Question 1: Palette 1/29

JEE Advanced 2026 MCQ

Let $y : (-\infty, \infty) \to (0, \infty)$ be the solution of the differential equation

$\frac{dy}{dx} = \frac{e^{5x} y^3 + y^3}{e^x + e^x y^4},$

satisfying $y(0) = \frac{1}{\sqrt{2}}$. Then the value of $y(\log_e 2)$ is

A

$\sqrt{\frac{5 + \sqrt{35}}{2}}$

B

$\sqrt{\frac{7 + \sqrt{53}}{2}}$

C

$\frac{7 + \sqrt{53}}{2}$

D

$\frac{5 + \sqrt{35}}{2}$

Question 2: Palette 2/29

JEE Advanced 2026 MSQ

Let $y = f(x)$ be the real valued function defined on the interval $(0, \infty)$, satisfying $y(1) = 0$ and the differential equation

$ x \frac{dy}{dx} = y - x^3. $

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

A

The function $f$ has a local minimum at $x = \frac{1}{\sqrt{3}}$

B

The function $f$ has a local maximum at $x = \frac{1}{\sqrt{3}}$

C

The function $f$ is increasing in the interval $(1, 2)$

D

If $g(x) = 4x^3 - 5x^2 + \frac{3}{2}x$ for $x > 0$, then the number of elements in the set $ \{x \in (0, \infty) : f(x) = g(x) \} $
is $2$

Question 3: Palette 3/29

JEE Advanced 2024 MCQ

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and

$ \lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1 $

for each $x>0$. Then, for all $x>0, f(x)$ is equal to :

A

$\frac{31}{11 x}-\frac{9}{11} x^{10}$

B

$\frac{9}{11 x}+\frac{13}{11} x^{10}$

C

$\frac{-9}{11 x}+\frac{31}{11} x^{10}$

D

$\frac{13}{11 x}+\frac{9}{11} x^{10}$

Question 4: Palette 4/29

JEE Advanced 2023 MCQ

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int\limits_1^x f(t) d t=x f(x)-\frac{x^3}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is :

A

$\frac{e^2+4}{3}$

B

$\frac{\log _e 4+e}{3}$

C

$\frac{4 e^2}{3}$

D

$\frac{e^2-4}{3}$

Question 5: Palette 5/29

JEE Advanced 2022 MCQ

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation

$ \frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0 $

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

A

$y(x)$ is an increasing function

B

$y(x)$ is a decreasing function

C

There exists a real number $\beta$ such that the line $y=\beta \quad$ intersects the curve $y=y(x)$ at infinitely many points

D

$y(x)$ is a periodic function

Question 6: Palette 6/29

JEE Advanced 2019 MSQ

Let $\Gamma $ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at Y<sub>P</sub>. If PY<sub>P</sub> has length 1 for each point P on I`, then which of the following options is/are correct?

A

$xy' + \sqrt {1 - {x^2}} = 0$

B

$xy' - \sqrt {1 - {x^2}} = 0$

C

$y = {\log _e}\left( {{{1 + \sqrt {1 - {x^2}} } \over x}} \right) - \sqrt {1 - {x^2}} $

D

$y = - {\log _e}\left( {{{1 + \sqrt {1 - {x^2}} } \over x}} \right) + \sqrt {1 - {x^2}} $

Question 7: Palette 7/29

JEE Advanced 2017 MCQ

If y = y(x) satisfies the differential equation

${8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \right)}^{ - 1}}}$

dx, x > 0 and y(0) = $\sqrt 7 $, then y(256) =

A

16

B

3

C

9

D

80

Question 8: Palette 8/29

JEE Advanced 2017 MSQ

If $g(x) = \int_{\sin x}^{\sin (2x)} {{{\sin }^{ - 1}}} (t)\,dt$, then

A

$g'\left( { - {\pi \over 2}} \right) = 0$

B

$g'\left( { - {\pi \over 2}} \right) = - 2\pi $

C

$g'\left( {{\pi \over 2}} \right) = 2\pi $

D

$g'\left( {{\pi \over 2}} \right) = 0$

Question 9: Palette 9/29

JEE Advanced 2016 MSQ

A solution curve of the differential equation

$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$ $x>0,$ passes through the

point $(1,3)$. Then the solution curve

A

intersects $y=x+2$ exactly at one point

B

intersects $y=x+2$ exactly at two points

C

intersects $y = {\left( {x + 2} \right)^2}$

D

does NOT intersect $\,y = {\left( {x + 3} \right)^2}$

Question 10: Palette 10/29

JEE Advanced 2016 MCQ

Let $f:(0,\infty ) \to R$ be a differentiable function such that $f'(x) = 2 - {{f(x)} \over x}$ for all $x \in (0,\infty )$ and $f(1) \ne 1$. Then

A

$\mathop {\lim }\limits_{x \to {0^ + }} f'\left( {{1 \over x}} \right) = 1$

B

$\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = 2$

C

$\mathop {\lim }\limits_{x \to {0^ + }} {x^2}f'(x) = 0$

D

$\left| {f(x)} \right| \le 2$ for all $x \in (0,2)$

Question 11: Palette 11/29

JEE Advanced 2015 MSQ

Let $y(x)$ be a solution of the differential equation
$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$
If $y(0)=2$, then which of the following statement is (are) true?

