How many questions from Application of Derivatives appear in JEE?

Application of Derivatives has appeared in 139 of the last 186 JEE Main and JEE Advanced papers — about 75% of them — contributing 207 questions in total across those papers.

Is Application of Derivatives an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 75% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Application of Derivatives questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Application of Derivatives JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

Consider the function

f : (0, \infty) \to (-\infty, \infty)

given by

f(x) = \sqrt{x}\, \log_e(x) - x + 1

. Then which one of the following statements is TRUE ?

(A)The derivative of the function $f$ is decreasing in the interval $(0, 1)$
(B)The function $f$ has a local maximum at some point $a \in (0, \infty)$
(C)The function $f$ has a local minimum at some point $b \in (0, \infty)$
(D)The function $f$ has NEITHER a point of local maximum NOR a point of local minimum in the interval $(0, \infty)$

Q2·MathematicsSingle correctJEE Main 2026

Let

f : \mathbb{R} \to \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'(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) $3e$

Q3·MathematicsSingle correctJEE Main 2026

\max_{0 \leq x \leq \pi}\left(16\sin\left(\frac{x}{2}\right)\cos^{3}\left(\frac{x}{2}\right)\right)

is equal to:

(A) $\frac{3\sqrt{3}}{2}$
(B) $3\sqrt{3}$
(C) $4\sqrt{3}$
(D) $6\sqrt{3}$

Q4·MathematicsSingle correctJEE Main 2026

Let

f(x)

be a polynomial of degree 5, and have extrema at

x = 1

and

x = -1

. If

\lim_{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) $112$

Q5·MathematicsSingle correctJEE Main 2026

The number of critical points of the function

f(x) = \begin{cases} \left|\frac{\sin x}{x}\right|, & 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

Q6·MathematicsNumericalJEE Main 2026

Let f be a differentiable function satisfying

f(x) = 1 - 2x + \int_0^x e^{(x-t)}\, f(t)\, dt

, x

\in

\mathbf{R}

and let

g(x) = \int_0^x (f(t) + 2)^{15}\, (t - 4)^6\, (t + 12)^{17}\, dt

, x

\in

\mathbf{R}

. If p and q are respectively the points of local minima and local maxima of g, then the value of

|p + q|

is equal to _________ .

Q7·MathematicsSingle correctJEE Main 2026

Consider the following three statements for the function

f : (0, \infty) \to \mathbb{R}

defined by

f(x) = |\log_{e} x| - |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)All (I), (II) and (III) are TRUE.
(B)Only (I) is TRUE.
(C)Only (II) and (III) are TRUE.
(D)Only (I) and (III) are TRUE.

Q8·MathematicsNumericalJEE Main 2026

Let (2α, α) be the largest interval in which the function

f(\mathrm{t}) = \frac{|\mathrm{t} + 1|}{\mathrm{t}^{2}}

, t < 0, is strictly decreasing. Then the local maximum value of the function

g(\mathrm{x}) = 2 \log_{e}(\mathrm{x} - 2) + \alpha \mathrm{x}^{2} + 4\mathrm{x} - \alpha

, x > 2, is _____

Q9·MathematicsSingle correctJEE Main 2026

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)−81
(B)−40
(C)−41
(D)−80

Q10·MathematicsSingle correctJEE Main 2026

Let

f : \mathbf{R} \to \mathbf{R}

be a twice differentiable function such that

f''(x) > 0

for all x∈

\mathbf{R}

and

f'(a - 1) = 0

, where a is 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)Neither (I) nor (II) is True
(B)Only (II) is True
(C)Only (I) is True
(D)Both (I) and (II) are True

Q11·MathematicsNumericalJEE Main 2026

Let f : R→R be a twice differentiable function such that the quadratic equation

f(x)m^{2}-2f'(x)m+f''(x)=0

in m, has two equal roots for every

x\in R

. If

f(0)=1

,

f'(0)=2

and (α, β) is the largest interval in which the function

f(\log_{e}x-x)

is increasing, then α + β is equal to

Q12·MathematicsMultiple correctJEE Advanced 2025

Let R denote the set of all real numbers. Let

f : R \rightarrow R

be defined by

f(x) = \begin{cases} \frac{6x + \sin x}{2x + \sin x} & \text{if } x \neq 0 \\ \frac{7}{3} & \text{if } x = 0 \end{cases}

Then which of the following statements is (are) TRUE?

(A)The point x = 0 is a point of local maxima of f
(B)The point x = 0 is a point of local minima of f
(C)Number of points of local maxima of f in the interval $[\pi, 6\pi]$ is 3
(D)Number of points of local minima of f in the interval $[2\pi, 4\pi]$ is 1

Q13·MathematicsSingle correctJEE Main 2025

Let the function

f(x)=\dfrac{x}{3}+\dfrac{3}{x}+3,\ x\ne0

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_{i=1}^{5}\alpha_i^2

is equal to:

(A) $48$
(B) $28$
(C) $40$
(D) $36$

Q14·MathematicsSingle correctJEE Main 2025

Let

x=-1

and

x=2

be the critical points of the function

f(x)=x^3+ax^2+b\log_e|x|+1

,

x\ne0

. Let

m

and

M

respectively be the absolute minimum and the absolute maximum values of

f

in the interval

\left[-2,-\dfrac{1}{2}\right]

. Then

|M+m|

is equal to (Take

\log_e2=0.7

):

(A)21.1
(B)19.8
(C)22.1
(D)20.9

Q15·MathematicsSingle correctJEE Main 2025

Let

a>0

. If the function

f(x)=6x^3-45ax^2+108a^2x+1

attains its local maximum and minimum values at the points

x_1

and

x_2

respectively such that

x_1x_2=54

, then

a+x_1+x_2

is equal to:

(A)15
(B)18
(C)24
(D)13

Q16·MathematicsSingle correctJEE Main 2025

If the function

f(x)=2x^3-9ax^2+12a^2x+1

, where

a>0

, attains its local maximum and minimum values at

p

and

q

, respectively, such that

p^2=q

, then

f(3)

is equal to:

(A)55
(B)10
(C)23
(D)37

Q17·MathematicsSingle correctJEE Main 2025

The sum of all local minimum values of the function

f(x)=\begin{cases}1-2x, & x<-1\\ \frac{1}{3}(7+2|x|), & -1\le x\le 2\\ \frac{11}{18}(x-4)(x-5), & x>2\end{cases}

is:

(A) $\frac{171}{72}$
(B) $\frac{131}{72}$
(C) $\frac{157}{72}$
(D) $\frac{167}{72}$

Q18·MathematicsSingle correctJEE Main 2025

Let

(2,3)

be the largest open interval in which the function

f(x)=2\log_e(x-2)-x^2+ax+1

is strictly increasing and

(b,c)

be the largest open interval, in which the function

g(x)=(x-1)^3(x+2-a)^2

is strictly decreasing, then

100(a+b-c)

is equal to:

(A)280
(B)360
(C)420
(D)160

Q19·MathematicsSingle correctJEE Main 2025

Let

f(x)=\displaystyle\int_0^{x^2}\dfrac{t^2-8t+15}{e^t}\,dt

,

x\in\mathbb{R}

. Then the numbers of local maximum and local minimum points of

f

, respectively, are:

(A)2 and 3
(B)3 and 2
(C)1 and 3
(D)2 and 2

Q20·MathematicsIntegerJEE Advanced 2024

Let the function

f : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = \frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)} + \frac{2}{e^{\pi x}} \frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)}

. Then the number of solutions of f(x) = 0 in

\mathbb{R}

is ______

Q21·MathematicsNumericalJEE Main 2024

Let the set of all values of

p

, for which

f(x)=(p^{2}-6p+8)\left(\sin^{2}2x-\cos^{2}2x\right)+2(2-p)x+\cos^{2}1

does not have any critical point, be the interval

(a, b)

. Then

16ab

is equal to _______.

Q22·MathematicsNumericalJEE Main 2024

Let the set of all positive values of

\lambda

, for which the point of local minimum of the function

\left(1 + x(\lambda^2 - x^2)\right)

satisfies

\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0

, be

(\alpha, \beta)

. Then

\alpha^2 + \beta^2

is equal to ________.

Q23·MathematicsSingle correctJEE Main 2024

For the function

f(x)=\cos x-x+1

,

x\in\mathbb{R}

, between the following two statements:

(S_1)

f(x)=0

for only one value of

x

in

[0,\pi]

.

(S_2)

f(x)

is decreasing in

\left[0,\dfrac\pi2\right]

and increasing in

\left[\dfrac\pi2,\pi\right]

.

(A)Both $(S_1)$ and $(S_2)$ are correct
(B)Only $(S_1)$ is correct
(C)Both $(S_1)$ and $(S_2)$ are incorrect
(D)Only $(S_2)$ is correct

Q24·MathematicsSingle correctJEE Main 2024

If the function

f(x)=2x^3-9ax^2+12a^2x+1

,

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) $x^2-6x+8=0$
(B) $8x^2-6x+8=0$
(C) $8x^2-6x+1=0$
(D) $x^2+6x+8=0$

Q25·MathematicsSingle correctJEE Main 2024

Let

f(x)=4\cos^3 x+3\sqrt3\cos^2 x-10

. The number of points of local maxima of

f

in interval

(0,2\pi)

is:

(A) $1$
(B) $2$
(C) $3$
(D) $4$

Q26·MathematicsNumericalJEE Main 2024

Let A be the region enclosed by the parabola

y^2=2x

and the line

x=24

. Then the maximum area of the rectangle inscribed in the region A is _______ .

