Question 1: Palette 1/57

JEE Advanced 2026 MCQ

The value of the definite integral

$\int\limits_{0}^{2} \frac{1}{3^x + 3} dx$

is

A

$ \frac{1}{2} $

B

$ \frac{1}{3} $

C

$ \frac{\log_e 3}{3} $

D

$ \frac{\log_e 3}{2} $

Question 2: Palette 2/57

JEE Advanced 2023 MCQ

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int\limits_{x^2}^x \sqrt{\frac{1-t}{t}} d t < g(x) < 2 \sqrt{x}$ for all $x \in(0,1)$. Then $\lim\limits_{x \rightarrow 0} f(x) g(x)$

A

does NOT exist

B

is equal to 1

C

is equal to 2

D

is equal to 3

Question 3: Palette 3/57

JEE Advanced 2022 MSQ

Consider the equation

$ \int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) $

Which of the following statements is/are TRUE?

A

No $a$ satisfies the above equation

B

An integer $a$ satisfies the above equation

C

An irrational number $a$ satisfies the above equation

D

More than one $a$ satisfy the above equation

Question 4: Palette 4/57

JEE Advanced 2021 MCQ

Which of the following statements is TRUE?

A

$f(\sqrt {\ln 3} ) + g(\sqrt {\ln 3} ) = {1 \over 3}$

B

For every x > 1, there exists an $\alpha$ $\in$ (1, x) such that ${\psi _1}(x) = 1 + \alpha x$

C

For every x > 0, there exists a $\beta$ $\in$ (0, x) such that ${\psi _2}(x) = 2x({\psi _1}(\beta ) - 1)$

D

f is an increasing function on the interval $\left[ {0,{3 \over 2}} \right]$

Question 5: Palette 5/57

JEE Advanced 2021 MCQ

Which of the following statements is TRUE?

A

${\psi _1}(x) \le 1$, for all x > 0

B

${\psi _2}(x) \le 0$, for all x > 0

C

$f(x) \ge 1 - {e^{ - {x^2}}} - {2 \over 3}{x^3} + {2 \over 5}{x^5}$, for all $x \in \left( {0,{1 \over 2}} \right)$

D

$g(x) \le {2 \over 3}{x^3} - {2 \over 5}{x^5} + {1 \over 7}{x^7}$, for all $x \in \left( {0,{1 \over 2}} \right)$

Question 6: Palette 6/57

JEE Advanced 2021 MSQ

Let $f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R$ be a continuous function such that $f(0) = 1$ and $\int_0^{{\pi \over 3}} {f(t)dt = 0} $. Then which of the following statements is(are) TRUE?

A

The equation $f(x) - 3\cos 3x = 0$ has at least one solution in $\left( {0,{\pi \over 3}} \right)$

B

The equation $f(x) - 3\sin 3x = - {6 \over \pi }$ has at least one solution in $\left( {0,{\pi \over 3}} \right)$

C

$\mathop {\lim }\limits_{x \to 0} {{x\int_0^x {f(t)dt} } \over {1 - {e^{{x^2}}}}} = - 1$

D

$\mathop {\lim }\limits_{x \to 0} {{\sin x\int_0^x {f(t)dt} } \over {{x^2}}} = - 1$

Question 7: Palette 7/57

JEE Advanced 2020 MSQ

Let b be a nonzero real number. Suppose f : R $ \to $ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $f'(x) = {{f(x)} \over {{b^2} + {x^2}}}$

for all x$ \in $R, then which of the following statements is/are TRUE?

A

If b > 0, then f is an increasing function

B

If b < 0, then f is a decreasing function

C

f(x) f($-$x) = 1 for all x$ \in $R

D

f(x) $-$f($-$x) = 0 for all x$ \in $R

Question 8: Palette 8/57

JEE Advanced 2020 MSQ

Which of the following inequalities is/are TRUE?

