2 \over 3}$ is a local maxima of f.

Question 106: Palette 106/196

JEE Mains 2020 MCQ

If the point P on the curve, 4x² + 5y² = 20 is
farthest from the point Q(0, -4), then PQ² is equal to:

A

36

B

48

C

21

D

29

Question 107: Palette 107/196

JEE Mains 2020 MCQ

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x²–1 below the x-axis, is :

A

${1 \over {3\sqrt 3 }}$

B

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

C

${4 \over {3\sqrt 3 }}$

D

${4 \over 3}$

Question 108: Palette 108/196

JEE Mains 2020 MCQ

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $ \ge $ 1 and f''(x) $ \ge $ 4, for all x $ \in $ (1, 6), then :

A

f(5) $ \le $ 10

B

f(5) + f'(5) $ \ge $ 28

C

f(5) + f'(5) $ \le $ 26

D

f'(5) + f''(5) $ \le $ 20

Question 109: Palette 109/196

JEE Mains 2020 MCQ

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec), when the length of a side of the cube is 10 cm, is :

A

9

B

10

C

18

D

20

Question 110: Palette 110/196

JEE Mains 2020 MCQ

The function, f(x) = (3x – 7)x^2/3, x $ \in $ R, is increasing for all x lying in :

A

$\left( { - \infty ,0} \right) \cup \left( {{3 \over 7},\infty } \right)$

B

$\left( { - \infty ,0} \right) \cup \left( {{{14} \over {15}},\infty } \right)$

C

$\left( { - \infty ,{{14} \over {15}}} \right)$

D

$\left( { - \infty ,{{14} \over {15}}} \right) \cup \left( {0,\infty } \right)$

Question 111: Palette 111/196

JEE Mains 2020 MCQ

The equation of the normal to the curve
y = (1+x)^2y + cos ²(sin^–1x) at x = 0 is :

A

y = 4x + 2

B

x + 4y = 8

C

y + 4x = 2

D

2y + x = 4

Question 112: Palette 112/196

JEE Mains 2020 MCQ

Let f : (–1, $\infty $) $ \to $ R be defined by f(0) = 1 and
f(x) = ${1 \over x}{\log _e}\left( {1 + x} \right)$, x $ \ne $ 0. Then the function f :

A

decreases in (–1, $\infty $)

B

decreases in (–1, 0) and increases in (0, $\infty $)

C

increases in (–1, $\infty $)

D

increases in (–1, 0) and decreases in (0, $\infty $)

Question 113: Palette 113/196

JEE Mains 2020 MCQ

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

A

6

B

12

C

-12

D

-24

Question 114: Palette 114/196

JEE Mains 2020 MCQ

Let P(h, k) be a point on the curve
y = x² + 7x + 2, nearest to the line, y = 3x – 3.
Then the equation of the normal to the curve at P is :

A

x – 3y – 11 = 0

B

x – 3y + 22 = 0

C

x + 3y – 62 = 0

D

x + 3y + 26 = 0

Question 115: Palette 115/196

JEE Mains 2020 MCQ

If the tangent to the curve y = x + sin y at a point
(a, b) is parallel to the line joining $\left( {0,{3 \over 2}} \right)$ and $\left( {{1 \over 2},2} \right)$, then :

A

b = a

B

|b - a| = 1

C

$b = {\pi \over 2}$ + a

D

|a + b| = 1

Question 116: Palette 116/196

JEE Mains 2020 MCQ

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm³/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is :

A

${1 \over {18\pi }}$

B

${1 \over {36\pi }}$

C

${1 \over {54\pi }}$

D

${5 \over {6\pi }}$

Question 117: Palette 117/196

JEE Mains 2020 MCQ

The length of the perpendicular from the origin, on the normal to the curve,
x² + 2xy – 3y² = 0 at the point (2,2) is

A

$\sqrt 2 $

B

$4\sqrt 2 $

C

2

D

$2\sqrt 2 $

Question 118: Palette 118/196

JEE Mains 2020 MCQ

Let ƒ(x) = xcos^–1(–sin|x|), $x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$, then which of the following is true?