Q27·MathematicsSingle correctJEE Main 2024

The number of critical points of the function

f(x)=(x-2)^{2/3}(2x+1)

is:

(A) $2$
(B) $0$
(C) $1$
(D) $3$

Q28·MathematicsSingle correctJEE Main 2024

If the function

f(x) = \left(\frac{1}{x}\right)^{2x}

;

x > 0

attains the maximum value at

x = \frac{1}{e}

then:

(A) $e^{\pi} < \pi^{e}$
(B) $e^{2\pi} < (2\pi)^{e}$
(C) $e^{\pi} > \pi^{e}$
(D) $(2e)^{\pi} > \pi^{(2e)}$

Q29·MathematicsSingle correctJEE Main 2024

The interval in which the function

f(x)=x^{x}

,

x>0

, is strictly increasing is

(A) $\left(0,\tfrac{1}{e}\right]$
(B) $\left[\tfrac{1}{e^{2}},1\right)$
(C) $(0,\infty)$
(D) $\left[\tfrac{1}{e},\infty\right)$

Q30·MathematicsSingle correctJEE Main 2024

For the function

f(x) = \sin x + 3x - \dfrac{2}{\pi}(x^2 + x)

, where

x \in \left[0, \dfrac{\pi}{2}\right]

, consider the following two statements: (I)

f

is increasing in

\left(0, \dfrac{\pi}{2}\right)

. (II)

f'

is decreasing in

\left(0, \dfrac{\pi}{2}\right)

. Between the above two statements,

(A)only (I) is true.
(B)only (II) is true.
(C)neither (I) nor (II) is true.
(D)both (I) and (II) are true.

Q31·MathematicsSingle correctJEE Main 2024

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) $72$
(B) $60$
(C) $80$
(D) $64$

Q32·MathematicsSingle correctJEE Main 2024

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)44
(B)42
(C)24
(D)38

Q33·MathematicsSingle correctJEE Main 2024

Let the sum of the maximum and the minimum values of the function

f(x)=\dfrac{2x^2-3x+8}{2x^2+3x+8}

be

\dfrac{m}{n}

, where

\gcd(m,n)=1

. Then

m+n

is equal to:

(A) $182$
(B) $217$
(C) $195$
(D) $201$

Q34·MathematicsSingle correctJEE Main 2024

If

5f(x)+4f\left(\dfrac1x\right)=x^2-2

,

\forall x\ne0

and

y=9x^2 f(x)

, then

y

is strictly increasing in:

(A) $\left(0,\dfrac{1}{\sqrt5}\right)$
(B) $\left(-\dfrac{1}{\sqrt5},0\right)\cup\left(\dfrac{1}{\sqrt5},\infty\right)$
(C) $\left(-\dfrac{1}{\sqrt5},0\right)\cup\left(0,\dfrac{1}{\sqrt5}\right)$
(D) $\left(-\infty,-\dfrac{1}{\sqrt5}\right)\cup\left(0,\dfrac{1}{\sqrt5}\right)$

Q35·MathematicsNumericalJEE Main 2024

Let

S=(-1,\infty)

and

f:S\to R

be defined as

f(x)=\displaystyle\int_{-1}^x \left(e^t-1\right)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61}dt

. Let p = Sum of square of the values of x, where

f(x)

attains local maxima on S, and q = Sum of the values of x, where

f(x)

attains local minima on S. Then, the value of

p^2+2q

is ______.

Q36·MathematicsSingle correctJEE Main 2024

The maximum area of a triangle whose one vertex is at

(0,0)

and the other two vertices are on the curve

y=-2x^2+54

at points

(x,y)

and

(-x,y)

where

y>0

is:

(A) $88$
(B) $122$
(C) $92$
(D) $108$

Q37·MathematicsSingle correctJEE Main 2024

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) $600$
(B) $392$
(C) $608$
(D) $108$

Q38·MathematicsSingle correctJEE Main 2024

Let

g(x)=3f\left(\dfrac{x}{3}\right)+f(3-x)

and

f''(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)24
(B)0
(C)18
(D)20

Q39·MathematicsMultiple correctJEE Advanced 2023

Let

S

be the set of all twice differentiable functions f from

\mathbb{R}

to

\mathbb{R}

such that

\frac{d^{2}f}{dx^{2}}(x) > 0

for all

x \in (-1, 1)

. For

f \in S

, let

X_{f}

be the number of points

x \in (-1, 1)

for which

f(x) = x

. Then which of the following statements is(are) true?

(A)There exists a function $f \in S$ such that $X_{f} = 0$
(B)For every function $f \in S$ , we have $X_{f} \le 2$
(C)There exists a function $f \in S$ such that $X_{f} = 2$
(D)There does NOT exist any function $f$ in $S$ such that $X_{f} = 1$

Q40·MathematicsSingle correctJEE Main 2023

\displaystyle\max_{0\le x\le\pi/2}\left\{x-2\sin x\cos x+\dfrac13\sin 3x\right\}=

(A) $\dfrac{5\pi+2+3\sqrt3}{6}$
(B) $\dfrac{5\pi+2-3\sqrt3}{6}$
(C) $\pi$
(D) $0$

Q41·MathematicsSingle correctJEE Main 2023

If the local maximum value of the function

f(x)=\left(\dfrac{\sqrt{3e}}{2\sin x}\right)^{\sin^2 x}

,

x\in\left(0,\dfrac{\pi}{2}\right)

, is

\dfrac{k}{e}

, then

\left(\dfrac{k}{e}\right)^8+\dfrac{k^8}{e^5}+k^5

is equal to

(A) $e^5+e^6+e^{11}$
(B) $e^3+e^6+e^{11}$
(C) $e^3+e^5+e^{11}$
(D) $e^5+e^6+e^{10}$

Q42·MathematicsSingle correctJEE Main 2023

Let

a

,

b

,

c

and

d

be positive real numbers such that

a+b+c+d=11

. If the maximum value of

a^5 b^3 c^2 d

is

3750\beta

, then the value of

\beta

is

(A) $90$
(B) $110$
(C) $55$
(D) $108$

Q43·MathematicsSingle correctJEE Main 2023

Let

f : [2, 4] \to \mathbb{R}

be a differentiable function such that

(x \log_e x) f'(x) + (\log_e x) f(x) + f(x) \ge 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) \le 1

, for all

x \in [2, 4]

(B):

f(x) \ge \frac{1}{8}

, for all

x \in [2, 4]

Then,

(A)Only statement (B) is true
(B)Neither statement (A) nor statement (B) is true
(C)Both the statement (A) and (B) are true
(D)Only statement (A) is true

Q44·MathematicsNumericalJEE Main 2023

Let a quadratic curve passing through the point

(-1,0)

and touching the line

y=x

at

(1,1)

be

y=f(x)

. Then the x-intercept of the normal to the curve at the point

(\alpha,\alpha+1)

in the first quadrant is _______ .

Q45·MathematicsSingle correctJEE Main 2023

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) $675$
(B) $1025$
(C) $800$
(D) $900$

Q46·MathematicsSingle correctJEE Main 2023

Let

g(x)=f(x)+f(1-x)

and

f''(x)>0,\ x\in(0,1)

. If

g

is decreasing in the interval

(0,\alpha)

and increasing in the interval

(\alpha,1)

, then

\tan^{-1}(2x)+\tan^{-1}\left(\frac{1}{\alpha}\right)+\tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)

is equal to:

(A) $\frac{3\pi}{2}$
(B) $\pi$
(C) $\frac{5\pi}{4}$
(D) $\frac{3\pi}{4}$

Q47·MathematicsNumericalJEE Main 2023

The number of points, where the curve

y=x^5-20x^3+50x+2

crosses the x-axis, is

Q48·MathematicsNumericalJEE Main 2023

If

f(x)=x^2+g'(1)x+g''(2)

and

g(x)=f(1)x^2+xf'(x)+f''(x)

, then the value of

f(4)-g(4)

is equal to _______ .

Q49·MathematicsSingle correctJEE Main 2023

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

f(x)=|x^2-5x+6|-3x+2

in the interval

[-1,3]

is equal to:

(A) $12$
(B) $13$
(C) $10$
(D) $24$

Q50·MathematicsSingle correctJEE Main 2023

Let

f(x)=2x+\tan^{-1}x

and

g(x)=\log_e\!\left(\sqrt{1+x^2}+x\right)

,

x\in[0,3]

. Then

(A) $\min f'(x)=1+\max g'(x)$
(B) $\max f'(x)>\max g'(x)$
(C)there exist $0<x_1<x_2<3$ such that $f(x)<g(x),\ \forall x\in(x_1,x_2)$
(D)there exists $\hat{x}\in[0,3]$ such that $f'(\hat{x})<g'(\hat{x})$

Q51·MathematicsSingle correctJEE Main 2023

A wire of length 20 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

2A_1+3A_2

is minimum then

(\pi l_1):l_2

is equal to :

(A) $1:6$
(B) $6:1$
(C) $3:1$
(D) $4:1$

Q52·MathematicsSingle correctJEE Main 2023

The absolute minimum value, of the function

f(x)=|x^2-x+1|+[x^2-x+1]

, where

[t]

denotes the greatest integer function, in the interval

[-1,2]

, is:

(A) $\dfrac14$
(B) $\dfrac32$
(C) $\dfrac54$
(D) $\dfrac34$

Q53·MathematicsNumericalJEE Main 2023

Let for

x \in \mathbb{R}

,

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

and

g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \ge 0 \end{cases}

. Then area bounded by the curve

y = (f \circ g)(x)

and the lines

y = 0,\ 2y - x = 15

is equal to

Q54·MathematicsSingle correctJEE Main 2023

If the functions

f(x)=\dfrac{x^3}{3} + 2bx + \dfrac{ax^2}{2}

and

g(x)=\dfrac{x^3}{3} + ax + bx^2

,

a \ne 2b

, have a common extreme point, then

a + 2b + 7

is equal to:

(A) $\dfrac{3}{2}$
(B) $3$
(C) $4$
(D) $6$

Q55·MathematicsSingle correctJEE Main 2023

The number of points on the curve

y=54x^5-135x^4-70x^3+180x^2+210x

at which the normal lines are parallel to

x+90y+2=0

is:

(A) $4$
(B) $2$
(C) $0$
(D) $3$

Q56·MathematicsNumericalJEE Main 2023

If the equation of the normal to the curve

y=\dfrac{x-a}{(x+b)(x-2)}

at the point

(1,-3)

is

x-4y=13

, then the value of

a+b

is equal to _____.