A

$y(-4)=0$

B

$y(-2)=0$

C

$y(x)$ has a critical point in the interval $(-1, 0)$

D

$y(x)$ has no critical point in the interval $(-1,0)$

Question 12: Palette 12/29

JEE Advanced 2015 MSQ

Consider the family of all circles whose centres lie on the straight line $y=x,$ If this family of circle is represented by the differential equation $Py'' + Qy' + 1 = 0,$ where $P, Q$ are functions of $x,y$ and $y'$ $\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)$ then which of the following statements is (are) true?

A

$P = y + x$

B

$\,P = y - x$

C

$\,P + Q = 1 - x + y + y' + {\left( {y'} \right)^2}$

D

$\,P - Q = 1 - x + y - y' - {\left( {y'} \right)^2}$

Question 13: Palette 13/29

JEE Advanced 2014 MCQ

The function $y=f(x)$ is the solution of the differential equation
${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,$ in $(-1,1)$ satisfying $f(0)=0$.
Then $\int\limits_{ - {{\sqrt 3 } \over 2}}^{{{\sqrt 3 } \over 2}} {f\left( x \right)} \,d\left( x \right)$ is

A

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

B

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

C

${\pi \over 6} - {{\sqrt 3 } \over 4}$

D

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

Question 14: Palette 14/29

JEE Advanced 2013 MCQ

A curve passes through the point $\left( {1,{\pi \over 6}} \right)$. Let the slope of
the curve at each point $(x,y)$ be ${y \over x} + \sec \left( {{y \over x}} \right),x > 0.$
Then the equation of the curve is

A

$sin\left( {{y \over x}} \right) = \log x + {1 \over 2}$

B

$cos\,ec\left( {{y \over x}} \right) = \log x + 2$

C

$\,s\,ec\left( {{{2y} \over x}} \right) = \log x + 2\,$

D

$\,cos\left( {{{2y} \over x}} \right) = \log x + {1 \over 2}$

Question 15: Palette 15/29

JEE Advanced 2012 MSQ

If $y(x)$ satisfies the differential equation $y' - y\,tan\,x = 2x\,secx$ and $y(0)=0,$ then

A

$y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {8\sqrt 2 }}$

B

$y'\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {18}}$

C

$y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over 9}$

D

$y'\left( {{\pi \over 3}} \right) = {{4\pi } \over 3} + {{2{\pi ^2}} \over {3\sqrt 3 }}$

Question 16: Palette 16/29

JEE Advanced 2009 MCQ

Match the statements/expressions in Column I with the values given in Column II:

	Column I		Column II
(A)	The number of solutions of the equation $x{e^{\sin x}} - \cos x = 0$ in the interval $\left( {0,{\pi \over 2}} \right)$	(P)	1
(B)	Value(s) of $k$ for which the planes $kx + 4y + z = 0,4x + ky + 2z = 0$ and $2x + 2y + z = 0$ intersect in a straight line	(Q)	2
(C)	Value(s) of $k$ for which $|x - 1| + |x - 2| + |x + 1| + |x + 2| = 4k$ has integer solution(s)	(R)	3
(D)	If $y' = y + 1$ and $y(0) = 1$ then value(s) of $y(\ln 2)$	(S)	4
		(T)	5

A

(A)$\to$(P); (B)$\to$(Q), (S); (C)$\to$(Q), (R), (S), (T); (D)$\to$(R)

B

(A)$\to$(T); (B)$\to$(Q), (S); (C)$\to$(Q), (S), (T); (D)$\to$(Q)

C

(A)$\to$(S); (B)$\to$(Q), (S); (C)$\to$(P), (R), (S), (T); (D)$\to$(R)

D

(A)$\to$(P); (B)$\to$(Q), (S); (C)$\to$(Q), (R), (T); (D)$\to$(S)

Question 17: Palette 17/29

JEE Advanced 2009 MCQ

Match the statements/expressions in Column I with the open intervals in Column II :

	Column I		Column II
(A)	Interval contained in the domain of definition of non-zero solutions of the differential equation ${(x - 3)^2}y' + y = 0$	(P)	$\left( { - {\pi \over 2},{\pi \over 2}} \right)$
(B)	Interval containing the value of the integral $\int\limits_1^5 {(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)dx} $	(Q)	$\left( {0,{\pi \over 2}} \right)$
(C)	Interval in which at least one of the points of local maximum of ${\cos ^2}x + \sin x$ lies	(R)	$\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$
(D)	Interval in which ${\tan ^{ - 1}}(\sin x + \cos x)$ is increasing	(S)	$\left( {0,{\pi \over 8}} \right)$
		(T)	$( - \pi ,\pi )$