A

ƒ' is decreasing in $\left( { - {\pi \over 2},0} \right)$ and increasing in $\left( {0,{\pi \over 2}} \right)$

B

ƒ '(0) = ${ - {\pi \over 2}}$

C

ƒ is not differentiable at x = 0

D

ƒ' is increasing in $\left( { - {\pi \over 2},0} \right)$ and decreasing in $\left( {0,{\pi \over 2}} \right)$

Question 119: Palette 119/196

JEE Mains 2020 MCQ

If c is a point at which Rolle's theorem holds for the function,
f(x) = ${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$ in the interval [3, 4], where a $ \in $ R, then ƒ''(c) is equal to

A

${1 \over {12}}$

B

${{\sqrt 3 } \over 7}$

C

$-{1 \over {12}}$

D

$-{1 \over {24}}$

Question 120: Palette 120/196

JEE Mains 2020 MCQ

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points.

If $\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4$, then which one of the following is not true?

A

ƒ(1) - 4ƒ(-1) = 4.

B

x = 1 is a point of minima and x = -1 is a point of maxima of ƒ.

C

x = 1 is a point of maxima and x = -1 is a point of minimum of ƒ.

D

ƒ is an odd function.

Question 121: Palette 121/196

JEE Mains 2020 MCQ

The value of c in the Lagrange's mean value theorem for the function
ƒ(x) = x³ - 4x² + 8x + 11, when x $ \in $ [0, 1] is:

A

${2 \over 3}$

B

${{\sqrt 7 - 2} \over 3}$

C

${{4 - \sqrt 5 } \over 3}$

D

${{4 - \sqrt 7 } \over 3}$

Question 122: Palette 122/196

JEE Mains 2020 MCQ

Let the function, ƒ:[-7, 0]$ \to $R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $ \le $ 2, for all x $ \in $ (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval:

A

$\left[ { - 6,20} \right]$

B

$\left( { - \infty ,\left. {20} \right]} \right.$

C

$\left[ { - 3,11} \right]$

D

$\left( { - \infty ,\left. {11} \right]} \right.$

Question 123: Palette 123/196

JEE Mains 2019 MCQ

If m is the minimum value of k for which the function f(x) = x$\sqrt {kx - {x^2}} $ is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :

A

$\left( {5,3\sqrt 6 } \right)$

B

$\left( {4,3\sqrt 3 } \right)$

C

$\left( {4,3\sqrt 2 } \right)$

D

$\left( {3,3\sqrt 3 } \right)$

Question 124: Palette 124/196

JEE Mains 2019 MCQ

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :

A

${{25} \over 3}$

B

25

C

25$\sqrt 3 $

D

${{25} \over {\sqrt 3 }}$

Question 125: Palette 125/196

JEE Mains 2019 MCQ

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³ /min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

A

${5 \over {6\pi }}$

B

${1 \over {9\pi }}$

C

${1 \over {36\pi }}$

D

${1 \over {18\pi }}$

Question 126: Palette 126/196

JEE Mains 2019 MCQ

If the tangent to the curve $y = {x \over {{x^2} - 3}}$ , $x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)$, at a point ($\alpha $, $\beta $) $ \ne $ (0, 0) on it is parallel to the line 2x + 6y – 11 = 0, then :

A

| 6$\alpha $ + 2$\beta $ | = 9

B

| 2$\alpha $ + 6$\beta $ | = 11

C

| 2$\alpha $ + 6$\beta $ | = 19

D

| 6$\alpha $ + 2$\beta $ | = 19

Question 127: Palette 127/196

JEE Mains 2019 MCQ

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$. Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-

A

${1 \over {15\pi }}$

B

${1 \over {5\pi }}$

C

${1 \over {10\pi }}$

D

${2 \over \pi }$

Question 128: Palette 128/196

JEE Mains 2019 MCQ

If ƒ(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set
S = {x $ \in $ R : ƒ(x) = ƒ(0)}
Contains exactly :

A

four rational numbers.