Q57·MathematicsSingle correctJEE Main 2023

Let

f

and

g

be twice differentiable functions on

\mathbb{R}

such that

f\,'\,'(x)=g\,'\,'(x)+6x,\,f\,'(1)=4g\,'(1)-3=9,\,f(2)=3g(2)=12

. Then which of the following is NOT true?

(A)There exists $x_0\in(1,3/2)$ such that $f(x_0)=g(x_0)$
(B) $|f\,'(x)-g\,'(x)|<6\Rightarrow -1<x<1$
(C)If $-1<x<2$ , then $|f(x)-g(x)|<8$
(D) $g(-2)-f(-2)=20$

Q58·MathematicsSingle correctJEE Main 2023

Let the function

f(x) = 2x^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) $\left(0, \dfrac{9}{2}\right)$
(B) $\left(-\infty, \dfrac{9}{2}\right)$
(C) $\left(-\dfrac{9}{2}, \dfrac{9}{2}\right)$
(D) $\left(\dfrac{9}{2}, \infty\right)$

Q59·MathematicsSingle correctJEE Main 2023

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 the local maximum value of the function

f

in

(-4,4)

, then

M=

(A) $18\sqrt{6}-\dfrac{31}{2}$
(B) $18\sqrt{6}+\dfrac{33}{2}$
(C) $12\sqrt{6}-\dfrac{31}{2}$
(D) $12\sqrt{6}-\dfrac{33}{2}$

Q60·MathematicsSingle correctJEE Main 2023

Let

f:(0,1)\to\mathbb{R}

be a function defined by

f(x)=\dfrac{1}{1-e^{-x}}

, and

g(x)=(f(-x)-f(x))

. 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 (I) is true
(D)Only (II) is true

Q61·MathematicsMultiple correctJEE Advanced 2022

Let α =

\sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)

. Let g : [0, 1] → R be the function defined by g(x) = 2

^{αx}

+ 2

^{α(1−x)}

. Then, which of the following statements is/are TRUE?

(A)The minimum value of g(x) is $2^{\frac{7}{6}}$
(B)The maximum value of g(x) is 1 + $2^{\frac{1}{3}}$
(C)The function g(x) attains its maximum at more than one point
(D)The function g(x) attains its minimum at more than one point

Q62·MathematicsNumericalJEE Main 2022

If the tangent to the curve

y = x^3 - x^2 + x

at the point

(a, b)

is also tangent to the curve

y = 5x^2 + 2x - 25

at the point

(2, -1)

, then

|2a + 9b|

is equal to ______.

Q63·MathematicsSingle correctJEE Main 2022

Let

f(x) = 3^{\left(x^{2}-2\right)^{3}+4}

,

x \in \mathbf{R}

. Then which of the following statements are true ? P :

x = 0

is a point of local minima of f Q :

x = \sqrt{2}

is a point of inflection of f R : f' 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

Q64·MathematicsSingle correctJEE Main 2022

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

f(x)=\tan^{-1}(\sin x-\cos x)

in the interval

[0,\pi]

is

(A)0
(B) $\tan^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi}{4}$
(C) $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi}{4}$
(D) $\frac{-\pi}{12}$

Q65·MathematicsSingle correctJEE Main 2022

If the minimum value of

f(x) = \frac{5x^{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

Q66·MathematicsSingle correctJEE Main 2022

The function

f(x)=xe^{x(1-x)}

,

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

Q67·MathematicsNumericalJEE Main 2022

Let

f : [0, 1] \to R

be a twice differentiable function in (0, 1) such that f(0) = 3 and f(1) = 5. If the line y = 2x + 3 intersects the graph of f at only two distinct points in (0, 1), then the least number of points

x \in (0, 1)

, at which

f''(x) = 0

, is _______.

Q68·MathematicsNumericalJEE Main 2022

Let

M

and

N

be the number of points on the curve

y^{5}-9xy+2x=0

, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals ________.

Q69·MathematicsNumericalJEE Main 2022

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is

\tan^{-1}\frac{3}{4}

. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ________.

Q70·MathematicsSingle correctJEE Main 2022

Let P and Q be any points on the curves

(x-1)^{2}+(y+1)^{2}=1

and

y = x^{2}

, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

(A) $\left(0,\frac{1}{4}\right)$
(B) $\left(\frac{1}{2},\frac{3}{4}\right)$
(C) $\left(\frac{1}{4},\frac{1}{2}\right)$
(D) $\left(\frac{3}{4},1\right)$

Q71·MathematicsSingle correctJEE Main 2022

Let

f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \le 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}

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

Q72·MathematicsSingle correctJEE Main 2022

If the maximum value of a, for which the function

f_{a}(x)=\tan^{-1}2x-3ax+7

is non-decreasing in

\left(-\frac{\pi}{6},\frac{\pi}{6}\right)

, is

\overline{a}

, then

f_{\overline{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) $8-\frac{\pi}{4}$

Q73·MathematicsSingle correctJEE Main 2022

The curve

y(x) = ax^{3} + bx^{2} + cx + 5

touches the x-axis at the point P

(-2, 0)

and cuts the y-axis at the point Q, where y' 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) $\frac{9}{2}$

Q74·MathematicsSingle correctJEE Main 2022

If the absolute maximum value of the function

f(x) = (x^{2} - 2x + 7)\, e^{(4x^{3} - 12x^{2} - 180x + 31)}

in the interval

[-3, 0]

is

f(\alpha)

, then :

(A) $\alpha = 0$
(B) $\alpha = -3$
(C) $\alpha \in (-1, 0)$
(D) $\alpha \in (-3, -1)$

Q75·MathematicsSingle correctJEE Main 2022

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) $\frac{22}{9 + 4\sqrt{3}}$
(B) $\frac{66}{9 + 4\sqrt{3}}$
(C) $\frac{22}{4 + 9\sqrt{3}}$
(D) $\frac{66}{4 + 9\sqrt{3}}$

Q76·MathematicsSingle correctJEE Main 2022

Let

f : \mathbb{R} \to \mathbb{R}

be a function defined by

f(x) = (x - 3)^{n_{1}}\,(x - 5)^{n_{2}}

,

n_{1}, n_{2} \in N

. The, which of the following is NOT true?

(A)For $n_{1} = 3$ , $n_{2} = 4$ , there exists $\alpha \in (3,5)$ where $f$ attains local maxima.
(B)For $n_{1} = 4$ , $n_{2} = 3$ , there exists $\alpha \in (3,5)$ where $f$ attains local manima.
(C)For $n_{1} = 3$ , $n_{2} = 5$ , there exists $\alpha \in (3,5)$ where $f$ attains local maxima.
(D)For $n_{1} = 4$ , $n_{2} = 6$ , there exists $\alpha \in (3,5)$ where $f$ attains local maxima.

Q77·MathematicsSingle correctJEE Main 2022

The number of real solutions of

x^{7}+5x^{3}+3x+1=0

is equal to ________.

(A)0
(B)1
(C)3
(D)5

Q78·MathematicsNumericalJEE Main 2022

Let

\ell

be a line which is normal to the curve

y = 2x^2 + x + 2

at a point P on the curve. If the point Q(6, 4) lies on the line

\ell

and O is origin, then the area of the triangle OPQ is equal to ______.

Q79·MathematicsSingle correctJEE Main 2022

If m and n respectively are the number of local maximum and local minimum points of the function

f(x)=\int_{0}^{x^{2}}\dfrac{t^{2}-5t+4}{2+e^{t}}dt

, then the ordered pair (m, n) is equal to

(A) $(3,\,2)$
(B) $(2,\,3)$
(C) $(2,\,2)$
(D) $(3,\,4)$

Q80·MathematicsSingle correctJEE Main 2022

The lengths of the sides of a triangle are

10 + x^{2}

,

10 + x^{2}

and

20 - 2x^{2}

. If for x = k, the area of the triangle is maximum, then

3k^{2}

is equal to :

(A)5
(B)8
(C)10
(D)12

Q81·MathematicsSingle correctJEE Main 2022

Let S be the set of all the natural numbers, for which the line

\frac{x}{a} + \frac{y}{b} = 2

is a tangent to the curve

\left(\frac{x}{a}\right)^{n} + \left(\frac{y}{b}\right)^{n} = 2

at the point (a, b),

ab \neq 0

. Then:

(A) $S = \phi$
(B) $n(S) = 1$
(C) $S = \{2k : k \in N\}$
(D) $S = N$

Q82·MathematicsSingle correctJEE Main 2022

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

Q83·MathematicsSingle correctJEE Main 2022

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

f(x) = |3x - x^{2} + 2| - x

in the interval

[-1, 2]

is :

(A) $\frac{\sqrt{17} + 3}{2}$
(B) $\frac{\sqrt{17} + 5}{2}$
(C)5
(D) $\frac{9 - \sqrt{17}}{2}$

Q84·MathematicsNumericalJEE Main 2022

Let

f(x) = \left|(x - 1)(x^{2} - 2x - 3)\right| + x - 3

,

x \in R

. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______

Q85·MathematicsSingle correctJEE Main 2022

Let

f : R \to R

and

g : R \to R

be two functions defined by

f(x) = \log_e(x^2 + 1) - e^{-x} + 1

and

g(x) = \frac{1 - 2e^{2x}}{e^x}

. Then, for which of the following range of α, the inequality

f\left(g\left(\frac{(\alpha - 1)^2}{3}\right)\right) > f\left(g\left(\alpha - \frac{5}{3}\right)\right)

holds?

