Question 1: Palette 1/35

JEE Advanced 2025 MCQ

Let ℝ denote the set of all real numbers. Then the area of the region

$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x}, 5x - 4y - 1 > 0, 4x + 4y - 17 < 0 \right\} $

is

A

$\frac{17}{16} - \log_e{4}$

B

$\frac{33}{8} - \log_e{4}$

C

$\frac{57}{8} - \log_e{4}$

D

$\frac{17}{2} - \log_e{4}$

Question 2: Palette 2/35

JEE Advanced 2024 MCQ

Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.$ and $\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal to

A

$\frac{17}{2}$

B

$\frac{17}{3}$

C

$\frac{17}{4}$

D

$\frac{17}{5}$

Question 3: Palette 3/35

JEE Advanced 2021 MCQ

The area of the region

$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } } \right\}$ is

A

${{11} \over {32}}$

B

${{35} \over {96}}$

C

${{37} \over {96}}$

D

${{13} \over {32}}$

Question 4: Palette 4/35

JEE Advanced 2021 MSQ

For any real numbers $\alpha$ and $\beta$, let ${y_{\alpha ,\beta }}(x)$, x$\in$R, be the solution of the differential equation ${{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1$. Let $S = \{ {y_{\alpha ,\beta }}(x):\alpha ,\beta \in R\} $. Then which of the following functions belong(s) to the set S?

A

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

B

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

C

$f(x) = {{{e^x}} \over 2}\left( {x - {1 \over 2}} \right) + \left( {e - {{{e^2}} \over 4}} \right){e^{ - x}}$

D

$f(x) = {{{e^x}} \over 2}\left( {{1 \over 2} - x} \right) + \left( {e + {{{e^2}} \over 4}} \right){e^{ - x}}$

Question 5: Palette 5/35

JEE Advanced 2020 MCQ

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

f(x) = e^{x $-$ 1} $-$ e^{$-$|x $-$ 1|}

and g(x) = ${1 \over 2}$(e^{x $-$ 1} + e^{1 $-$ x}).

The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is

A

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

B

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

C

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

D

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

Question 6: Palette 6/35

JEE Advanced 2019 MCQ

The area of the region

{(x, y) : xy $ \le $ 8, 1 $ \le $ y $ \le $ x²} is

A

$8{\log _e}2 - {{14} \over 3}$

B

$8{\log _e}2 - {{7} \over 3}$

C

$16{\log _e}2 - {{14} \over 3}$

D

$16{\log _e}2 - 6$

Question 7: Palette 7/35

JEE Advanced 2018 MSQ

Let f : [0, $\infty $) $ \to $ R be a continuous function such that

$f(x) = 1 - 2x + \int_0^x {{e^{x - t}}f(t)dt} $ for all x $ \in $ [0, $\infty $). Then, which of the following statement(s) is (are) TRUE?

A

The curve y = f(x) passes through the point (1, 2)

B

The curve y = f(x) passes through the point (2, $-$1)

C

The area of the region $\{ (x,y) \in [0,1] \times R:f(x) \le y \le \sqrt {1 - {x^2}} \} $ is ${{\pi - 2} \over 4}$

D

The area of the region $\{ (x,y) \in [0,1] \times R:f(x) \le y \le \sqrt {1 - {x^2}} \} $ is ${{\pi - 1} \over 4}$

Question 8: Palette 8/35

JEE Advanced 2017 MSQ

If the line x = $\alpha $ divides the area of region R = {(x, y) $ \in $R² : x³ $ \le $ y $ \le $ x, 0 $ \le $ x $ \le $ 1} into two equal parts, then

A

2$\alpha $⁴ $-$ 4$\alpha $² + 1 =0

B

$\alpha $⁴ + 4$\alpha $² $-$ 1 =0

C

${1 \over 2} < \alpha < 1$

D

0 < $\alpha $ $ \le $ ${1 \over 2}$

Question 9: Palette 9/35

JEE Advanced 2016 MCQ

Area of the region

$\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} \right\}$

is equal to

A

${1 \over 6}$

B

${4 \over 3}$

C

${3 \over 2}$

D

${5 \over 3}$

Question 10: Palette 10/35

JEE Advanced 2015 MSQ

Let $F:R \to R$ be a thrice differentiable function. Suppose that
$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$ and $F\left( x \right) < 0$ for all $x \in \left( {{1 \over 2},3} \right).$ Let $f\left( x \right) = xF\left( x \right)$ for all $x \in R.$