B

four irrational numbers.

C

two irrational and one rational number.

D

two irrational and two rational numbes.

Question 129: Palette 129/196

JEE Mains 2019 MCQ

Let S be the set of all values of x for which the tangent to the curve
y = ƒ(x) = x³ – x² – 2x at (x, y) is parallel to the line segment joining the points (1, ƒ(1)) and (–1, ƒ(–1)), then S is equal to :

A

$\left\{ { {1 \over 3}, - 1} \right\}$

B

$\left\{ { - {1 \over 3}, 1} \right\}$

C

$\left\{ { - {1 \over 3}, - 1} \right\}$

D

$\left\{ { {1 \over 3}, 1} \right\}$

Question 130: Palette 130/196

JEE Mains 2019 MCQ

If the tangent to the curve, y = x³ + ax – b at the point (1, –5) is perpendicular to the line, –x + y + 4 = 0, then which one of the following points lies on the curve ?

A

(2, –2)

B

(2, –1)

C

(–2, 2)

D

(–2, 1)

Question 131: Palette 131/196

JEE Mains 2019 MCQ

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

A

$\sqrt 3 $

B

$2\sqrt 3 $

C

$\sqrt 6 $

D

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

Question 132: Palette 132/196

JEE Mains 2019 MCQ

Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $2y \over x^2$. If the curve passes through the centre of the circle x² + y² – 2x – 2y = 0, then its equation is :

A

x log_e|y| = 2(x – 1)

B

x² log_e|y| = –2(x – 1)

C

x log_e|y| = x – 1

D

x log_e|y| = –2(x – 1)

Question 133: Palette 133/196

JEE Mains 2019 MCQ

Let ƒ : [0, 2] $ \to $ R be a twice differentiable function such that ƒ''(x) > 0, for all x $ \in $ (0, 2). If $\phi $(x) = ƒ(x) + ƒ(2 – x), then $\phi $ is :

A

decreasing on (0, 2)

B

decreasing on (0, 1) and increasing on (1, 2)

C

increasing on (0, 2)

D

increasing on (0, 1) and decreasing on (1, 2)

Question 134: Palette 134/196

JEE Mains 2019 MCQ

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function,

ƒ(x) = 9x⁴ + 12x³ – 36x² + 25, x $ \in $ R, then :

A

S₁ = {–1}; S₂ = {0, 2}

B

S₁ = {–2}; S₂ = {0, 1}

C

S₁ = {–2, 0}; S₂ = {1}

D

S₁ = {–2, 1}; S₂ = {0}

Question 135: Palette 135/196

JEE Mains 2019 MCQ

The tangent to the curve y = x² – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :

A

$\left\{ {{1 \over 4},{7 \over 2}} \right\}$

B

$\left( { - {1 \over 8},7} \right)$

C

$\left( {{7 \over 2},{1 \over 4}} \right)$

D

$\left( {{1 \over 8}, - 7} \right)$

Question 136: Palette 136/196

JEE Mains 2019 MCQ

If the function f given by f(x) = x³ – 3(a – 2)x² + 3ax + 7, for some a$ \in $R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, ${{f\left( x \right) - 14} \over {{{\left( {x - 1} \right)}^2}}} = 0\left( {x \ne 1} \right)$ is :

A

$-$ 7

B

5

C

7

D

6

Question 137: Palette 137/196

JEE Mains 2019 MCQ

Let f(x) = ${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$ x $\, \in $ R, where a, b and d are non-zero real constants. Then :

A

f is an increasing function of x

B

f is neither increasing nor decreasing function of x

C

f ' is not a continuous function of x

D

f is a decreasing function of x

Question 138: Palette 138/196

JEE Mains 2019 MCQ

The maximum value of the function f(x) = 3x³ – 18x² + 27x – 40 on the set S = $\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}$ is :