(A) $(2, 3)$
(B) $(-2, -1)$
(C) $(1, 2)$
(D) $(-1, 1)$

Q86·MathematicsSingle correctJEE Main 2022

If the angle made by the tangent at the point

(x_0, y_0)

on the curve

x = 12(t + \sin t \cos t)

,

y = 12(1 + \sin t)^2, 0 < t < \frac{\pi}{2}

, with the positive x-axis is

\frac{\pi}{3}

, then

y_0

is equal to

(A) $6\left(3 + 2\sqrt{2}\right)$
(B) $3\left(7 + 4\sqrt{3}\right)$
(C)27
(D)48

Q87·MathematicsSingle correctJEE Main 2022

Water is being filled at the rate of

1\,\mathrm{cm}^3/\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

\mathrm{cm}^2/\sec

) at which the wet conical surface area of the vessel increases is

(A)5
(B) $\frac{\sqrt{21}}{5}$
(C) $\frac{\sqrt{26}}{5}$
(D) $\frac{\sqrt{26}}{10}$

Q88·MathematicsSingle correctJEE Main 2022

f(x)=4\log_e(x-1)-2x^{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 $(e,e+1)$

Q89·MathematicsSingle correctJEE Main 2022

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

f(x)=|2x^{2}+3x-2|+\sin x\cos x

in the interval

[0,1]

is :

(A)(A)
(B)(B)
(C)(C)
(D)(D)

Q90·MathematicsSingle correctJEE Main 2022

The number of distinct real roots of the equation

x^{7} - 7x - 2 = 0

is

(A)5
(B)7
(C)1
(D)3

Q91·MathematicsSingle correctJEE Main 2022

the tangent at the point

(x_{1},\ y_{1})

on the curve

y=x^{3}+3x^{2}+5

passes through the origin, then

(x_{1},y_{1})

does NOT lie on the curve :

(A) $x^{2}+\dfrac{y^{2}}{81}=2$
(B) $\dfrac{y^{2}}{9}-x^{2}=8$
(C) $y=4x^{2}+5$
(D) $\dfrac{x}{3}-y^{2}=2$

Q92·MathematicsSingle correctJEE Main 2022

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

Q93·MathematicsSingle correctJEE Main 2022

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

Q94·MathematicsSingle correctJEE Advanced 2021

Let

\psi_1 : [0, \infty) \to \mathbb{R}

,

\psi_2 : [0, \infty) \to \mathbb{R}

,

f : [0, \infty) \to \mathbb{R}

and

g : [0, \infty) \to \mathbb{R}

be functions such that

f(0) = g(0) = 0,

\psi_1(x) = e^{-x} + x, \quad x \ge 0,

\psi_2(x) = x^2 - 2x - 2e^{-x} + 2, \quad x \ge 0,

f(x) = \int_{-x}^{x} \left( |t| - t^2 \right) e^{-t^2} \, dt, \quad x > 0

and

g(x) = \int_{0}^{x^2} \sqrt{t} \; e^{-t} \, dt, \quad x > 0

Which of the following statements is TRUE ?

(A) $f\left(\sqrt{\ln 3}\right) + g\left(\sqrt{\ln 3}\right) = \frac{1}{3}$
(B)For every x > 1, there exists an α ∈ (1, x) such that $\psi_1(x) = 1 + \alpha x$
(C)For every x > 0, there exists a β ∈ (0, x) such that $\psi_2(x) = 2x(\psi_1(\beta) - 1)$
(D) $f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$

Q95·MathematicsNumericalJEE Advanced 2021

Let

f_1 : (0, \infty) \to \mathbb{R}

and

f_2 : (0, \infty) \to \mathbb{R}

be defined by

f_1(x) = \int_{0}^{x} \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0

and

f_2(x) = 98(x - 1)^{50} - 600(x - 1)^{49} + 2450, \quad x > 0,

where, for any positive integer n and real numbers

a_1, a_2, \ldots, a_n

,

\prod_{i=1}^{n} a_i

denotes the product of

a_1, a_2, \ldots, a_n

. Let

m_i

and

n_i

, respectively, denote the number of points of local minima and the number of points of local maxima of function

f_i

,

i = 1, 2

, in the interval

(0, \infty)

The value of

2m_1 + 3n_1 + m_1n_1

is _____.

Q96·MathematicsMultiple correctJEE Advanced 2021

Let

f : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}

. Then which of the following statements is (are) TRUE ?

(A) $f$ is decreasing in the interval $(-2, -1)$
(B) $f$ is increasing in the interval $(1, 2)$
(C) $f$ is onto
(D)Range of $f$ is $\left[-\frac{3}{2}, 2\right]$

Q97·MathematicsNumericalJEE Advanced 2021

Let

f_1 : (0, \infty) \to \mathbb{R}

and

f_2 : (0, \infty) \to \mathbb{R}

be defined by

f_1(x) = \int_{0}^{x} \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0

and

f_2(x) = 98(x - 1)^{50} - 600(x - 1)^{49} + 2450, \quad x > 0,

where, for any positive integer n and real numbers

a_1, a_2, \ldots, a_n

,

\prod_{i=1}^{n} a_i

denotes the product of

a_1, a_2, \ldots, a_n

. Let

m_i

and

n_i

, respectively, denote the number of points of local minima and the number of points of local maxima of function

f_i

,

i = 1, 2

, in the interval

(0, \infty)

The value of

6m_2 + 4n_2 + 8m_2n_2

is _____.

Q98·MathematicsSingle correctJEE Main 2021

The function

f(x) = x^{3} - 6x^{2} + ax + b

is such that f(2) = f(4) = 0. Consider two statements. (S1) there exists

x_{1}, x_{2} \in (2, 4)

,

x_{1} < x_{2}

, such that

f'(x_{1}) = -1

and

f'(x_{2}) = 0

. (S2) there exists

x_{3}, x_{4} \in (2, 4)

,

x_{3} < x_{4}

, such that f is decreasing in

(2, x_{4})

, increasing in

(x_{4}, 4)

and

2f'(x_{3}) = \sqrt{3}f\left(x_{4}\right)

. Then

(A)both (S1) and (S2) are true
(B)(S1) is false and (S2) is true
(C)both (S1) and (S2) are false
(D)(S1) is true and (S2) is false

Q99·MathematicsNumericalJEE Main 2021

If 'R' is the least value of 'a' such that the function

f(x) = x^{2} + ax + 1

is increasing on

[1, 2]

and 'S' is the greatest value of 'a' such that the function

f(x) = x^{2} + ax + 1

is decreasing on

[1, 2]

, then the value of

\left| R - S \right|

is ______.

Q100·MathematicsNumericalJEE Main 2021

Let f(x) be a cubic polynomial with f(1) = -10, f(-1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = -1. Then f(3) is equal to ______.

Q101·MathematicsSingle correctJEE Main 2021

An angle of intersection of the curves,

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

and

x^2 + y^2 = ab

,

a > b

, is :

(A) $\tan^{-1}\left(\frac{a + b}{\sqrt{ab}}\right)$
(B) $\tan^{-1}\left(\frac{a - b}{2\sqrt{ab}}\right)$
(C) $\tan^{-1}\left(\frac{a - b}{\sqrt{ab}}\right)$
(D) $\tan^{-1}\left(2\sqrt{ab}\right)$

Q102·MathematicsSingle correctJEE Main 2021

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) $\frac{5}{2+\sqrt{3}}$
(B) $\frac{10}{2+3\sqrt{3}}$
(C) $\frac{5}{3+\sqrt{3}}$
(D) $\frac{10}{3+2\sqrt{3}}$

Q103·MathematicsSingle correctJEE Main 2021

A box open from top is made from a rectangular sheet of dimension a × 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) $\frac{a + b - \sqrt{a^{2} + b^{2} - ab}}{12}$
(B) $\frac{a + b - \sqrt{a^{2} + b^{2} + ab}}{6}$
(C) $\frac{a + b - \sqrt{a^{2} + b^{2} - ab}}{6}$
(D) $\frac{a + b + \sqrt{a^{2} + b^{2} - ab}}{6}$

Q104·MathematicsSingle correctJEE Main 2021

Let M and m respectively be the maximum and minimum values of the function

f(x) = \tan^{-1}(\sin x + \cos x)

in

\left[0, \frac{\pi}{2}\right]

, Then the value of tan(M − m) is equal to:

(A) $2 + \sqrt{3}$
(B) $2 - \sqrt{3}$
(C) $3 + 2\sqrt{2}$
(D) $3 - 2\sqrt{2}$

Q105·MathematicsSingle correctJEE Main 2021

The local maximum value of the function

f(x) = \left(\frac{2}{x}\right)^{x^{2}}

,

x > 0

, is

(A) $\left(2\sqrt{e}\right)^{\frac{1}{e}}$
(B) $\left(\frac{4}{\sqrt{e}}\right)^{\frac{e}{4}}$
(C) $\left(e\right)^{\frac{2}{e}}$
(D) $1$

Q106·MathematicsNumericalJEE Main 2021

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then

\left(\frac{4}{\pi} + 1\right)k

is equal to ________.

Q107·MathematicsSingle correctJEE Main 2021

Let f : (a,b) → R be twice differentiable such that

f(x) = \int_a^x g(t) dt

for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then

g(x)g'(x) = 0

has at least :

(A)seven roots in (a, b)
(B)twelve roots in (a, b)
(C)five roots in (a, b)
(D)three roots in (a, b)

Q108·MathematicsNumericalJEE Main 2021

If a rectangle is inscribed in an equilateral triangle of side length

2\sqrt{2}

as shown in the figure, then the square of the largest area of such a rectangle is .........

Q109·MathematicsSingle correctJEE Main 2021

Let

f(x)=3\sin^{4}x+10\sin^{3}x+6\sin^{2}x-3

,

x \in \left[-\frac{\pi}{6},\frac{\pi}{2}\right]

. Then, f is :

(A)decreasing in $\left(0,\frac{\pi}{2}\right)$
(B)increasing in $\left(-\frac{\pi}{6},\frac{\pi}{2}\right)$
(C)decreasing in $\left(-\frac{\pi}{6},0\right)$
(D)increasing in $\left(-\frac{\pi}{6},0\right)$

Q110·MathematicsSingle correctJEE Main 2021

Let

f : R \rightarrow R

be defined as

f(x) = \begin{cases} -\dfrac{4}{3}x^3 + 2x^2 + 3x, & x > 0 \\ 3xe^{x}, & x \leq 0 \end{cases}

Then f is increasing function in the interval.