A

$\int_0^1 {x\cos xdx\, \ge \,{3 \over 8}} $

B

$\int_0^1 {x\sin xdx\, \ge \,{3 \over {10}}} $

C

$\int_0^1 {{x^2}\cos xdx\, \ge \,{1 \over 2}} $

D

$\int_0^1 {{x^2}\sin xdx\, \ge \,{2 \over 9}} $

Question 9: Palette 9/57

JEE Advanced 2017 MSQ

If $I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx} $, then

A

$I > {\log _e}99$

B

$I < {\log _e}99$

C

$I < {{49} \over {50}}$

D

$I > {{49} \over {50}}$

Question 10: Palette 10/57

JEE Advanced 2016 MCQ

The value of $\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $ is equal to

A

${{{\pi ^2}} \over 4} - 2$

B

${{{\pi ^2}} \over 4} + 2$

C

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

D

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

Question 11: Palette 11/57

JEE Advanced 2016 MSQ

Let
$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$ for

all $x>0.$ Then

A

$f\left( {{1 \over 2}} \right) \ge f\left( 1 \right)$

B

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

C

$\,f'\left( 2 \right) \le 0$

D

$\,{{f'\left( 3 \right)} \over {f\left( 3 \right)}} \ge {{f'\left( 2 \right)} \over {f\left( 2 \right)}}$

Question 12: Palette 12/57

JEE Advanced 2015 MCQ

Let $f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$ for all $x \in R\,\,$ with $f\left( {{1 \over 2}} \right) = 0$.
If $m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $ then the possible values of $m$ and $M$ are

A

$m=13,$ $M=24$

B

$\,m = {1 \over 4},M = {1 \over 2}$

C

$m=-11,$ $M=0$

D

$m=1,$ $M=12$

Question 13: Palette 13/57

JEE Advanced 2015 MSQ

Let $f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$ for all $x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$
Then the correct expression(s) is (are)

A

$\int\limits_0^{\pi /4} {xf\left( x \right)dx = {1 \over {12}}} $

B

$\int\limits_0^{\pi /4} {f\left( x \right)dx = 0} $

C

$\int\limits_0^{\pi /4} {xf\left( x \right)dx = {1 \over {6}}} $

D

$\int\limits_0^{\pi /4} {f\left( x \right)dx = 1} $

Question 14: Palette 14/57

JEE Advanced 2015 MSQ

The option(s) with the values of a and $L$ that satisfy the following equation is (are) $${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} }} = L?$$

A

$a = 2,L = {{{e^{4\pi }} - 1} \over {{e^\pi } - 1}}$

B

$a = 2,L = {{{e^{4\pi }} + 1} \over {{e^\pi } + 1}}$

C

$a = 4,L = {{{e^{4\pi }} - 1} \over {{e^\pi } - 1}}$

D

$a = 4,L = {{{e^{4\pi }} + 1} \over {{e^\pi } + 1}}$

Question 15: Palette 15/57

JEE Advanced 2014 MCQ

List - $I$
P.$\,\,\,\,$ The number of polynomials $f(x)$ with non-negative integer coefficients of degree $ \le 2$, satisfying $f(0)=0$ and $\int_0^1 {f\left( x \right)dx = 1,} $ is
Q.$\,\,\,\,$ The number of points in the interval $\left[ { - \sqrt {13} ,\sqrt {13} } \right]$
at which $f\left( x \right) = \sin \left( {{x^2}} \right) + \cos \left( {{x^2}} \right)$ attains its maximum value, is
R.$\,\,\,\,$ $\int\limits_{ - 2}^2 {{{3{x^2}} \over {\left( {1 + {e^x}} \right)}}dx} $ equals
S.$\,\,\,\,$ ${{\left( {\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)} \over {\left( {\int\limits_0^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)}}$

List $II$
1.$\,\,\,\,$ $8$
2.$\,\,\,\,$ $2$
3.$\,\,\,\,$ $4$
4.$\,\,\,\,$ $0$

A

$P = 3,Q = 2,R = 4,S = 1$

B

$P = 2,Q = 3,R = 4,S = 1$

C

$P = 3,Q = 2,R = 1,S = 4$

D

$P = 2,Q = 3,R = 1,S = 4$

Question 16: Palette 16/57

JEE Advanced 2014 MCQ

The following integral $\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx} $ is equal to

A

$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {2{{\left( {{e^u} + {e^{ - u}}} \right)}^{16}}\,du} $

B

$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {{{\left( {{e^u} + {e^{ - u}}} \right)}^{17}}\,du} $

C

$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {{{\left( {{e^u} - {e^{ - u}}} \right)}^{17}}\,du} $

D

$\int\limits_0^{\log \left( {1 + \sqrt 2 } \right)} {2{{\left( {{e^u} - {e^{ - u}}} \right)}^{16}}\,du} $