If $\int_1^3 {{x^2}F'\left( x \right)dx = - 12} $ and $\int_1^3 {{x^3}F''\left( x \right)dx = 40,} $ then the correct expression(s) is (are)

A

$9f'\left( 3 \right) + f'\left( 1 \right) - 32 = 0$

B

$\int_1^3 {f\left( x \right)dx = 12} $

C

$9f'\left( 3 \right) - f'\left( 1 \right) + 32 = 0$

D

$\int_1^3 {f\left( x \right)dx = -12} $

Question 11: Palette 11/35

JEE Advanced 2013 MCQ

The area enclosed by the curves $y = \sin x + {\mathop{\rm cosx}\nolimits} $ and $y = \left| {\cos x - \sin x} \right|$ over the interval $\left[ {0,{\pi \over 2}} \right]$ is

A

$4\left( {\sqrt 2 - 1} \right)$

B

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

C

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

D

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

Question 12: Palette 12/35

JEE Advanced 2012 MSQ

Let $S$ be the area of the region enclosed by $y = {e^{ - {x^2}}}$, $y=0$, $x=0$, and $x=1$; then

A

$S \ge {1 \over e}$

B

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

C

$S \le {1 \over 4}\left( {1 + {1 \over {\sqrt e }}} \right)$

D

$S \le {1 \over {\sqrt 2 }} + {1 \over {\sqrt e }}\left( {1 - {1 \over {\sqrt 2 }}} \right)$

Question 13: Palette 13/35

JEE Advanced 2011 MCQ

Let the straight line $x=b$ divide the area enclosed by
$y = {\left( {1 - x} \right)^2},y = 0,$ and $x=0$ into two parts ${R_1}\left( {0 \le x \le b} \right)$ and
${R_2}\left( {b \le x \le 1} \right)$ such that ${R_1} - {R_2} = {1 \over 4}.$ Then $b$ equals

A

${3 \over 4}$

B

${ 1\over 2}$

C

${1 \over 3}$

D

${1 \over 4}$

Question 14: Palette 14/35

JEE Advanced 2011 MCQ

Let f $:$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$ be a continuous function such that
$f\left( x \right) = f\left( {1 - x} \right)$ for all $x \in \left[ { - 1,2} \right]$

Let ${R_1} = \int\limits_{ - 1}^2 {xf\left( x \right)dx,} $ and ${R_2}$ be the area of the region bounded by $y=f(x),$ $x=-1,$ $x=2,$ and the $x$-axis. Then

A

${R_1} = 2{R_2}$

B

${R_1} = 3{R_2}$

C

${2R_1} = {R_2}$

D

${3R_1} = {R_2}$

Question 15: Palette 15/35

JEE Advanced 2010 MCQ

Consider the polynomial
$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$
Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t = \left| s \right|.$

The area bounded by the curve $y=f(x)$ and the lines $x=0,$ $y=0$ and $x=t,$ lies in the interval

A

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

B

$\left( {{{21} \over {64}},{{11} \over {16}}} \right)$

C

$\left( {9,10} \right)$

D

$\left( {0,{{21} \over {64}}} \right)$

Question 16: Palette 16/35

JEE Advanced 2010 MSQ

Let $f$ be a real-valued function defined on the interval $\left( {0,\infty } \right)$
by $\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt {1 + \sin t\,} dt.} $ then which of the following
statement(s) is (are) true?

A

$f''(x)$ exists for all $x \in \left( {0,\infty } \right)$

B

$f'(x)$ exists for all $x \in \left( {0,\infty } \right)$ and $f'$ is continuous on $\left( {0,\infty } \right)$, but not differentiable on $\left( {0,\infty } \right)$

C

there exists $\,\,\alpha > 1$ such that $\left| {f'\left( x \right)} \right| < \left| {f\left( x \right)} \right|$ for all $x \in \left( {\alpha ,\infty } \right)\,$

D

there exists $\beta > 0$ such that $\left| {f\left( x \right)} \right| + \left| {f'\left( x \right)} \right| \le \beta $ for all $x \in \left( {0,\infty } \right)$

Question 17: Palette 17/35

JEE Advanced 2009 MCQ

Let $f$ be a non-negative function defined on the interval $[0,1]$.