(A) $\left(-\dfrac{1}{2}, 2\right)$
(B) $\left(-1, \dfrac{3}{2}\right)$
(C) $(-3, -1)$
(D) $(0, 2)$

Q111·MathematicsSingle correctJEE Main 2021

Let 'a' be a real number such that the function

f(x) = ax^2 + 6x - 15

, x ∈ R is increasing in

\left(-\infty, \frac{3}{4}\right)

and decreasing in

\left(\frac{3}{4}, \infty\right)

. Then the function

g(x) = ax^2 - 6x + 15

, x ∈ R has a :

(A)local minimum at $x = \frac{3}{4}$
(B)local maximum at $x = \frac{3}{4}$
(C)local minimum at $x = -\frac{3}{4}$
(D)local maximum at $x = -\frac{3}{4}$

Q112·MathematicsNumericalJEE Main 2021

If the point on the curve

y^{2}=6x

, nearest to the point

\left(3,\frac{3}{2}\right)

is

(\alpha,\beta)

, then

2(\alpha+\beta)

is equal to………

Q113·MathematicsSingle correctJEE Main 2021

The sum of all the local minimum values of the twice differentiable function

f:R\to R

defined by

f(x)=x^{3}-3x^{2}-\frac{3f''(2)}{2}x+f''(1)

is :

(A) $-22$
(B) $5$
(C) $0$
(D) $-27$

Q114·MathematicsSingle correctJEE Main 2021

Let

A = \left[a_{ij}\right]

be a 3 x 3 matrix, where

a_{ij} = \begin{cases} 1 , & \text{if } i = j \\ -x , & \text{if } |i - j| = 1 \\ 2x+1 , & \text{otherwise} \end{cases}

Let a function

f : R \rightarrow R

be defined as

f(x) = \det(A)

. Then the sum of maximum and minimum values of f on R is equal to :

(A) $-\frac{88}{27}$
(B) $\frac{20}{27}$
(C) $\frac{88}{27}$
(D) $-\frac{20}{27}$

Q115·MathematicsNumericalJEE Main 2021

Let P(x) be a real polynomial of degree 3 which vanishes at x = −3. Let P(x) have local minima at x = 1, local maxima at x = −1 and

\int_{-1}^{1} P(x)\,dx = 18

, then the sum of all the coefficients of the polynomial P(x) is equal to ________ .

Q116·MathematicsSingle correctJEE Main 2021

Consider the function

f : R \rightarrow R

defined by

f(x) = \begin{cases} \left(2-\sin\left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}

. Then

f

is :

(A)monotonic on $(-\infty, 0) \cup (0, \infty)$
(B)not monotonic on $(-\infty, 0)$ and $(0, \infty)$
(C)monotonic on $(0, \infty)$ only
(D)monotonic on $(-\infty, 0)$ only

Q117·MathematicsNumericalJEE Main 2021

The maximum value of z in the following equation

z = 6xy + y^2

, where

3x + 4y \leq 100

and

4x + 3y \leq 75

for

x \geq 0

and

y \geq 0

is _______ .

Q118·MathematicsNumericalJEE Main 2021

Let

f : [-1, 1] \rightarrow R

be defined as

f(x) = ax^{2}+bx+c

for all

x \in [-1, 1]

, where

a, b, c \in R

such that

f(-1) = 2

,

f'(-1) = 1

and for

x \in (-1, 1)

the maximum value of

f''(x)

is

\frac{1}{2}

. If

f(x) \le \alpha

,

x \in [-1, 1]

, then the least value of

\alpha

is equal to ______.

Q119·MathematicsSingle correctJEE Main 2021

Let f be a real valued function, defined on

R - \{-1, 1\}

and given by

f(x) = 3\log_{e}\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}.

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

(A) $(-\infty,-1)\cup\left(\left[\frac{1}{2},\infty\right)-\{1\}\right)$
(B) $(-\infty, \infty)-\{-1, 1\}$
(C) $\left(-1,\frac{1}{2}\right]$
(D) $\left(-\infty,\frac{1}{2}\right]-\{-1\}$

Q120·MathematicsNumericalJEE Main 2021

If the normal to the curve

y(x) = \int_{0}^{x}(2t^{2} - 15t + 10)dt

at a point (a,b) is parallel to the line x + 3y = −5, a > 1, then the value of |a + 6b| is equal to ________ .

Q121·MathematicsSingle correctJEE Main 2021

The range of a ∈ ℝ for which the function

f(x) = (4a - 3)(x + \log_{e} 5) + 2(a - 7)\cot\left(\frac{x}{2}\right)\sin^{2}\left(\frac{x}{2}\right)

, x ≠ 2nπ, n ∈ ℕ, has critical points, is :

(A)(−3, 1)
(B) $\left[-\frac{4}{3}, 2\right]$
(C)[1, ∞)
(D)(−∞, −1]

Q122·MathematicsSingle correctJEE Main 2021

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is:

(A)A right angle triangle having two of its sides of length 2r and r.
(B)An equilateral triangle of height $\frac{2r}{3}$ .
(C)An isosceles triangle with base equal to 2r.
(D)An equilateral triangle having each of its side of length $\sqrt{3}$ r.

Q123·MathematicsSingle correctJEE Main 2021

The maximum slope of the curve

y = \frac{1}{2}x^{4} - 5x^{3} + 18x^{2} - 19x

occurs at the point:

(A)(2, 9)
(B)(2,2)
(C) $\left( 3, \frac{21}{2} \right)$
(D)(0, 0)

Q124·MathematicsNumericalJEE Main 2021

Let a be an integer such that all the real roots of the polynomial

2x^{5}+5x^{4}+10x^{3}+10x^{2}+10x+10

lie in the interval (a, a + 1). Then,

|a|

is equal to________________.

Q125·MathematicsSingle correctJEE Main 2021

The minimum value of f(x) = a

^{a}^{x}

+a

^{1–a}^{x}

, where a, x ∈ R and a > 0, is equal to:

(A)a + 1a
(B)a + 1
(C)2a
(D)2 a

Q126·MathematicsSingle correctJEE Main 2021

If the curves,

\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1

and

\frac{x^{2}}{c} + \frac{y^{2}}{d} = 1

intersect each other at an angle of

90^{\circ}

, then which of the following relations is true ?

(A) $a + b = c + d$
(B) $a - b = c - d$
(C) $ab = \frac{c+d}{a+b}$
(D) $a - c = b + d$

Q127·MathematicsSingle correctJEE Main 2021

If Rolle's theorem holds for the function

f(x) = x^{3} - ax^{2} + bx - 4

,

x \in [1, 2]

with

f'\left(\frac{4}{3}\right) = 0

, then ordered pair (a, b) is equal to :

(A) $(-5, 8)$
(B) $(5, 8)$
(C) $(5, -8)$
(D) $(-5, -8)$

Q128·MathematicsNumericalJEE Main 2021

If the curves x = y

^{4}

and xy = k cut at right angles, then (4k)

^{6}

is equal to ______.

Q129·MathematicsNumericalJEE Main 2021

Let

f(x)

be a polynomial of degree 6 in

x

, in which the coefficient of

x^6

is unity and it has extrema at

x = -1

and

x = 1

. If

\lim\limits_{x \to 0} \dfrac{f(x)}{x^3} = 1

, then

5 \cdot f(2)

is equal to __________

Q130·MathematicsSingle correctJEE Main 2021

Let

f:\mathbf{R}\to\mathbf{R}

be defined as

f(x)=\begin{cases} -55x, & \text{if } x<-5 \\ 2x^{3}-3x^{2}-120x, & \text{if } -5\le x\le 4 \\ 2x^{3}-3x^{2}-36x-336, & \text{if } x>4 \end{cases}

Let

A=\{x\in R: f\ \text{is increasing}\}

. Then A is equal to :

(A) $\left(-5,-4\right)\cup\left(4,\infty\right)$
(B) $\left(-5,\infty\right)$
(C) $\left(-\infty,-5\right)\cup\left(4,\infty\right)$
(D) $\left(-\infty,-5\right)\cup\left(-4,\infty\right)$

Q131·MathematicsNumericalJEE Main 2021

The minimum value of

\alpha

for which the equation

\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha

has at least one solution in

\left(0, \frac{\pi}{2}\right)

is ______

Q132·MathematicsSingle correctJEE Main 2021

If the tangent to the curve

y = x^3

at the point

P(t, t^3)

meets the curve again at

Q

, then the ordinate of the point which divides

PQ

internally in the ratio

1 : 2

is :

(A) $-2t^3$
(B) $-t^3$
(C) $0$
(D) $2t^3$

Q133·MathematicsSingle correctJEE Main 2021

If P is a point on the parabola

y=x^{2}+4

which is closest to the straight line

y=4x-1,

then the co-ordinates of P are :

(A) $(-2, 8)$
(B) $(1, 5)$
(C) $(3, 13)$
(D) $(2, 8)$

Q134·MathematicsSingle correctJEE Main 2021

The function

f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x

:

(A)increases in $\left[\frac{1}{2}, \infty\right)$
(B)decreases $\left(-\infty, \frac{1}{2}\right]$
(C)increases in $\left(-\infty, \frac{1}{2}\right]$
(D)decreases $\left[\frac{1}{2}, \infty\right)$

Q135·MathematicsSingle correctJEE Main 2021

If the curve

y=ax^{2}+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=0
(B)a = $-1$ , b=1, c =1
(C)a =1, b=0, c =1
(D) $a=\frac{1}{2}, b=\frac{1}{2}, c=1$

Q136·MathematicsNumericalJEE Advanced 2020

For a polynomial g(x) with real coefficient, let

m_{g}

denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by

S = \left\{\left(x^{2} - 1\right)^{2}\left(a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}\right) : a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right\}

. For a polynomial f, let

f'

and

f''

denote its first and second order derivatives, respectively. Then the minimum possible value of

\left(m_{f'} + m_{f''}\right)

, where

f \in S

, is ______

Q137·MathematicsSingle correctJEE Advanced 2020

Consider all rectangles lying in the region

\left\{(x, y) \in \mathbb{R} \times \mathbb{R} : 0 \leq x \leq \frac{\pi}{2} \text{ and } 0 \leq y \leq 2\sin(2x)\right\}

and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

(A) $\frac{3\pi}{2}$
(B) $\pi$
(C) $\frac{\pi}{2\sqrt{3}}$
(D) $\frac{\pi\sqrt{3}}{2}$

