Question 1: Palette 1/71

JEE Advanced 2026 MCQ

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 $(0, \infty)$

Question 2: Palette 2/71

JEE Advanced 2025 MSQ

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by

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

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

Question 3: Palette 3/71

JEE Advanced 2023 MCQ

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d\left(\ell_1, \ell_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_1, \ell_2\right)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is :

A

$\frac{1}{\sqrt{6}}$

B

$\frac{1}{\sqrt{8}}$

C

$\frac{1}{\sqrt{3}}$

D

$\frac{1}{\sqrt{12}}$

Question 4: Palette 4/71

JEE Advanced 2022 MSQ

Let

$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$ g(x)=2^{\alpha x}+2^{\alpha(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

Question 5: Palette 5/71

JEE Advanced 2020 MCQ

Consider the rectangles lying the region

$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$ and $\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$

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

A

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

B

$\pi $

C

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

D

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

Question 6: Palette 6/71

JEE Advanced 2019 MSQ

Let f : R $ \to $ R be given by

$f(x) = (x - 1)(x - 2)(x - 5)$. Define

$F(x) = \int\limits_0^x {f(t)dt} $, x > 0

Then which of the following options is/are correct?

A

F(x) $ \ne $ 0 for all x $ \in $ (0, 5)

B

F has a local maximum at x = 2

C

F has two local maxima and one local minimum in (0, $\infty $)

D

F has a local minimum at x = 1

Question 7: Palette 7/71

JEE Advanced 2019 MSQ

Let, $f(x) = {{\sin \pi x} \over {{x^2}}}$, x > 0

Let x₁ < x₂ < x₃ < ... < x_n < ... be all the points of local maximum of f and y₁ < y₂ < y₃ < ... < y_n < ... be all the points of local minimum of f.

Then which of the following options is/are correct?

A

$|{x_n} - {y_n}|\, > 1$ for every n

B

${x_{n + 1}} - {x_n}\, > 2$ for every n

C

x₁ < y₁

D

${x_n} \in \left( {2n,\,2n + {1 \over 2}} \right)$ for every n

Question 8: Palette 8/71

JEE Advanced 2017 MCQ

Which of the following options is the only INCORRECT combination?

A

(I) (iii) (P)

B

(II) (iv) (Q)

C

(II) (ii) (P)

D

(III) (i) (R)

Question 9: Palette 9/71

JEE Advanced 2017 MCQ

Which of the following options is the only CORRECT combination?

A

(I) (ii) (R)

B

(III) (iv) (P)

C

(II) (iii) (S)

D

(IV) (i) (S)

Question 10: Palette 10/71

JEE Advanced 2017 MCQ

Which of the following options is the only CORRECT combination?

A

(III) (iii) (R)

B

(IV) (iv) (S)

C

(II) (ii) (Q)

D

(I0 (i) (P)

Question 11: Palette 11/71

JEE Advanced 2017 MSQ

f : R $ \to $ R is a differentiable function such that f'(x) > 2f(x) for all x$ \in $R, and f(0) = 1 then

A

f(x) > e^2x in (0, $\infty $)

B

f'(x) < e^2x in (0, $\infty $)

C

f(x) is increasing in (0, $\infty $)

D

f(x) is decreasing in (0, $\infty $)

Question 12: Palette 12/71

JEE Advanced 2017 MSQ

If $f(x) = \left| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right|$,

then

A

f(x) attains its minimum at x = 0

B

f(x) attains its maximum at x = 0

C

f'(x) = 0 at more than three points in ($-$$\pi $, $\pi $)

D

f'(x) = 0 at exactly three points in ($-$$\pi $, $\pi $)

Question 13: Palette 13/71

JEE Advanced 2016 MCQ

The least value of a $ \in R$ for which $4a{x^2} + {1 \over x} \ge 1,$, for all $x>0$. is