Question 17: Palette 17/57

JEE Advanced 2014 MCQ

Given that for each $a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt} $ exists. Let this limit be $g(a).$ In addition, it is given that the function $g(a)$ is differentiable on $(0,1).$

The value of $g'\left( {{1 \over 2}} \right)$ is

A

${\pi \over 2}$

B

$\pi $

C

$-{\pi \over 2}$

D

$0$

Question 18: Palette 18/57

JEE Advanced 2014 MCQ

Given that for each $a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt} $ exists. Let this limit be $g(a).$ In addition, it is given that the function $g(a)$ is differentiable on $(0,1).$

The value of $g\left( {{1 \over 2}} \right)$ is

A

$\pi $

B

$2\pi $

C

${\pi \over 2}$

D

${\pi \over 4}$

Question 19: Palette 19/57

JEE Advanced 2014 MSQ

Let $f:\left( {0,\infty } \right) \to R$ be given by $f\left( x \right) $= $\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt$. Then

A

$f(x)$ is monotonically increasing on $\left[ {1,\infty } \right)$

B

$f(x)$ is monotonically decreasing on $(0,1)$

C

$f(x)$ $ + f\left( {{1 \over x}} \right) = 0$, for all $x \in \left( {0,\infty } \right)$

D

$f\left( {{2^x}} \right)$ is an odd function of $x$ on $R$

Question 20: Palette 20/57

JEE Advanced 2014 MSQ

Let a $\in$ R and f : R $\to$ R be given by f(x) = x⁵ $-$ 5x + a. Then,

A

f(x) has three real roots , if a > 4

B

f(x) has only one real root, if a > 4

C

f(x) has three real roots, if a < $-$4

D

f(x) has three real roots, if $-$4 < a < 4

Question 21: Palette 21/57

JEE Advanced 2013 MCQ

Let $f$ $:\,\,\left[ {{1 \over 2},1} \right] \to R$ (the set of all real number) be a positive,
non-constant and differentiable function such that
$f'\left( x \right) < 2f\left( x \right)$ and $f\left( {{1 \over 2}} \right) = 1.$ Then the value of $\int\limits_{1/2}^1 {f\left( x \right)} \,dx$ lies in the interval

A

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

B

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

C

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

D

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

Question 22: Palette 22/57

JEE Advanced 2012 MCQ

The value of the integral $\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx} $ is

A

$0$

B

${{{\pi ^2}} \over 2} - 4$

C

${{{\pi ^2}} \over 2} + 4$

D

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

Question 23: Palette 23/57

JEE Advanced 2011 MCQ

The value of $\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx} $ is

A

${1 \over 4}\,\ell n{3 \over 2}$

B

$\,{1 \over 2}\,\ell n{3 \over 2}$

C

$\ell n{3 \over 2}$

D

$\,\,{1 \over 6}\,\ell n{3 \over 2}$

Question 24: Palette 24/57

JEE Advanced 2010 MCQ

The value of $\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {{t^4} + 4}}} dt$ is

A

$0$

B

${1 \over 12}$

C

${1 \over 24}$

D

${1 \over 64}$

Question 25: Palette 25/57

JEE Advanced 2010 MCQ

The value of $\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx} $ is (are)

A

${{22} \over 7} - \pi $

B

${2 \over {105}}$

C

$0$

D

${{71} \over {15}} - {{3\pi } \over 2}$

Question 26: Palette 26/57

JEE Advanced 2010 MCQ

Let $f$ be a real-valued function defined on the interval $(-1, 1)$ such that
${e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,} $ for all $x \in \left( { - 1,1} \right)$,
and let ${f^{ - 1}}$ be the inverse function of $f$. Then $\left( {{f^{ - 1}}} \right)'\left( 2 \right)$ is equal to

A

$1$

B

${{1 \over 3}}$

C

${{1 \over 2}}$

D

${{1 \over e}}$

Question 27: Palette 27/57

JEE Advanced 2009 MSQ

If ${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,} $ .... then

A

${I_n} = {I_{n + 2}}$

B

$\sum\limits_{m = 1}^{10} {{I_{2m + 1}}} = 10\pi $

C

$\sum\limits_{m = 1}^{10} {{I_{2m}}} = 0$

D

${I_n} = {I_{n + 1}}$

Question 28: Palette 28/57

JEE Advanced 2008 MCQ

Let $g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$

Which of the following is true?