Q138·MathematicsNumericalJEE Advanced 2020

Let the function

f : (0,\pi) \to \mathbb{R}

be defined by

f(\theta) = (\sin\theta + \cos\theta)^{2} + (\sin\theta - \cos\theta)^{4}

Suppose the function

f

has a local minimum at

\theta

precisely when

\theta \in \{\lambda_{1}\pi, \ldots, \lambda_{r}\pi\}

, where

0 < \lambda_{1} < \cdots < \lambda_{r} < 1

. Then the value of

\lambda_{1} + \cdots + \lambda_{r}

is ________

Q139·MathematicsSingle correctJEE Main 2020

For all twice differentiable functions

f : R \to R

, with

f(0) = f(1) = f'(0) = 0

,

(A) $f''(x) \neq 0$ , 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 $x \in (0, 1)$

Q140·MathematicsSingle correctJEE Main 2020

The set of all real values of

\lambda

for which the function

f(x) = (1 - \cos^{2} x)\cdot(\lambda + \sin x)

,

x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

, has exactly one maxima and exactly one minima, is

(A) $\left(-\frac{1}{2}, \frac{1}{2}\right) - \{0\}$
(B) $\left(-\frac{3}{2}, \frac{3}{2}\right)$
(C) $\left(-\frac{1}{2}, \frac{1}{2}\right)$
(D) $\left(-\frac{3}{2}, \frac{3}{2}\right) - \{0\}$

Q141·MathematicsSingle correctJEE Main 2020

The position of a moving car at time t is given by

f(t) = at^2+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

[t_1, t_2]

is attained at the point:

(A) $\dfrac{(t_2-t_1)}{2}$
(B) $a(t_2-t_1)+b$
(C) $\dfrac{(t_1+t_2)}{2}$
(D) $2a(t_1+t_2)+b$

Q142·MathematicsNumericalJEE Main 2020

Let AD and BC be two vertical poles at A and b respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that

MD^{2}+MC^{2}

is minimum is ____.

Q143·MathematicsSingle correctJEE Main 2020

If the tangent to the curve,

y = f(x) = x\log_{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) $\frac{e-1}{e}$
(B) $\frac{1}{e-1}$
(C) $e^{\left(\frac{1}{e-1}\right)}$
(D) $e^{\left(\frac{1}{1-e}\right)}$

Q144·MathematicsSingle correctJEE Main 2020

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

x^4e^y+2\sqrt{y+1}=3

at the point

(1,0)

?

(A) $(2,2)$
(B) $(-2,6)$
(C) $(-2,4)$
(D) $(2,6)$

Q145·MathematicsSingle correctJEE Main 2020

If the point P on the curve,

4x^{2}+5y^{2}=20

is farthest from the point Q(0, –4) then

PQ^{2}

is equal to:

(A)21
(B)48
(C)36
(D)29

Q146·MathematicsNumericalJEE Main 2020

If the lines x + y = a and x - y = b touch the curve

y = x^2 - 3x + 2

at the points where the curve intersects the x-axis, then

\dfrac{a}{b}

is equal to __________.

Q147·MathematicsSingle correctJEE Main 2020

Let

f

be a twice differentiable function on

(1, 6)

. If

f(2) = 8

,

f'(2) = 5

,

f'(x) \geq 1

and

f''(x) \geq 4

, for all

x \in (1, 6)

, then:

(A) $f(5) + f'(5) \leq 26$
(B) $f(5) \leq 10$
(C) $f(5) + f'(5) \geq 28$
(D) $f'(5) + f''(5) \leq 20$

Q148·MathematicsSingle correctJEE Main 2020

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 = x^{2} - 1

below the x-axis, is:

(A) $\frac{4}{3\sqrt{3}}$
(B) $\frac{1}{3\sqrt{3}}$
(C) $\frac{4}{3}$
(D) $\frac{2}{3\sqrt{3}}$

Q149·MathematicsSingle correctJEE Main 2020

If the surface area of a cube is increasing at a rate of 3.6

cm^2

/sec, retaining its shape; then the rate of change of its volume (in

cm^3

/sec), when the length of a side of the cube is 10 cm, is:

(A) $9$
(B) $18$
(C) $10$
(D) $20$

Q150·MathematicsNumericalJEE Main 2020

If the tangent to the curve,

y = e^x

at a point

\left( c, e^c \right)

and the normal to the parabola,

y^2 = 4x

at the point

(1, 2)

intersect at the same point on the x

-

axis, then the value of c is __________.

Q151·MathematicsSingle correctJEE Main 2020

The function,

f(x) = (3x - 7)x^{2/3}

,

x \in R

, is increasing for all x lying in:

(A) $\left(-\infty, \frac{14}{15}\right)$
(B) $(-\infty, 0) \cup \left(\frac{3}{7}, \infty\right)$
(C) $\left(-\infty, -\frac{14}{15}\right) \cup (0, \infty)$
(D) $(-\infty, 0) \cup \left(\frac{14}{15}, \infty\right)$

Q152·MathematicsSingle correctJEE Main 2020

Suppose

f(x)

is a polynomial of degree four, having critical points at

-1, 0, 1

. If

T = \left\{ x \in R \,\middle|\, f(x) = f(0) \right\}

, then the sum of squares of all the elements of T is:

(A)8
(B)6
(C)2
(D)4

Q153·MathematicsSingle correctJEE Main 2020

Let f be any function continuous on

[a,b]

and twice differentiable on

(a,b)

. If for all

x \in (a,b), f'(x)>0

and

f''(x)<0

, then for any

c \in (a,b)

,

\dfrac{f(c)-f(a)}{f(b)-f(c)}

is greater than:

(A) $\dfrac{c-a}{b-c}$
(B) $\dfrac{b+a}{b-a}$
(C) $\dfrac{b-c}{c-a}$
(D) $1$

Q154·MathematicsSingle correctJEE Main 2020

A spherical iron ball of 10 cm 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 (in cm/min) at which of the thickness of ice decreases, is:

(A) $\dfrac{5}{6\pi}$
(B) $\dfrac{1}{36\pi}$
(C) $\dfrac{1}{18\pi}$
(D) $\dfrac{1}{54\pi}$

Q155·MathematicsSingle correctJEE Main 2020

Let a function

f : [0, 5] \rightarrow R

be continuous,

f(1) = 3

and F be defined as:

F(x) = \int\limits_{1}^{x} t^{2} g(t)\,dt

, where

g(t) = \int\limits_{1}^{t} f(u)\,du

. Then for the function F, the point x = 1 is:

(A)a point of local minima
(B)a point of inflection.
(C)not a critical point
(D)a point of local maxima

Q156·MathematicsNumericalJEE Main 2020

Let the normal at a point P on the curve

y^{2}-3x^{2}+y+10=0

intersect the y-axis at

\left(0,\dfrac{3}{2}\right)

. If m is the slope of the tangent at P to the curve, then

|m|

is equal to

Q157·MathematicsSingle correctJEE Main 2020

The length of the perpendicular from the origin, 0n the normal to the curve,

x^{2}+2xy-3y^{2}=0

at the point (2, 2) is:

(A) $2\sqrt{2}$
(B) $4\sqrt{2}$
(C) $\sqrt{2}$
(D) $2$

Q158·MathematicsSingle correctJEE Main 2020

If c is a point at which Rolle's theorem holds for the function,

f(x)=\log_{e}\left(\dfrac{x^{2}+\alpha}{7x}\right)

in the interval

[3,4]

, where

\alpha\in R

, then

f''(c)

is equal to:

(A) $-\dfrac{1}{12}$
(B) $-\dfrac{1}{24}$
(C) $\dfrac{\sqrt{3}}{7}$
(D) $\dfrac{1}{12}$

Q159·MathematicsNumericalJEE Main 2020

Let

f(x)

be a polynomial of degree 3 such that

f(-1)=10, f(1)=-6

,

f(x)

has a critical point at

x=-1

and

f'(x)

has a critical point at

x=1

. Then

f(x)

has a local minima at x=_______.

Q160·MathematicsSingle correctJEE Main 2020

The value of c in the Lagrange's mean value theorem for the function

f(x)=x^{3}-4x^{2}+8x+11

, when

x\in[0,1]

is:

(A) $\dfrac{\sqrt{7}-2}{3}$
(B) $\dfrac{4-\sqrt{5}}{3}$
(C) $\dfrac{4-\sqrt{7}}{3}$
(D) $\dfrac{2}{3}$

Q161·MathematicsSingle correctJEE Main 2020

Let

f(x)

be a polynomial of degree 5 such that

x=\pm1

are its critical points. If

\displaystyle\lim_{x\to0}\left(2+\dfrac{f(x)}{x^{3}}\right)=4

, then which one of the following is not true?

(A)x = 1 is a point of minima and x = -1 is a point of maxims of f.
(B)x = 1 is a point of maxima and x = -1 is a point of minimum of f
(C)f is an odd function
(D) $f(1)-4f(-1)=4$

Q162·MathematicsSingle correctJEE Main 2020

Let the function,

f:[-7,0]\to R

be continuous on

[-7,0]

and differentiable on

(-7,0)

. If

f(-7)=-3

and

f'(x)\le 2

, for all

x\in(-7,0)

, then for all such functions f,

f(-1)+f(0)

lies in the interval:

(A) $[-3,11]$
(B) $(-\infty, 20]$
(C) $[-6,20]$
(D) $(-\infty,11]$

Q163·MathematicsMultiple correctJEE Advanced 2019

Let

f : \mathbb{R} \to \mathbb{R}

be given by

f(x) = (x - 1)(x - 2)(x - 5)

. Define

F(x) = \displaystyle\int_{0}^{x} f(t)\, dt

,

x > 0

Then which of the following options is/are correct?