A

${1 \over {64}}$

B

${1 \over {32}}$

C

${1 \over {27}}$

D

${1 \over {25}}$

Question 14: Palette 14/71

JEE Advanced 2016 MSQ

Let f: R $ \to \left( {0,\infty } \right)$ and g : R $ \to $ R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'$(2)$ $=$ g$(2)=0$, f''$(2)$$ \ne 0$ and g'$(2)$ $ \ne 0$. If
$\mathop {\lim }\limits_{x \to 2} {{f\left( x \right)g\left( x \right)} \over {f'\left( x \right)g'\left( x \right)}} = 1,$ 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 $x \in R$

Question 15: Palette 15/71

JEE Advanced 2015 MSQ

Let $f, g :$ $\left[ { - 1,2} \right] \to R$ 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:

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

In each of the intervals $(-1, 0)$ and $(0, 2)$ the function $(f-3g)''$ never vanishes. Then the correct statement(s) is (are)

A

$f'\left( x \right) - 3g'\left( x \right) = 0$ has exactly three solutions in $\left( { - 1,0} \right) \cup \left( {0,2} \right)$

B

$f'\left( x \right) - 3g'\left( x \right) = 0$ has exactly one solution in $(-1, 0)$

C

$f'\left( x \right) - 3g'\left( x \right) = 0$ has exactly one solution in $(0, 2)$

D

$f'\left( x \right) - 3g'\left( x \right) = 0$ has exactly two solutions in $(-1, 0)$ and exactly two solutions in $(0, 2)$

Question 16: Palette 16/71

JEE Advanced 2013 MCQ

Let $f:\left[ {0,1} \right] \to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable,
$f(0) = f(1)=0$ and satisfies $f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$.

Which of the following is true for $0 < x < 1?$

A

$0 < f\left( x \right) < \infty $

B

$ - {1 \over 2} < f\left( x \right) < {1 \over 2}$

C

$ - {1 \over 4} < f\left( x \right) < 1$

D

$ - \infty < f\left( x \right) < 0$

Question 17: Palette 17/71

JEE Advanced 2013 MCQ

Let $f:\left[ {0,1} \right] \to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable,
$f(0) = f(1)=0$ and satisfies $f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$.

If the function ${e^{ - x}}f\left( x \right)$ assumes its minimum in the interval $\left[ {0,1} \right]$ at $x = {1 \over 4}$, which of the following is true?

A

$f'\left( x \right) < f\left( x \right),{1 \over 4} < x < {3 \over 4}$

B

$f'\left( x \right) > f\left( x \right),0 < x < {1 \over 4}$

C

$f'\left( x \right) < f\left( x \right),0 < x < {1 \over 4}$

D

$f'\left( x \right) < f\left( x \right),{3 \over 4} < x < 1$

Question 18: Palette 18/71

JEE Advanced 2013 MSQ

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

A

$-$2

B

${{ - 2} \over 3}$

C

2

D

${{ 2} \over 3}$

Question 19: Palette 19/71

JEE Advanced 2013 MSQ

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 vsides of the rectangular sheet are

A

$24$

B

$32$

C

$45$

D

$60$

Question 20: Palette 20/71

JEE Advanced 2012 MCQ

Let $f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$ for all $x \in IR$ and let
$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $ for all $x \in \left( {1,\,\infty } \right)$.

Which of the following is true?

A

$g$ is increasing on $\left( {1,\infty } \right)$

B

$g$ is decreasing on $\left( {1,\infty } \right)$

C

$g$ is increasing on $(1, 2)$ and decreasing on $\left( {2,\infty } \right)$

D

$g$ is decreasing on $(1, 2)$ and increasing on $\left( {2,\infty } \right)$

Question 21: Palette 21/71

JEE Advanced 2012 MCQ

Let $f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$ for all $x \in IR$ and let
$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $ for all $x \in \left( {1,\,\infty } \right)$.