A

$g'(x)$ is positive on $\left( { - \infty ,0} \right)$ and negative on $\left( {0,\infty } \right)$

B

$g'(x)$ is negative on $\left( { - \infty ,0} \right)$ and positive on $\left( {0,\infty } \right)$

C

$g'(x)$ changes sign on both $\left( { - \infty ,0} \right)$ and $\left( {0,\infty } \right)$

D

$g'(x)$ does not change sign on $\left( { - \infty ,0} \right)$

Question 29: Palette 29/57

JEE Advanced 2008 MCQ

$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $

A

$2g(-1)$

B

$0$

C

$-2g(1)$

D

$2g(1)$

Question 30: Palette 30/57

JEE Advanced 2007 MCQ

$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{\int\limits_2^{{{\sec }^2}x} {f(t)\,dt} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}$ equal

A

${8 \over \pi }f(2)$

B

${2 \over \pi }f(2)$

C

${2 \over \pi }f\left( {{1 \over 2}} \right)$

D

$4f(2)$

Question 31: Palette 31/57

JEE Advanced 2007 MCQ

Match the integrals in Column I with the values in Column II.

	Column I		Column II
(A)	$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $	(P)	${1 \over 2}\log \left( {{2 \over 3}} \right)$
(B)	$\int\limits_0^1 {{{dx} \over {\sqrt {1 + {x^2}} }}} $	(Q)	$2\log \left( {{2 \over 3}} \right)$
(C)	$\int\limits_2^3 {{{dx} \over {1 + {x^2}}}} $	(R)	${\pi \over 3}$
(D)	$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $	(S)	${\pi \over 2}$

A

A - s, B - s, C - r, D - p

B

A - s, B - q, C - p, D - r

C

A - s, B - s, C - p, D - r

D

A - s, B - q, C - s, D - r

Question 32: Palette 32/57

JEE Advanced 2005 MCQ

$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx} $ is equal to

A

$-4$

B

$0$

C

$4$

D

$6$

Question 33: Palette 33/57

JEE Advanced 2005 MCQ

Evatuate:

$\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x$

A

${e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$

B

$24{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$

C

$12{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$

D

$5{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$

Question 34: Palette 34/57

JEE Advanced 2004 MCQ

The value of the integral $\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx} $ is

A

${\pi \over 2} + 1$

B

${\pi \over 2} - 1$

C

$-1$

D

$1$

Question 35: Palette 35/57

JEE Advanced 2004 MCQ

If $f(x)$ is differentiable and $\int\limits_0^{{t^2}} {xf\left( x \right)dx = {2 \over 5}{t^5},} $ then $f\left( {{4 \over {25}}} \right)$ equals

A

$2/5$

B

$-5/2$

C

$1$

D

$5/2$

Question 36: Palette 36/57

JEE Advanced 2003 MCQ

If $l\left( {m,n} \right) = \int\limits_0^1 {{t^m}{{\left( {1 + t} \right)}^n}dt,} $ then the expression for $l(m, n)$ in terms of $l(m+n, n-1)$ is

A

${{{2^n}} \over {m + 1}} - {n \over {m + 1}}l\left( {m + 1,n - 1} \right)$

B

${n \over {m + 1}}l\left( {m + 1,n - 1} \right)$

C

${{{2^n}} \over {m + 1}} + {n \over {m + 1}}l\left( {m + 1,n - 1} \right)$

D

${m \over {n + 1}}l\left( {m + 1,n - 1} \right)$

Question 37: Palette 37/57

JEE Advanced 2003 MCQ

If $f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,$ then $f(x)$ increases in

A

$(-2, 2)$

B

no value of $x$

C

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

D

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

Question 38: Palette 38/57

JEE Advanced 2002 MCQ

The integral $\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx} $ equal to

A

$ - {1 \over 2}$

B

$0$

C

$1$

D

$2\ell n\left( {{1 \over 2}} \right)$

Question 39: Palette 39/57

JEE Advanced 2002 MCQ

Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.

If $I = \int\limits_0^T {f\left( x \right)dx} $ then the value of $\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $ is

A

$3/2I$

B

$2I$

C

$3I$

D

$6I$

Question 40: Palette 40/57

JEE Advanced 2002 MCQ

Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.