A

$-$ 222

B

$-$ 122

C

$122$

D

222

Question 139: Palette 139/196

JEE Mains 2019 MCQ

The tangent to the curve, y = xe^x2 passing through the point (1, e) also passes through the point

A

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

B

(3, 6e)

C

(2, 3e)

D

$\left( {{5 \over 3},2e} \right)$

Question 140: Palette 140/196

JEE Mains 2019 MCQ

A helicopter is flying along the curve given by y – x^3/2 = 7, (x $ \ge $ 0). A soldier positioned at the point $\left( {{1 \over 2},7} \right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is -

A

${1 \over 6}\sqrt {{7 \over 3}} $

B

${{\sqrt 5 } \over 6}$

C

${1 \over 2}$

D

${1 \over 3}$$\sqrt {{7 \over 3}} $

Question 141: Palette 141/196

JEE Mains 2019 MCQ

The shortest distance between the point  $\left( {{3 \over 2},0} \right)$   and the curve y = $\sqrt x $, (x > 0), is -

A

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

B

${5 \over 4}$

C

${3 \over 2}$

D

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

Question 142: Palette 142/196

JEE Mains 2019 MCQ

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :

A

2$\sqrt3$$\pi $

B

3$\sqrt3$$\pi $

C

6$\pi $

D

${4 \over 3}\pi $

Question 143: Palette 143/196

JEE Mains 2018 MCQ

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x³ $-$ 9x² + 12x + 5 in the interval [0, 3]. Then M $-$m is equal to :

A

5

B

9

C

4

D

1

Question 144: Palette 144/196

JEE Mains 2018 MCQ

If the curves y² = 6x, 9x² + by² = 16 intersect each other at right angles, then the value of b is :

A

${9 \over 2}$

B

6

C

${7 \over 2}$

D

4

Question 145: Palette 145/196

JEE Mains 2018 MCQ

Let $f\left( x \right) = {x^2} + {1 \over {{x^2}}}$ and $g\left( x \right) = x - {1 \over x}$,
$x \in R - \left\{ { - 1,0,1} \right\}$.
If $h\left( x \right) = {{f\left( x \right)} \over {g\left( x \right)}}$, then the local minimum value of h(x) is

A

$2\sqrt 2 $

B

3

C

-3

D

$-2\sqrt 2 $

Question 146: Palette 146/196

JEE Mains 2018 MCQ

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm²) of this cone is :

A

$6\sqrt 2 \pi $

B

$6\sqrt 3 \pi $

C

$8\sqrt 2 \pi $

D

$8\sqrt 3 \pi $

Question 147: Palette 147/196

JEE Mains 2018 MCQ

If $\beta $ is one of the angles between the normals to the ellipse, x² + 3y² = 9 at the points (3 cos $\theta $, $\sqrt 3 \sin \theta $) and ($-$ 3 sin $\theta $, $\sqrt 3 \,\cos \theta $); $\theta \in \left( {0,{\pi \over 2}} \right);$ then ${{2\,\cot \beta } \over {\sin 2\theta }}$ is equal to :

A

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

B

${1 \over {\sqrt 3 }}$

C

$\sqrt 2 $

D

${{\sqrt 3 } \over 4}$

Question 148: Palette 148/196

JEE Mains 2017 MCQ

The function f defined by

f(x) = x³ $-$ 3x² + 5x + 7 , is :

A

increasing in R.

B

decreasing in R.