Consider the statements:
$P:$ There exists some $x \in R$ such that $f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)$
$Q:\,\,$ There exists some $x \in R$ such that $2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)$
Then

A

both $P$ and $Q$ are true

B

$P$ is true and $Q$ is false

C

$P$ is false and $Q$ is true

D

both $P$ and $Q$ are false

Question 22: Palette 22/71

JEE Advanced 2012 MSQ

If $f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$ for all $x \in \left( {0,\infty } \right),$ then

A

$f$ has a local maximum at $x=2$

B

$f$ is decreasing on $(2, 3)$

C

there exists some $c \in \left( {0,\infty } \right),$ such that $f'(c)=0$

D

$f$ has a local minimum at $x=3$

Question 23: Palette 23/71

JEE Advanced 2009 MSQ

For the function $$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,$$

A

for at least one $x$ in the interval $\left[ {1,\infty } \right)$, $f\left( {x + 2} \right) - f\left( x \right) < 2$

B

$\mathop {\lim }\limits_{x \to \infty } f'\left( x \right) = 1$

C

for all $x$ in the interval $\left[ {1,\infty } \right)f\left( {x + 2} \right) - f\left( x \right) > 2$

D

$f'(x)$ is strictly decreasing in the interval $\left[ {1,\infty } \right)$

Question 24: Palette 24/71

JEE Advanced 2008 MCQ

The total number of local maxima and local minima of the function

$f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.$ is

A

0

B

1

C

2

D

3

Question 25: Palette 25/71

JEE Advanced 2007 MCQ

The tangent to the curve $y = {e^x}$ drawn at the point $\left( {c,{e^c}} \right)$ intersects the line joining the points $\left( {c - 1,{e^{c - 1}}} \right)$ and $\left( {c + 1,{e^{c + 1}}} \right)$

A

on the left of $x=c$

B

on the right of $x=c$

C

at no point

D

at all points

Question 26: Palette 26/71

JEE Advanced 2007 MCQ

If a continuous function $f$ defined on the real line $R$, assumes positive and negative values in $R$ then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative then the equation $f(x)=0$ has a root in $R$.
Consider $f\left( x \right) = k{e^x} - x$ for all real $x$ where $k$ is real constant.

The positive value of $k$ for which $k{e^x} - x = 0$ has only one root is

A

${1 \over e}$

B

$1$

C

$e$

D

${\log _e}2$

Question 27: Palette 27/71

JEE Advanced 2007 MCQ

If a continuous function $f$ defined on the real line $R$, assumes positive and negative values in $R$ then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative then the equation $f(x)=0$ has a root in $R$.
Consider $f\left( x \right) = k{e^x} - x$ for all real $x$ where $k$ is real constant.

The line $y=x$ meets $y = k{e^x}$ for $k \le 0$ at

A

no point

B

one point

C

two points

D

more than two points

Question 28: Palette 28/71

JEE Advanced 2007 MCQ

If a continuous function $f$ defined on the real line $R$, assumes positive and negative values in $R$ then the equation $f(x)=0$ has a root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative then the equation $f(x)=0$ has a root in $R$.
Consider $f\left( x \right) = k{e^x} - x$ for all real $x$ where $k$ is real constant.

For $k>0$, the set of all values of $k$ for which $k{e^x} - x = 0$ has two distinct roots is

A

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

B

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

C

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

D

$\left( {0,1} \right)$

Question 29: Palette 29/71

JEE Advanced 2007 MCQ

Let $f(x)$ be differentiable on the interval (0, $\infty$) such that $f(1)=1$, and $\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1$ for each $x > 0$. Then $f(x)$ is

A

${1 \over {3x}} + {{2{x^2}} \over 3}$

B

$ - {1 \over {3x}} + {{4{x^2}} \over 3}$

C

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

D

${1 \over x}$

Question 30: Palette 30/71

JEE Advanced 2006 MSQ

A tangent drawn to the curve $y=f(x)$ at $\mathrm{P}(x, y)$ cuts the X -axis and Y -axis at A and B respectively such that $\mathrm{BP}: \mathrm{AP}=3: 1$, given that $f(1)=1$, then