(A) $F(x) \neq 0$ for all $x \in (0, 5)$
(B)F has a local minimum at $x = 1$
(C)F has a local maximum at $x = 2$
(D)F has two local maxima and one local minimum in $(0, \infty)$

Q164·MathematicsMultiple correctJEE Advanced 2019

Let

f(x) = \dfrac{\sin \pi x}{x^{2}}

,

x > 0

Let

x_1 < x_2 < x_3 < ... < x_n < ....

be all the points of local maximum of f and

y_1 < y_2 < y_3 < .... < y_n < ....

be all the points of local minimum of f. Then which of the following options is/are correct?

(A) $x_1 < y_1$
(B) $|x_n - y_n| > 1$ for every n
(C) $x_n \in \left(2n,\ 2n + \dfrac{1}{2}\right)$ for every n
(D) $x_{n+1} - x_n > 2$ for every n

Q165·MathematicsSingle correctJEE Main 2019

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
(B) $\frac{25}{3}$
(C) $25\sqrt{3}$
(D) $\frac{25}{\sqrt{3}}$

Q166·MathematicsSingle correctJEE Main 2019

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) $(5, 3\sqrt{6})$
(B) $(4, 3\sqrt{2})$
(C) $(3, 3\sqrt{3})$
(D) $(4, 3\sqrt{3})$

Q167·MathematicsSingle correctJEE Main 2019

The tangents to the curve

y = (x-2)^{2} -1

at its points of intersection with the line

x - y = 3

, intersect at the point :

(A) $\left(\dfrac{5}{3},1\right)$
(B) $\left(-\dfrac{5}{2},-1\right)$
(C) $\left(-\dfrac{5}{2},1\right)$
(D) $\left(\dfrac{5}{2},-1\right)$

Q168·MathematicsSingle correctJEE Main 2019

If the tangent to the curve

y = \dfrac{x}{x^{2} - 3}

,

x \in R, \left(x \neq \pm\sqrt{3}\right)

at a point (α, β) ≠ (0, 0) on it is parallel to the line 2x + 6y − 11 = 0 then

(A) $|2\alpha + 6\beta| = 11$
(B) $|2\alpha + 6\beta| = 19$
(C) $|6\alpha + 2\beta| = 19$
(D) $|6\alpha + 2\beta| = 9$

Q169·MathematicsSingle correctJEE Main 2019

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

cm^{3}

/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of ice decreases is

(A) $\dfrac{1}{36\pi}$
(B) $\dfrac{5}{6\pi}$
(C) $\dfrac{1}{9\pi}$
(D) $\dfrac{1}{18\pi}$

Q170·MathematicsSingle correctJEE Main 2019

Let f(x) =

e^{x}-x

and g(x) =

x^{2}-x

,

\forall

x

\in

R. Then the set of all x

\in

R, where the function h(x) = (fog) (x) is increasing is:

(A) $\left[0,\dfrac{1}{2}\right] \cup [1,\infty)$
(B) $\left[1,\dfrac{1}{2}\right] \cup \left[\dfrac{1}{2},\infty\right)$
(C) $\left[\dfrac{-1}{2},0\right] \cup [1,\infty)$
(D) $[0,\infty)$

Q171·MathematicsSingle correctJEE Main 2019

If f(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; f(x)=f(0)\}

contains exactly:

(A)four irrational numbers
(B)four rational numbers
(C)two irrational and one rational number
(D)two irrational and two rational numbers

Q172·MathematicsSingle correctJEE Main 2019

A water tank has the shape of an inverted right circular cone, whose semi vertical angle is

\tan^{-1}\left(\dfrac{1}{2}\right)

. Water is poured in at a constant rage of 5 cubic meter per minute. Then 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 10 m is:

(A) $\dfrac{2}{\pi}$
(B) $\dfrac{1}{5\pi}$
(C) $\dfrac{1}{10\pi}$
(D) $\dfrac{1}{15\pi}$

Q173·MathematicsSingle correctJEE Main 2019

Let S be the set of all values of x for which the tangent to the curve

y=f(x)=x^{3}-x^{2}-2x

at

(x, y)

is parallel to the line segment joining the points

(1, f(1))

and

(-1, f(-1))

, then S is equal to:

(A) $\left\{\dfrac{1}{3},-1\right\}$
(B) $\left\{-\dfrac{1}{3},-1\right\}$
(C) $\left\{\dfrac{1}{3},1\right\}$
(D) $\left\{-\dfrac{1}{3},1\right\}$

Q174·MathematicsSingle correctJEE Main 2019

If the tangent to the curve,

y=x^{3}+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) $(-2,2)$
(C) $(-2,1)$
(D) $(2,-1)$

Q175·MathematicsSingle correctJEE Main 2019

If

S_{1}

and

S_{2}

are respectively the sets of local minimum and local maximum points of the function.

f(x)=9x^{4}+12x^{3}-36x^{2}+25, x\in R

, then

(A) $S_{1}=\{-2,1\};S_{2}=\{0\}$
(B) $S_{1}=\{-2,0\};S_{2}=\{1\}$
(C) $S_{1}=\{-2\};S_{2}=\{0,1\}$
(D) $S_{1}=\{-1\};S_{2}=\{0,2\}$

Q176·MathematicsSingle correctJEE Main 2019

The shortest distance between the line

y=x

and the curve

y^{2}=x-2

is:

(A) $\frac{11}{4\sqrt{2}}$
(B) $2$
(C) $\frac{7}{4\sqrt{2}}$
(D) $\frac{7}{8}$

Q177·MathematicsSingle correctJEE Main 2019

Let

f:[0,2]\to R

be a twice differentiable function such that

f''(x)>0

, for all

x\in(0,2)

. If

\phi(x)=f(x)+f(2-x)

, then

\phi

is:

(A)increasing on (0, 2)
(B)decreasing on (0, 2)
(C)decreasing on (0, 1) and increasing on (1, 2)
(D)increasing on (0, 1) and decreasing on (1, 2)

Q178·MathematicsSingle correctJEE Main 2019

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

(A) $\sqrt{3}$
(B) $\sqrt{6}$
(C) $2\sqrt{3}$
(D) $\frac{2}{3}\sqrt{3}$

Q179·MathematicsSingle correctJEE Main 2019

If the function f given by

f(x) = x^{3} - 3(a-2)x^{2} + 3ax + 7

, for some

a \in R

is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation,

\frac{f(x) - 14}{(x-1)^{2}} = 0

(x \neq 1)

is :

(A)−7
(B)5
(C)7
(D)6

Q180·MathematicsSingle correctJEE Main 2019

The tangent to the curve

y = x^{2} - 5x + 5

, parallel to the line

2y = 4x + 1

, also passes through the point :

(A) $\left(\dfrac{7}{2}, \dfrac{1}{4}\right)$
(B) $\left(\dfrac{1}{8}, -7\right)$
(C) $\left(-\dfrac{1}{8}, 7\right)$
(D) $\left(\dfrac{1}{4}, \dfrac{7}{2}\right)$

Q181·MathematicsSingle correctJEE Main 2019

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola,

y = 12 - x^{2}

such that the rectangle lies inside the parabola, is:

(A)36
(B) $20\sqrt{2}$
(C)32
(D) $18\sqrt{3}$

Q182·MathematicsSingle correctJEE Main 2019

Let

f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}, r\in R f

, where a, b and d are non – zero real constant. Then:

(A)f is an increasing function of x
(B)f is a decreasing function of x
(C)f is not a continuous function of x
(D)f is neither increasing nor decreasing function of x

Q183·MathematicsSingle correctJEE Main 2019

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression

\dfrac{x^{m}y^{n}}{\left(1+x^{2m}\right)\left(1+y^{2n}\right)}

is:

(A)1
(B) $\dfrac{1}{2}$
(C) $\dfrac{1}{4}$
(D) $\dfrac{m+n}{6mn}$

Q184·MathematicsSingle correctJEE Main 2019

The maximum value of the function

f(x)=3x^{3}-18x^{2}+27x-40

on the set S =

\{x\in R : x^{2}+30\le 11x\}

is

(A)-122
(B)-222
(C)122
(D)222

Q185·MathematicsSingle correctJEE Main 2019

The shortest distance between the point

\left(\frac{3}{2},0\right)

and the curve

y=\sqrt{x},(x>0)

, is:

(A) $\frac{\sqrt{5}}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{3}{2}$
(D) $\frac{5}{4}$

Q186·MathematicsSingle correctJEE Main 2019

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

(A) $6\pi$
(B) $3\sqrt{3}\,\pi$
(C) $\dfrac{4}{3}\pi$
(D) $2\sqrt{3}\pi$

Q187·MathematicsSingle correctJEE Main 2019

If

\theta

denotes the acute angle between the curves,

y=10-x^{2}

and

y=2+x^{2}

at a point of their intersection, then

|\tan\theta|

is equal to

(A) $\dfrac{4}{9}$
(B) $\dfrac{8}{15}$
(C) $\dfrac{7}{17}$
(D) $\dfrac{8}{17}$

Q188·MathematicsMultiple correctJEE Advanced 2018

For every twice differentiable function

f : R \to [-2, 2]

with

(f(0))^{2} + (f'(0))^{2} = 85

, which of the following statement(s) is (are) TRUE ?

(A)There exist r, s $\in$ R, where $r < s$ , such that f is one-one on the open interval (r, s)
(B)There exists $x_{0} \in (-4, 0)$ such that $|f'(x_{0})| \le 1$
(C) $\lim_{x \to \infty} f(x) = 1$
(D)There exist $\alpha \in (-4, 4)$ such that $f(\alpha) + f''(\alpha) = 0$ and $f'(\alpha) \ne 0$

Q189·MathematicsSingle correctJEE Advanced 2017

Answer by appropriately matching the information given in the three columns of the following table. Let

f(x) = x + \log_{e} x - x\log_{e} x

,

x \in (0, \infty)

. • Column 1 contains information about zeros of

f(x)

,

f'(x)

and

f''(x)

. • Column 2 contains information about the limiting behavior of

f(x)

,

f'(x)

and

f''(x)

at infinity. • Column 3 contains information about increasing/decreasing nature of

f(x)

and

f'(x)

. Which of the following options is the only CORRECT combination ?