C

decreasing in (0, $\infty $) and increasing in ($-$ $\infty $, 0)

D

increasing in (0, $\infty $) and decreasing in ($-$ $\infty $, 0)

Question 149: Palette 149/196

JEE Mains 2017 MCQ

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :

A

$\left( {{1 \over 3},24} \right)$

B

$\left( {{1 \over 2},4} \right)$

C

$\left( {2,{1 \over 8}} \right)$

D

$\left( {3,{1 \over 28}} \right)$

Question 150: Palette 150/196

JEE Mains 2017 MCQ

The tangent at the point (2, $-$2) to the curve, x²y² $-$ 2x = 4(1 $-$ y) does not pass through the point :

A

$\left( {4,{1 \over 3}} \right)$

B

(8, 5)

C

($-$4, $-$9)

D

($-$2, $-$7)

Question 151: Palette 151/196

JEE Mains 2017 MCQ

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point :

A

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

B

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

C

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

D

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

Question 152: Palette 152/196

JEE Mains 2017 MCQ

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

A

10

B

25

C

30

D

12.5

Question 153: Palette 153/196

JEE Mains 2016 MCQ

Let C be a curve given by y(x) = 1 + $\sqrt {4x - 3} ,x > {3 \over 4}.$ If P is a point on C, such that the tangent at P has slope ${2 \over 3}$, then a point through which the normal at P passes, is :

A

(2, 3)

B

(4, $-$3)

C

(1, 7)

D

(3, $-$ 4),

Question 154: Palette 154/196

JEE Mains 2016 MCQ

Let f(x) = sin⁴x + cos⁴ x. Then f is an increasing function in the interval :

A

$] 0, \frac{\pi}{4}[$

B

$] \frac{\pi}{4}, \frac{\pi}{2}[$

C

$] \frac{\pi}{2}, \frac{5 \pi}{8}[$

D

$] \frac{5 \pi}{8}, \frac{3 \pi}{4}[$

Question 155: Palette 155/196

JEE Mains 2016 MCQ

The minimum distance of a point on the curve y = x²−4 from the origin is :

A

${{\sqrt {19} } \over 2}$

B

$\sqrt {{{15} \over 2}} $

C

${{\sqrt {15} } \over 2}$

D

$\sqrt {{{19} \over 2}} $

Question 156: Palette 156/196

JEE Mains 2016 MCQ

If the tangent at a point P, with parameter t, on the curve x = 4t² + 3, y = 8t³−1, t $ \in $ R, meets the curve again at a point Q, then the coordinates of Q are :

A

(t² + 3, − t³ −1)

B

(4t² + 3, − 8t³ −1)

C

(t² + 3, t³ −1)

D

(16t² + 3, − 64t³ −1)

Question 157: Palette 157/196

JEE Mains 2016 MCQ

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then:

A

$x=2r$

B

$2x=r$

C

$2x = \left( {\pi + 4} \right)r$

D

$\left( {4 - \pi } \right)x = \pi \,\, r$

Question 158: Palette 158/196

JEE Mains 2016 MCQ

Consider :
f $\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right).$

A normal to $y = $ f$\left( x \right)$ at $x = {\pi \over 6}$ also passes through the point:

A

$\left( {{\pi \over 6},0} \right)$

B

$\left( {{\pi \over 4},0} \right)$

C

$(0,0)$

D

$\left( {0,{{2\pi } \over 3}} \right)$

Question 159: Palette 159/196

JEE Mains 2015 MCQ

Let $f(x)$ be a polynomial of degree four having extreme values
at $x=1$ and $x=2$. If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3$, then f$(2)$ is equal to :

A

$0$

B

$4$

C

$-8$

D

$-4$

Question 160: Palette 160/196

JEE Mains 2015 MCQ

The normal to the curve, ${x^2} + 2xy - 3{y^2} = 0$, at $(1,1)$

A

meets the curve again in the third quadrant.

B

meets the curve again in the fourth quadrant.

C

does not meet the curve again.

D

meets the curve again in the second quadrant.