If $\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} } $, and $f(0) = 0$, then

A

$f\left( {{1 \over 2}} \right) < {1 \over 2}$ and $f\left( {{1 \over 3}} \right) > {1 \over 3}$

B

$f\left( {{1 \over 2}} \right) > {1 \over 2}$ and $f\left( {{1 \over 3}} \right) > {1 \over 3}$

C

$f\left( {{1 \over 2}} \right) < {1 \over 2}$ and $f\left( {{1 \over 3}} \right) < {1 \over 3}$

D

$f\left( {{1 \over 2}} \right) > {1 \over 2}$ and $f\left( {{1 \over 3}} \right) < {1 \over 3}$

Question 18: Palette 18/35

JEE Advanced 2009 MSQ

Area of the region bounded by the curve $y = {e^x}$ and lines $x=0$ and $y=e$ is

A

$e-1$

B

$\int\limits_1^e {\ln \left( {e + 1 - y} \right)dy} $

C

$e - \int\limits_0^1 {{e^x}dx} $

D

$\int\limits_1^e {\ln y\,dy} $

Question 19: Palette 19/35

JEE Advanced 2008 MCQ

The area of the region between the curves $y = \sqrt {{{1 + \sin x} \over {\cos x}}} $
and $y = \sqrt {{{1 - \sin x} \over {\cos x}}} $ bounded by the lines $x=0$ and $x = {\pi \over 4}$ is

A

$\int\limits_0^{\sqrt 2 - 1} {{t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt} $

B

$\int\limits_0^{\sqrt 2 - 1} {{4t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt} $

C

$\int\limits_0^{\sqrt 2 + 1} {{4t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt} $

D

$\int\limits_0^{\sqrt 2 + 1} {{t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt} $

Question 20: Palette 20/35

JEE Advanced 2008 MCQ

The area of the region bounded by the curve $y=f(x),$ the
$x$-axis, and the lines $x=a$ and $x=b$, where $ - \infty < a < b < - 2,$ is :

A

$\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)$

B

$ - \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)$

C

$\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)$

D

$ - \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)$

Question 21: Palette 21/35

JEE Advanced 2006 MCQ

$\int_\limits{0}^{\pi / 2} \sin x d x$ is equal to:

A

$\frac{\pi}{8}(1+\sqrt{2})$

B

$\frac{\pi}{4}(1+\sqrt{2})$

C

$\frac{\pi}{8 \sqrt{2}}$

D

$\frac{\pi}{4 \sqrt{2}}$

Question 22: Palette 22/35

JEE Advanced 2006 MCQ

If $\lim_\limits{t \rightarrow a} \frac{\int_{a}^{t} f(x) d x-\frac{(t-a)}{2}\{f(t)+f(a)\}}{(t-a)^{3}}=0$ then the degree of polynomial function $f(x)$ almost is:

A

0

B

1

C

3

D

2

Question 23: Palette 23/35

JEE Advanced 2006 MCQ

$f''(x) < 0 \forall x \in(a, b)$ and $c$ is a point such that $a < c < b$, and $(c, f(C))$ is the point lying on the curve for which $\mathrm{F}(C)$ is maximum, then $f'(C)$ is equal to:

A

$\frac{f(b)-f(a)}{b-a}$

B

$\frac{2(f(b)-f(a))}{b-a}$

C

$\frac{2 f(b)-f(a)}{2 b-a}$

D

0

Question 24: Palette 24/35

JEE Advanced 2006 MCQ

$ \text { Match the following : } $

(i)	$ \int_0^{\pi / 2}(\sin x)^{\cos x}\left(\cos x \cot x-\log \left(\sin ^x\right)^{\sin } x\right) \mathrm{d} x $	(A)	1
(ii)	$ \text { Area bounded by }-4 y^2=x \text { and } x-1=-5 y^2 $	(B)	0
(iii)	Cosine of the angle of intersection of $y=3^{x-1} \log x$ and $y=x^{x-1}$ is	(C)	6 In 2
(iv)	$ \frac{d y}{d x}=\frac{2}{(x+y)} ; y\left(-\frac{2}{3}\right)=0 \text {, then value of constant }(\mathrm{k})= $	(D)	4/3

A

$ \begin{aligned} & \text { (i)-(A); (ii)-(D); (iii)-(B); }\text { (iv)-(D) } \end{aligned} $

B

$ \begin{aligned} & \text { (i)-(A); (ii)-(C); (iii)-(B); }\text { (iv)-(D) } \end{aligned} $

C

$ \begin{aligned} & \text { (i)-(A); (ii)-(D); (iii)-(A); }\text { (iv)-(D) } \end{aligned} $

D

$ \begin{aligned} & \text { (i)-(A); (ii)-(B); (iii)-(C); }\text { (iv)-(D) } \end{aligned} $

Question 25: Palette 25/35

JEE Advanced 2005 MCQ

The area bounded by the parabola $y = {\left( {x + 1} \right)^2}$ and
$y = {\left( {x - 1} \right)^2}$ and the line $y=1/4$ is

A

$4$ sq. units

B

$1/6$ sq. units

C

$4/3$ sq. units

D

$1/3$ sq. units

Question 26: Palette 26/35

JEE Advanced 2005 MCQ

If length of tangent at any point on the curve $y = f(x)$ intercepted between the point and the X-axis is of length 1. Find the equation of the curve.