A

equation of curve is $x \frac{d y}{d x}-3 y=0$

B

normal at $(1,1)$ is $x+3 y=4$

C

curve passes through $(2,1 / 8)$

D

equation of curve is $x \frac{d y}{d x}+3 y=0$

Question 31: Palette 31/71

JEE Advanced 2006 MSQ

$f(x)$ is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18, f(1)=-1$ and $f(x)$ has local minima at $x=0$, then

A

the distance between $(-1,2)$ and $(a, f(A)$, where $x=a$ is the point of local minima is $2 \sqrt{5}$

B

$f(x)$ is increasing for $x \in[1,2 \sqrt{5}]$

C

$f(x)$ has local minima at $x=1$

D

the value of $f(0)=5$

Question 32: Palette 32/71

JEE Advanced 2006 MSQ

$ \begin{aligned} & f(x)=\left\{\begin{array}{cc} e^x, & 0 \leq x \leq 1 \\ 2-e^{x-1}, & 1 < x \leq 2 \\ x-e, & 2 < x \leq 3 \end{array} \quad\right. \text { and } \\ & g(x)=\int_0^x f(t) d t, x \in[1,3] \text { then } g(x) \text { has } \end{aligned} $

A

local maxima at $x=1+\ln 2$ and local $\operatorname{minima}$ at $x=e$

B

local maxima at $x=1$ and local minima at $x=2$

C

no local maxima

D

no local minima

Question 33: Palette 33/71

JEE Advanced 2005 MCQ

If $P(x)$ is a polynomial of degree less than or equal to $2$ and $S$ is the set of all such polynomials so that $P(0)=0$, $P(1)=1$ and $P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$ then

A

$S = \phi $

B

$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,2} \right)$

C

$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,\infty } \right)$

D

$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,1} \right)$

Question 34: Palette 34/71

JEE Advanced 2005 MCQ

If $\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}$, for all $x_{1}, x_{2} \in$ $\mathbb{R}$. Find the equation of tangent to the curve $y=f(x)$ at the point $(1,2)$.

A

$y-2=0$

B

$3y-2=0$

C

$3y-5=0$

D

$5y-3=0$

Question 35: Palette 35/71

JEE Advanced 2005 MCQ

If $p(x)$ be a polynomial of degree 3 satisfying $p(-1)=10, p(1)=-6$ and $p(x)$ has maximum at $x=-1$ and $p'(x)$ has minima at $x=1$. Find the distance between the local maximum and local minimum of the curve.

A

$2\sqrt{65}$

B

$\sqrt{65}$

C

$4\sqrt{65}$

D

$4\sqrt{75}$

Question 36: Palette 36/71

JEE Advanced 2004 MCQ

If $f\left( x \right) = {x^a}\log x$ and $f\left( 0 \right) = 0,$ then the value of $\alpha $ for which Rolle's theorem can be applied in $\left[ {0,1} \right]$ is

A

$-2$

B

$-1$

C

$0$

D

$1/2$

Question 37: Palette 37/71

JEE Advanced 2004 MCQ

If $f\left( x \right) = {x^3} + b{x^2} + cx + d$ and $0 < {b^2} < c,$ then in $\left( { - \infty ,\infty } \right)$

A

$f\left( x \right)$ is a strictly increasing function

B

$f\left( x \right)$ has a local maxima

C

$f\left( x \right)$ is a strictly decreasing function

D

$f\left( x \right)$ is bounded

Question 38: Palette 38/71

JEE Advanced 2003 MCQ

In $\left[ {0,1} \right]$ Languages Mean Value theorem is NOT applicable to

A

$f\left( x \right) = \left\{ {\matrix{ {{1 \over 2} - x} & {x < {1 \over 2}} \cr {{{\left( {{1 \over 2} - x} \right)}^2}} & {x \ge {1 \over 2}} \cr } } \right.$

B

$f\left( x \right) = \left\{ {\matrix{ {\sin x,} & {x \ne 0} \cr {1,} & {x = 0} \cr } } \right.$

C

$f\left( x \right) = x\left| x \right|$

D

$f\left( x \right) = \left| x \right|$

Question 39: Palette 39/71

JEE Advanced 2003 MCQ

Tangent is drawn to ellipse
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.