Question 161: Palette 161/196

JEE Mains 2014 MCQ

If $x=-1$ and $x=2$ are extreme points of $f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x$ then

A

$\alpha = 2,\beta = - {1 \over 2}$

B

$\alpha = 2,\beta = {1 \over 2}$

C

$\alpha = - 6,\beta = {1 \over 2}$

D

$\alpha = - 6,\beta = -{1 \over 2}$

Question 162: Palette 162/196

JEE Mains 2014 MCQ

If $f$ and $g$ are differentiable functions in $\left[ {0,1} \right]$ satisfying
$f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) = 0$ and $f\left( 1 \right) = 6,$ then for some $c \in \left] {0,1} \right[$

A

$f'\left( c \right) = g'\left( c \right)$

B

$f'\left( c \right) = 2g'\left( c \right)$

C

$2f'\left( c \right) = g'\left( c \right)$

D

$2f'\left( c \right) = 3g'\left( c \right)$

Question 163: Palette 163/196

JEE Mains 2013 MCQ

The intercepts on $x$-axis made by tangents to the curve,
$y = \int\limits_0^x {\left| t \right|dt,x \in R,} $ which are parallel to the line $y=2x$, are equal to :

A

$ \pm 1$

B

$ \pm 2$

C

$ \pm 3$

D

$ \pm 4$

Question 164: Palette 164/196

JEE Mains 2013 MCQ

The real number $k$ for which the equation, $2{x^3} + 3x + k = 0$ has two distinct real roots in $\left[ {0,\,1} \right]$

A

lies between 1 and 2

B

lies between 2 and 3

C

lies between $ - 1$ and 0

D

does not exist.

Question 165: Palette 165/196

JEE Mains 2012 MCQ

A line is drawn through the point $(1, 2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ,$ where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is :

A

$-{1 \over 4}$

B

$-4$

C

$-2$

D

$-{1 \over 2}$

Question 166: Palette 166/196

JEE Mains 2012 MCQ

Let $a,b \in R$ be such that the function $f$ given by $f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0$ has extreme values at $x=-1$ and $x=2$

Statement-1 : $f$ has local maximum at $x=-1$ and at $x=2$.

Statement-2 : $a = {1 \over 2}$ and $b = {-1 \over 4}$

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 167: Palette 167/196

JEE Mains 2012 MCQ

A spherical balloon is filled with $4500\pi $ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi $ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is :

A

${{9 \over 7}}$

B

${{7 \over 9}}$

C

${{2 \over 9}}$

D

${{9 \over 2}}$

Question 168: Palette 168/196

JEE Mains 2011 MCQ

For $x \in \left( {0,{{5\pi } \over 2}} \right),$ define $f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.} $ Then $f$ has

A

local minimum at $\pi $ and $2\pi $

B

local minimum at $\pi $ and local maximum at $2\pi $

C

local maximum at $\pi $ and local minimum at $2\pi $

D

local maximum at $\pi $ and $2\pi $

Question 169: Palette 169/196

JEE Mains 2011 MCQ

The shortest distance between line $y-x=1$ and curve $x = {y^2}$ is

A

${{3\sqrt 2 } \over 8}$

B

${8 \over {3\sqrt 2 }}$

C

${4 \over {\sqrt 3 }}$

D

${{\sqrt 3 } \over 4}$

Question 170: Palette 170/196

JEE Mains 2010 MCQ

Let $f:R \to R$ be a continuous function defined by $$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$

Statement - 1 : $f\left( c \right) = {1 \over 3},$ for some $c \in R$.

Statement - 2 : $0 < f\left( x \right) \le {1 \over {2\sqrt 2 }},$ for all $x \in R$

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 171: Palette 171/196

JEE Mains 2010 MCQ

The equation of the tangent to the curve $y = x + {4 \over {{x^2}}}$, that
is parallel to the $x$-axis, is

A

$y=1$

B

$y=2$

C

$y=3$

D

$y=0$

Question 172: Palette 172/196

JEE Mains 2010 MCQ

Let $f:R \to R$ be defined by $$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.$$

If $f$has a local minimum at $x=-1$, then a possible value of $k$ is

A

$0$

B

$ - {1 \over 2}$

C

$-1$

D

$1$

Question 173: Palette 173/196

JEE Mains 2009 MCQ

Given $P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$ such that $x=0$ is the only
real root of $P'\,\left( x \right) = 0.$ If $P\left( { - 1} \right) < P\left( 1 \right),$ then in the interval $\left[ { - 1,1} \right]:$

A

$P(-1)$ is not minimum but $P(1)$ is the maximum of $P$

B

$P(-1)$ is the minimum but $P(1)$ is not the maximum of $P$

C

Neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$

D

$P(-1)$ is the minimum and $P(1)$ is the maximum of $P$

Question 174: Palette 174/196

JEE Mains 2008 MCQ

How many real solutions does the equation
${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$ have?