A

(A)$\to$(P), (Q), (S); (B)$\to$(P), (T), (S); (C)$\to$(P), (Q), (R), (T); (D)$\to$(S)

B

(A)$\to$(P), (Q), (S); (B)$\to$(P), (T), (R); (C)$\to$(P), (Q), (R), (T); (D)$\to$(R)

C

(A)$\to$(P), (Q), (S); (B)$\to$(P), (T), (S); (C)$\to$(S), (Q), (R), (T); (D)$\to$(S)

D

(A)$\to$(P), (T), (S); (B)$\to$(P), (T), (S); (C)$\to$(P), (Q), (R), (T); (D)$\to$(S)

Question 18: Palette 18/29

JEE Advanced 2008 MCQ

Let a solution $y=y(x)$ of the differential equation,

$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$ satify $y\left( 2 \right) = {2 \over {\sqrt 3 }}.$

STATEMENT-1 : $y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - {\pi \over 6}} \right)$ and

STATEMENT-2 : $y\left( x \right)$ given by ${1 \over y} = {{2\sqrt 3 } \over x} - \sqrt {1 - {1 \over {{x^2}}}} $

A

STATEMENT-1 is True, STATEMENT-2 is True;STATEMENT-2 is a correct explanation for STATEMENT-1

B

STATEMENT-1 is True, STATEMENT-2 is True;STATEMENT-2 is NOT a correct explanation for STATEMENT-1

C

STATEMENT-1 is True, STATEMENT-2 is False

D

STATEMENT-1 is False , STATEMENT-2 is True

Question 19: Palette 19/29

JEE Advanced 2007 MCQ

The differential equation $\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}$ determines a family of circles with :

A

variable radii and a fixed centre at $(0,1)$

B

variable radii and a fixed centre at $(0,-1)$

C

fixed radius 1 and variable centres along the $x$-axis

D

fixed radius 1 and variable centres along the $y$-axis

Question 20: Palette 20/29

JEE Advanced 2005 MCQ

The differential equation ${{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}$ determines a family of circles with

A

variable radii and a fixed centre at $(0,1)$

B

variable radii and a fixed centre at $(0,-1)$

C

fixed radius $1$ and variable centres along the $x$-axis.

D

fixed radius $1$ and variable centrs along the $y$-axis.

Question 21: Palette 21/29

JEE Advanced 2005 MCQ

For the primitive integral equation $ydx + {y^2}dy = x\,dy;$
$x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,$ then $y(-3)$ is

A

$3$

B

$2$

C

$1$

D

$5$

Question 22: Palette 22/29

JEE Advanced 2005 MCQ

The solution of primitive integral equation $\left( {{x^2} + {y^2}} \right)dy = xy$
$dx$ is $y=y(x),$ If $y(1)=1$ and $\left( {{x_0}} \right) = e$, then ${{x_0}}$ is equal to

A

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

B

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

C

$\sqrt 3 \,e$

D

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

Question 23: Palette 23/29

JEE Advanced 2005 MCQ

If $y=y(x)$ and it follows the relation $x\cos \,y + y\,cos\,x = \pi $ then $y''(0)=$

A

$1$

B

$-1$

C

${\pi}$

D

$ - \pi $

Question 24: Palette 24/29

JEE Advanced 2004 MCQ

If $y=y(x)$ and ${{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,$
then $y\left( {{\pi \over 2}} \right)$ equals

A

$1/3$

B

$2/3$

C

$-1/3$

D

$1$

Question 25: Palette 25/29

JEE Advanced 2003 MCQ

If $y(t)$ is a solution of $\left( {1 + t} \right){{dy} \over {dt}} - ty = 1$ and $y\left( 0 \right) = - 1,$ then $y(1)$ is equal to

A

$ - 1/2$

B

$e+1/2$

C

$e-1/2$

D

$ 1/2$

Question 26: Palette 26/29

JEE Advanced 2000 MCQ

If ${x^2} + {y^2} = 1,$ then

A

$yy'' - 2{\left( {y'} \right)^2} + 1 = 0$

B

$yy'' + {\left( {y'} \right)^2} + 1 = 0$

C

$yy'' + {\left( {y'} \right)^2} - 1 = 0$

D

$yy'' + 2{\left( {y'} \right)^2} + 1 = 0$

Question 27: Palette 27/29

JEE Advanced 1999 MCQ

A solution of the differential equation
${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$ is

A

$y=2$

B

$y=2x$

C

$y=2x-4$

D

$y = 2{x^2} - 4$

Question 28: Palette 28/29

JEE Advanced 1999 MSQ

The differential equation representing the family of curves
${y^2} = 2c\left( {x + \sqrt c } \right),$ where $c$ is a positive parameter, is of

A

order $1$

B

order $2$

C

degree $3$

D

degree $4$

Question 29: Palette 29/29

JEE Advanced 1998 MCQ

The order of the differential equation whose general solution is given by
$y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},$ where
${C_1},{C_2},{C_3},{C_4},{C_5},$ are arbitrary constants, is

A

$5$

B

$4$

C

$3$

D

$2$