Then the value of $\theta $ such that sum of intercepts on axes made by this tangent is minimum, is

A

$\pi /3$

B

$\pi /6$

C

$\pi /8$

D

$\pi /4$

Question 40: Palette 40/71

JEE Advanced 2002 MCQ

The length of a longest interval in which the function $3\,\sin x - 4{\sin ^3}x$ is increasing, is

A

${\pi \over 3}$

B

${\pi \over 2}$

C

${3\pi \over 2}$

D

$\pi $

Question 41: Palette 41/71

JEE Advanced 2002 MCQ

The point(s) in the curve ${y^3} + 3{x^2} = 12y$ where the tangent is vertical, is (are)

A

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

B

$\left( { \pm \sqrt {{{11} \over 3}} ,1} \right)$

C

$(0,0)$

D

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

Question 42: Palette 42/71

JEE Advanced 2001 MCQ

Let $f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$ and let $m(b)$ be the minimum value of $f(x)$. As $b$ varies, the range of $m(b)$ is

A

$\left[ {0,1} \right]$

B

$\left( {0,\,1/2} \right]$

C

$\left[ {1/2,\,1} \right]$

D

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

Question 43: Palette 43/71

JEE Advanced 2001 MCQ

The triangle formed by the tangent to the curve $f\left( x \right) = {x^2} + bx - b$ at the point $(1, 1)$ and the coordinate axex, lies in the first quadrant. If its area is $2$, then the value of $b$ is

A

$-1$

B

$3$

C

$-3$

D

$1$

Question 44: Palette 44/71

JEE Advanced 2001 MCQ

If $f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$ then $f(x)$ is

A

increasing on $\left[ { - 1/2,1} \right]$

B

decreasing on $R$

C

increasing on $R$

D

decreasing on $\left[ { - 1/2,1} \right]$

Question 45: Palette 45/71

JEE Advanced 2000 MCQ

Consider the following statements in $S$ and $R$
$S:$ $\,\,\,$$ Both $\sin \,\,x$ and $\cos \,\,x$ are decreasing functions in the interval $\left( {{\pi \over 2},\pi } \right)$
$R:$$\,\,\,$ If a differentiable function decreases in an interval $(a, b)$, then its derivative also decreases in $(a, b)$.
Which of the following is true ?

A

Both $S$ and $R$ are wrong

B

Both $S$ and $R$ are correct, but $R$ is not the correct explanation of $S$

C

$S$ is correct and $R$ is the correct explanation for $S$

D

$S$ is correct and $R$ is wrong

Question 46: Palette 46/71

JEE Advanced 2000 MCQ

Let $f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $ Then $f$ decreases in the interval

A

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

B

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

C

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

D

$\left( {2, + \infty } \right)$

Question 47: Palette 47/71

JEE Advanced 2000 MCQ

Let $f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {x = 0} \cr } } \right.$ then at $x=0$, $f$ has

A

a local maximum

B

no local maximum

C

a local minimum

D

no extremum

Question 48: Palette 48/71

JEE Advanced 2000 MCQ

If the normal to the curve $y = f\left( x \right)$ and the point $(3, 4)$ makes an angle ${{{3\pi } \over 4}}$ with the positive $x$-axis, then $f'\left( 3 \right) = $