A

$7$

B

$1$

C

$3$

D

$5$

Question 175: Palette 175/196

JEE Mains 2008 MCQ

Suppose the cubic ${x^3} - px + q$ has three distinct real roots
where $p>0$ and $q>0$. Then which one of the following holds?

A

The cubic has minima at $\sqrt {{p \over 3}} $ and maxima at $-\sqrt {{p \over 3}} $

B

The cubic has minima at $-\sqrt {{p \over 3}} $ and maxima at $\sqrt {{p \over 3}} $

C

The cubic has minima at both $\sqrt {{p \over 3}} $ and $-\sqrt {{p \over 3}} $

D

The cubic has maxima at both $\sqrt {{p \over 3}} $ and $-\sqrt {{p \over 3}} $

Question 176: Palette 176/196

JEE Mains 2007 MCQ

The function $f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$ is an incresing function in

A

$\left( {0,{\pi \over 2}} \right)$

B

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

C

$\left( { {\pi \over 4},{\pi \over 2}} \right)$

D

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

Question 177: Palette 177/196

JEE Mains 2007 MCQ

A value of $c$ for which conclusion of Mean Value Theorem holds for the function $f\left( x \right) = {\log _e}x$ on the interval $\left[ {1,3} \right]$ is

A

${\log _3}e$

B

${\log _e}3$

C

$2\,\,{\log _3}e$

D

${1 \over 2}{\log _3}e$

Question 178: Palette 178/196

JEE Mains 2007 MCQ

If $p$ and $q$ are positive real numbers such that ${p^2} + {q^2} = 1$, then the maximum value of $(p+q)$ is

A

${1 \over 2}$

B

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

C

${\sqrt 2 }$

D

$2$

Question 179: Palette 179/196

JEE Mains 2006 MCQ

The function $f\left( x \right) = {x \over 2} + {2 \over x}$ has a local minimum at

A

$x=2$

B

$x=-2$

C

$x=0$

D

$x=1$

Question 180: Palette 180/196

JEE Mains 2006 MCQ

Angle between the tangents to the curve $y = {x^2} - 5x + 6$ at the points $(2,0)$ and $(3,0)$ is

A

$\pi $

B

${\pi \over 2}$

C

${\pi \over 6}$

D

${\pi \over 4}$

Question 181: Palette 181/196

JEE Mains 2006 MCQ

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

A

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

B

$\sqrt {{{{x^3}} \over 8}} $

C

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

D

$\pi {x^2}$

Question 182: Palette 182/196

JEE Mains 2005 MCQ

Let f be differentiable for all x. If f(1) = -2 and f'(x) $ \ge $ 2 for
x $ \in \left[ {1,6} \right]$, then

A

f(6) $ \ge $ 8

B

f(6) < 8

C

f(6) < 5

D

f(6) = 5

Question 183: Palette 183/196

JEE Mains 2005 MCQ

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

A

Interval	Function
(- $\infty $, $\infty $)	x3 - 3x2 + 3x + 3

B

Interval	Function
[2, $\infty $)	2x3 - 3x2 - 12x + 6

C

Interval	Function
$\left( { - \infty ,{1 \over 3}} \right]$	3x2 - 2x + 1

D

Interval	Function
($ - \infty $, - 4 )	x3 + 6x2 + 6

Question 184: Palette 184/196

JEE Mains 2005 MCQ

The normal to the curve
$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$ at any point
$\theta\, '$ is such that