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f(x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x \to \infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f'(x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0, 1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^{2})$

(A)(III) (iii) (R)
(B)(I) (i) (P)
(C)(IV) (iv) (S)
(D)(II) (ii) (Q)

Q190·MathematicsMultiple correctJEE Advanced 2017

If

f : \mathbb{R} \to \mathbb{R}

is a differentiable function such that

f'(x) > 2f(x)

for all

x \in \mathbb{R}

, and

f(0) = 1

, then

(A) $f(x)$ is increasing in $(0, \infty)$
(B) $f(x)$ is decreasing in $(0, \infty)$
(C) $f(x) > e^{2x}$ in $(0, \infty)$
(D) $f'(x) < e^{2x}$ in $(0, \infty)$

Q191·MathematicsSingle correctJEE Advanced 2017

If

f : \mathbb{R} \to \mathbb{R}

is a twice differentiable function such that

f''(x) > 0

for all

x \in \mathbb{R}

, and

f\left(\dfrac{1}{2}\right) = \dfrac{1}{2}

,

f(1) = 1

, then

(A) $f'(1) \le 0$
(B) $0 < f'(1) \le \dfrac{1}{2}$
(C) $\dfrac{1}{2} < f'(1) \le 1$
(D) $f'(1) > 1$

Q192·MathematicsSingle correctJEE Advanced 2017

Answer by appropriately matching the information given in the three columns of the following table. Let

f(x) = x + \log_{e} x - x\log_{e} x

,

x \in (0, \infty)

. • Column 1 contains information about zeros of

f(x)

,

f'(x)

and

f''(x)

. • Column 2 contains information about the limiting behavior of

f(x)

,

f'(x)

and

f''(x)

at infinity. • Column 3 contains information about increasing/decreasing nature of

f(x)

and

f'(x)

. Which of the following options is the only INCORRECT combination ?

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f(x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x \to \infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f'(x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0, 1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^{2})$

(A)(II) (iii) (P)
(B)(II) (iv) (Q)
(C)(I) (iii) (P)
(D)(III) (i) (R)

Q193·MathematicsSingle correctJEE Advanced 2017

Answer by appropriately matching the information given in the three columns of the following table. Let

f(x) = x + \log_{e} x - x\log_{e} x

,

x \in (0, \infty)

. • Column 1 contains information about zeros of

f(x)

,

f'(x)

and

f''(x)

. • Column 2 contains information about the limiting behavior of

f(x)

,

f'(x)

and

f''(x)

at infinity. • Column 3 contains information about increasing/decreasing nature of

f(x)

and

f'(x)

. Which of the following options is the only CORRECT combination ?

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f(x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x \to \infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f'(x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0, 1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^{2})$

(A)(IV) (i) (S)
(B)(I) (ii) (R)
(C)(III) (iv) (P)
(D)(II) (iii) (S)

Q194·MathematicsMultiple correctJEE Advanced 2016

Let

f(x) = \lim_{n \to \infty} \left( \frac{n^{n} (x + n) \left( x + \frac{n}{2} \right) \ldots \left( x + \frac{n}{n} \right)}{n! \left( x^{2} + n^{2} \right) \left( x^{2} + \frac{n^{2}}{4} \right) \ldots \left( x^{2} + \frac{n^{2}}{n^{2}} \right)} \right)^{\frac{x}{n}}

, for all

x > 0

. Then

(A) $f\left( \frac{1}{2} \right) \geq f(1)$
(B) $f\left( \frac{1}{3} \right) \leq f\left( \frac{2}{3} \right)$
(C) $f'(2) \leq 0$
(D) $\frac{f'(3)}{f(3)} \geq \frac{f'(2)}{f(2)}$

Q195·MathematicsMultiple correctJEE Advanced 2016

Let

f : \mathbb{R} \to (0, \infty)

and

g : \mathbb{R} \to \mathbb{R}

be twice differentiable functions such that

f''

and

g''

are continuous functions on

\mathbb{R}

. Suppose

f'(2) = g(2) = 0

,

f''(2) \neq 0

and

g'(2) \neq 0

. If

\lim_{x \to 2} \frac{f(x) g(x)}{f'(x) g'(x)} = 1

, then

(A)f has a local minimum at $x = 2$
(B)f has a local maximum at $x = 2$
(C) $f''(2) > f(2)$
(D) $f(x) - f''(x) = 0$ for at least one $x \in \mathbb{R}$

Q196·MathematicsSingle correctJEE Advanced 2016

The least value of

\alpha \in \mathbb{R}

for which

4\alpha x^{2} + \frac{1}{x} \geq 1

, for all

x > 0

, is

(A) $\frac{1}{64}$
(B) $\frac{1}{32}$
(C) $\frac{1}{27}$
(D) $\frac{1}{25}$

Q197·MathematicsMultiple correctJEE Advanced 2015

Let

F : \mathbb{R} \to \mathbb{R}

be a thrice differentiable function. Suppose that

F(1) = 0

,

F(3) = -4

and

F'(x) < 0

for all

x \in (1/2, 3)

. Let

f(x) = xF(x)

for all

x \in \mathbb{R}

. The correct statement(s) is(are)

(A) $f'(1) < 0$
(B) $f(2) < 0$
(C) $f'(x) \ne 0$ for any $x \in (1, 3)$
(D) $f'(x) = 0$ for some $x \in (1, 3)$

Q198·MathematicsMultiple correctJEE Advanced 2015

Let

f

,

g : [-1, 2] \to \mathbb{R}

be continuous functions which are twice differentiable on the interval

(-1, 2)

. Let the values of f and g at the points

-1

, 0 and 2 be as given in the following table: In each of the intervals

(-1, 0)

and

(0, 2)

the function

(f - 3g)''

never vanishes. Then the correct statement(s) is(are)

	$x = -1$	$x = 0$	$x = 2$
$f(x)$	3	6	0
$g(x)$	0	1	$-1$

(A) $f'(x) - 3g'(x) = 0$ has exactly three solutions in $(-1, 0) \cup (0, 2)$
(B) $f'(x) - 3g'(x) = 0$ has exactly one solution in $(-1, 0)$
(C) $f'(x) - 3g'(x) = 0$ has exactly one solution in $(0, 2)$
(D) $f'(x) - 3g'(x) = 0$ has exactly two solutions in $(-1, 0)$ and exactly two solutions in $(0, 2)$

Q199·MathematicsIntegerJEE Advanced 2015

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm

^{3}

, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of

\dfrac{V}{250\pi}

is

Q200·MathematicsIntegerJEE Advanced 2014

The slope of the tangent to the curve

(y - x^5)^2 = x(1 + x^2)^2

at the point

(1, 3)

is __________

Q201·MathematicsSingle correctJEE Advanced 2013

A line

L : y = mx + 3

meets y-axis at

E(0, 3)

and the arc of the parabola

y^{2} = 16x

,

0 \leq y \leq 6

at the point

F(x_{0}, y_{0})

. The tangent to the parabola at

F(x_{0}, y_{0})

intersects the y-axis at

G(0, y_{1})

. The slope

m

of the line L is chosen such that the area of the triangle EFG has a local maximum. Match List I with List II and select the correct answer using the code given below the lists :

List-I	List-II
P. $m =$	1. $\frac{1}{2}$
Q.Maximum area of $\Delta EFG$ is	2. $4$
R. $y_{0} =$	3. $2$
S. $y_{1} =$	4. $1$

(A)P-4, Q-1, R-2, S-3
(B)P-3, Q-4, R-1, S-2
(C)P-1, Q-3, R-2, S-4
(D)P-1, Q-3, R-4, S-2

Q202·MathematicsSingle correctJEE Advanced 2013

Let

f : [0, 1] \to \mathbb{R}

(the set of all real numbers) be a function. Suppose the function

f

is twice differentiable,

f(0) = f(1) = 0

and satisfies

f''(x) - 2f'(x) + f(x) \geq e^{x}

,

x \in [0, 1]

. Which of the following is true for

0 < x < 1

?

(A) $0 < f(x) < \infty$
(B) $-\frac{1}{2} < f(x) < \frac{1}{2}$
(C) $-\frac{1}{4} < f(x) < 1$
(D) $-\infty < f(x) < 0$

Q203·MathematicsSingle correctJEE Advanced 2013

The number of points in

(-\infty,\infty)

, for which

x^2-x\sin x-\cos x=0

, is

(A) $6$
(B) $4$
(C) $2$
(D) $0$

Q204·MathematicsSingle correctJEE Advanced 2013

Let

f : [0, 1] \to \mathbb{R}

(the set of all real numbers) be a function. Suppose the function

f

is twice differentiable,

f(0) = f(1) = 0

and satisfies

f''(x) - 2f'(x) + f(x) \geq e^{x}

,

x \in [0, 1]

. If the function

e^{-x} f(x)

assumes its minimum in the interval

[0, 1]

at

x = \frac{1}{4}

, which of the following is true ?

(A) $f'(x) < f(x), \frac{1}{4} < x < \frac{3}{4}$
(B) $f'(x) > f(x), 0 < x < \frac{1}{4}$
(C) $f'(x) < f(x), 0 < x < \frac{1}{4}$
(D) $f'(x) < f(x), \frac{3}{4} < x < 1$

Q205·MathematicsMultiple correctJEE Advanced 2013

Let

f(x)=x\sin\pi x

,

x>0

. Then for all natural numbers

n

,

f'(x)

vanishes at

(A)a unique point in the interval $(n,n+\frac{1}{2})$
(B)a unique point in the interval $(n+\frac{1}{2},n+1)$
(C)a unique point in the interval $(n,n+1)$
(D)two points in the interval $(n,n+1)$

Q206·MathematicsMultiple correctJEE Advanced 2013

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio

8:15

is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

(A) $24$
(B) $32$
(C) $45$
(D) $60$

Q207·MathematicsMultiple correctJEE Advanced 2013

The function

f(x) = 2|x| + |x + 2| - \left||x + 2| - 2|x|\right|

has a local minimum or a local maximum at

x =

(A) $-2$
(B) $\frac{-2}{3}$
(C) $2$
(D) $\frac{2}{3}$

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