A

$\sqrt{1-y^{2}}-\frac{1}{2} \log \left|\frac{1+\sqrt{1-y^{2}}}{1-\sqrt{1-y^{2}}}\right|= \pm x+c$

B

$\sqrt{1-y^{2}}- \log \left|\frac{1+\sqrt{1-y^{2}}}{1-\sqrt{1-y^{2}}}\right|= \pm x+c$

C

$\sqrt{1-y^{2}}+\frac{1}{2} \log \left|\frac{1+\sqrt{1-y^{2}}}{1-\sqrt{1-y^{2}}}\right|= \pm x+c$

D

$\sqrt{1-y^{2}}-\frac{1}{2} \log \left|\frac{1+\sqrt{1-y^{2}}}{1-\sqrt{1-y^{2}}}\right|= \pm 5x+c$

Question 27: Palette 27/35

JEE Advanced 2005 MCQ

Find the area bounded by the curves $x^{2}=y, x^{2}=-y$ and $y^{2}=4 x-3$.

A

$\frac{1}{3}$

B

$\frac{1}{5}$

C

$\frac{2}{3}$

D

$\frac{1}{7}$

Question 28: Palette 28/35

JEE Advanced 2005 MCQ

If $\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{array}\right], \quad f(x)$

is a quadratic function and its maximum value occurs at a point $\mathrm{V}$. If A is a point of intersection of $y=f(x)$ with $x$-axis and point B is such that chord AB subtends a right angle at point $\mathrm{V}$. Find the area enclosed by $f(x)$ and chord AB.

A

${{125} \over 3}$

B

${{125} \over 7}$

C

${{25} \over 3}$

D

${{23} \over 6}$

Question 29: Palette 29/35

JEE Advanced 2004 MCQ

The area enclosed between the curves $y = a{x^2}$ and
$x = a{y^2}\left( {a > 0} \right)$ is $1$ sq. unit, then the value of $a$ is

A

$1/\sqrt 3 $

B

$1/2$

C

$1$

D

$1/3$

Question 30: Palette 30/35

JEE Advanced 2003 MCQ

The area bounded by the curves $y = \sqrt x ,2y + 3 = x$ and
$x$-axis in the 1^st quadrant is

A

$9$

B

$27/4$

C

$36$

D

$18$

Question 31: Palette 31/35

JEE Advanced 2002 MCQ

Let $f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.} $ Then the real roots of the equation
${x^2} - f'\left( x \right) = 0$ are

A

$ \pm 1$

B

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

C

$ \pm {1 \over 2}$

D

$0$ and $1$

Question 32: Palette 32/35

JEE Advanced 2002 MCQ

The area bounded by the curves $y = \left| x \right| - 1$ and $y = - \left| x \right| + 1$ is

A

$1$

B

$2$

C

$2\sqrt 2 $

D

$4$

Question 33: Palette 33/35

JEE Advanced 1999 MSQ

For which of the following values of $m$, is the area of the region bounded by the curve $y = x - {x^2}$ and the line $y=mx$ equals $9/2$?

A

$-4$

B

$-2$

C

$2$

D

$4$

Question 34: Palette 34/35

JEE Advanced 1997 MCQ

If $g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $ then $g\left( {x + \pi } \right)$ equals

A

$g\left( x \right) + g\left( \pi \right)$

B

$g\left( x \right) - g\left( \pi \right)$

C

$g\left( x \right) g\left( \pi \right)$

D

${{g\left( x \right)} \over {g\left( \pi \right)}}$

Question 35: Palette 35/35

JEE Advanced 1982 MCQ

The area bounded by the curves $y=f(x)$, the $x$-axis and the ordinates $x=1$ and $x=b$ is $(b-1)$ sin $(3b+4)$. Then $f(x)$ is

A

$\left( {x - 1} \right)\cos \left( {3x + 4} \right)$

B

$\sin \left( {3x + 4} \right)$

C

$\sin \left( {3x + 4} \right) + 3\left( {x - 1} \right)\cos \left( {3x + 4} \right)$

D

none of these