A

it passes through the origin

B

it makes an angle ${\pi \over 2} + \theta $ with the $x$-axis

C

it passes through $\left( {a{\pi \over 2}, - a} \right)$

D

it is at a constant distance from the origin

Question 185: Palette 185/196

JEE Mains 2005 MCQ

A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50$ cm$^3$ /min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases is

A

${1 \over {36\pi }}$ cm/min

B

${1 \over {18\pi }}$ cm/min

C

${1 \over {54\pi }}$ cm/min

D

${5 \over {6\pi }}$ cm/min

Question 186: Palette 186/196

JEE Mains 2005 MCQ

If the equation ${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$
${a_1} \ne 0,n \ge 2,$ has a positive root $x = \alpha $, then the equation
$n{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + ........... + {a_1} = 0$ has a positive root, which is

A

greater than $\alpha $

B

smaller than $\alpha $

C

greater than or equal to smaller than $\alpha $

D

equal to smaller than $\alpha $

Question 187: Palette 187/196

JEE Mains 2005 MCQ

Area of the greatest rectangle that can be inscribed in the
ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$

A

$2ab$

B

$ab$

C

$\sqrt {ab} $

D

${a \over b}$

Question 188: Palette 188/196

JEE Mains 2005 MCQ

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after :

A

20 s

B

1 s

C

21 s

D

24 s

Question 189: Palette 189/196

JEE Mains 2004 MCQ

The normal to the curve x = a(1 + cos $\theta $), $y = a\sin \theta $ at $'\theta '$ always passes through the fixed point

A

$(a, a)$

B

$(0, a)$

C

$(0, 0)$

D

$(a, 0)$

Question 190: Palette 190/196

JEE Mains 2004 MCQ

A function $y=f(x)$ has a second order derivative $f''\left( x \right) = 6\left( {x - 1} \right).$ If its graph passes through the point $(2, 1)$ and at that point the tangent to the graph is $y = 3x - 5$, then the function is :

A

${\left( {x + 1} \right)^2}$

B

${\left( {x - 1} \right)^3}$

C

${\left( {x + 1} \right)^3}$

D

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

Question 191: Palette 191/196

JEE Mains 2004 MCQ

If $2a+3b+6c=0$, then at least one root of the equation
$a{x^2} + bx + c = 0$ lies in the interval

A

$(1, 3)$

B

$(1, 2)$

C

$(2, 3)$

D

$(0, 1)$

Question 192: Palette 192/196

JEE Mains 2004 MCQ

A point on the parabola ${y^2} = 18x$ at which the ordinate increases at twice the rate of the abscissa is

A

$\left( {{9 \over 8},{9 \over 2}} \right)$

B

$(2, -4)$

C

$\left( {{-9 \over 8},{9 \over 2}} \right)$

D

$(2, 4)$

Question 193: Palette 193/196

JEE Mains 2003 MCQ

The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :

A

-2

B

2

C

1

D

-1

Question 194: Palette 194/196

JEE Mains 2003 MCQ

If the function $f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$ where $a>0,$ attains its maximum and minimum at $p$ and $q$ respectively such that ${p^2} = q$ , then $a$ equals

A

${1 \over 2}$

B

$3$

C

$1$

D

$2$

Question 195: Palette 195/196

JEE Mains 2002 MCQ

The maximum distance from origin of a point on the curve
$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$
$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$ both $a,b > 0$ is

A

$a-b$

B

$a+b$

C

$\sqrt {{a^2} + {b^2}} $

D

$\sqrt {{a^2} - {b^2}} $

Question 196: Palette 196/196

JEE Mains 2002 MCQ

If $2a+3b+6c=0,$ $\left( {a,b,c \in R} \right)$ then the quadratic equation $a{x^2} + bx + c = 0$ has

A

at least one root in $\left[ {0,1} \right]$

B

at least one root in $\left[ {2,3} \right]$

C

at least one root in $\left[ {4,5} \right]$

D

none of these