If $I = \int\limits_0^T {f\left( x \right)dx} $ then the value of $\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $ is

A

$3/2I$

B

$2I$

C

$3I$

D

$6I$

Question 41: Palette 41/57

JEE Advanced 2001 MCQ

The value of $\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $ is

A

$\pi $

B

$a\pi $

C

$\pi /2$

D

$2\pi $

Question 42: Palette 42/57

JEE Advanced 2000 MCQ

If $f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} & {otherwise,} \cr } } \right.$ then $\int\limits_{ - 2}^3 {f\left( x \right)dx = } $

A

$0$

B

$1$

C

$2$

D

$3$

Question 43: Palette 43/57

JEE Advanced 2000 MCQ

The value of the integral $\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $ is :

A

$3/2$

B

$5/2$

C

$3$

D

$5$

Question 44: Palette 44/57

JEE Advanced 2000 MCQ

Let $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,} $ where f is such that
${1 \over 2} \le f\left( t \right) \le 1,$ for $t \in \left[ {0,1} \right]$ and $\,0 \le f\left( t \right) \le {1 \over 2},$ for $t \in \left[ {1,2} \right]$.
Then $g(2)$ satisfies the inequality

A

$ - {3 \over 2} \le g\left( 2 \right) < {1 \over 2}$

B

$0 \le g\left( 2 \right) < 2$

C

${3 \over 2} < g\left( 2 \right) \le {5 \over 2}$

D

$2 < g\left( 2 \right) < 4$

Question 45: Palette 45/57

JEE Advanced 1999 MCQ

If for a real number $y$, $\left[ y \right]$ is the greatest integer less than or
equal to $y$, then the value of the integral $\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx} $ is

A

$ - \pi $

B

$0$

C

$ - \pi /2$

D

$ \pi /2$

Question 46: Palette 46/57

JEE Advanced 1999 MCQ

$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $ is equal to

A

$2$

B

$-2$

C

$1/2$

D

$-1/2$

Question 47: Palette 47/57

JEE Advanced 1998 MCQ

Let $f\left( x \right) = x - \left[ x \right],$ for every real number $x$, where $\left[ x \right]$ is the integral part of $x$. Then $\int_{ - 1}^1 {f\left( x \right)\,dx} $ is

A

$1$

B

$2$

C

$0$

D

$1/2$

Question 48: Palette 48/57

JEE Advanced 1998 MCQ

If $\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $ then the value of $f(1)$ is

A

$1/2$

B

$0$

C

$1$

D

$-1/2$

Question 49: Palette 49/57

JEE Advanced 1995 MCQ

If $f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2 $ and
$\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },} $ then constants $A$ and $B$ are

A

${\pi \over 2}$ and ${\pi \over 2}$

B

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

C

$0$ and ${-4 \over \pi }$

D

${4 \over \pi }$ and $0$

Question 50: Palette 50/57

JEE Advanced 1995 MCQ

The value of $\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $ where [ . ] represents the greatest integer function is

A

${{ - 5\pi } \over 3}$

B

$\pi $

C

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

D

$ - 2\pi $

Question 51: Palette 51/57

JEE Advanced 1993 MCQ

The value of $\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $ is

A

$0$

B

$1$

C

$\pi /2$

D

$\pi /4$

Question 52: Palette 52/57

JEE Advanced 1990 MCQ

Let $f:R \to R$ and $\,\,g:R \to R$ be continuous functions. Then the value of the integral
$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $ is

A

$\pi $

B

$1$

C

$-1$

D

$0$

Question 53: Palette 53/57

JEE Advanced 1988 MCQ

The value of the integral $\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $ is equal to $a$.

A

TRUE

B

FALSE

Question 54: Palette 54/57

JEE Advanced 1985 MCQ

For any integer $n$ the integral ...........
$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $ has the value

A

$\pi $

B

$1$

C

$0$

D

none of these

Question 55: Palette 55/57

JEE Advanced 1983 MCQ

The value of the integral $\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $ is

A

$\pi /4$

B

$\pi /2$

C

$\pi $

D

none of these

Question 56: Palette 56/57

JEE Advanced 1981 MCQ

Let $a, b, c$ be non-zero real numbers such that
$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $
Then the quadratic equation $a{x^2} + bx + c = 0$ has

A

no root in $(0, 2)$

B

at least one root in $(0, 2)$

C

a double root in $(0, 2)$

D

two imaginary roots

Question 57: Palette 57/57

JEE Advanced 1981 MCQ

The value of the definite integral $\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$

A

$-1$

B

$2$

C

$1 + {e^{ - 1}}$

D

none of these