A

$-1$

B

$ - {3 \over 4}$

C

${4 \over 3}$

D

$1$

Question 49: Palette 49/71

JEE Advanced 2000 MCQ

For all $x \in \left( {0,1} \right)$

A

${e^x} < 1 + x$

B

${\log _e}\left( {1 + x} \right) < x$

C

$\sin x > x$

D

${\log _e}x > x$

Question 50: Palette 50/71

JEE Advanced 1999 MCQ

The function $f(x)=$ ${\sin ^4}x + {\cos ^4}x$ increases if

A

$0 < x < \pi /8$

B

$\pi /4 < x < 3\pi /8$

C

$3\pi /8 < x < 5\pi /8$

D

$5\pi /8 < x < 3\pi /4$

Question 51: Palette 51/71

JEE Advanced 1999 MSQ

The function $f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}} $ $dt$ has a local minimum at $x=$

A

$0$

B

$1$

C

$2$

D

$3$

Question 52: Palette 52/71

JEE Advanced 1998 MCQ

If $f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$ for every real number $x$, then the minimum value of $f$

A

does not exist because $f$ is unbounded

B

is not attained even though $f$ is bounded

C

is equal to 1

D

is equal to -1

Question 53: Palette 53/71

JEE Advanced 1998 MCQ

The number of values of $x$ where the function
$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$ attains its maximum is

A

$0$

B

$1$

C

$2$

D

infinite

Question 54: Palette 54/71

JEE Advanced 1998 MSQ

Let $h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$ for every real number $x$. Then

A

$h$ is increasing whenever $f$ is increasing

B

$h$ is increasing whenever $f$ is decreasing

C

$h$ is decreasing whenever $f$ is decreasing

D

nothing can be said in general.

Question 55: Palette 55/71

JEE Advanced 1997 MCQ

If $f\left( x \right) = {x \over {\sin x}}$ and $g\left( x \right) = {x \over {\tan x}}$, where $0 < x \le 1$, then in this interval

A

both $f(x)$ and $g(x)$ are increasing functions

B

both $f(x)$ and $g(x)$ are decreasing functions

C

$f(x)$ is an increasing functions

D

$g(x)$ is an increasing functions

Question 56: Palette 56/71

JEE Advanced 1995 MCQ

The function $f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$ is

A

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

B

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

C

increasing on $\left( {0,\pi /e} \right),$ decreasing on $\left( {\pi /e,\infty } \right)$

D

decreasing on $\left( {0,\pi /e} \right),$ increasing on $\left( {\pi /e,\infty } \right)$

Question 57: Palette 57/71

JEE Advanced 1995 MCQ

The slope of the tangent to a curve $y = f\left( x \right)$ at $\left[ {x,\,f\left( x \right)} \right]$ is $2x+1$. If the curve passes through the point $\left( {1,2} \right)$, then the area bounded by the curve, the $x$-axis and the line $x=1$ is

A

${5 \over 6}$

B

${6 \over 5}$

C

${1 \over 6}$

D

$6$

Question 58: Palette 58/71

JEE Advanced 1995 MCQ

On the interval $\left[ {0,1} \right]$ the function ${x^{25}}{\left( {1 - x} \right)^{75}}$ takes its maximum value at the point

A

$0$

B

${1 \over 4}$

C

${1 \over 2}$

D

${1 \over 3}$

Question 59: Palette 59/71

JEE Advanced 1994 MCQ

Which one of the following curves cut the parabola ${y^2} = 4ax$ at right angles?

A

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

B

$y = {e^{ - x/2a}}$

C

$y = ax$

D

${x^2} = 4ay$

Question 60: Palette 60/71

JEE Advanced 1994 MCQ

The function defined by $f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$

A

decreasing for all $x$

B

decreasing in $\left( { - \infty , - 1} \right)$ and increasing in $\left( { - 1,\infty } \right)$

C

increasing for all $x$

D

decreasing in $\left( { - 1,\infty } \right)$ and increasing in $\left( { - \infty , - 1} \right)$

Question 61: Palette 61/71

JEE Advanced 1993 MSQ

If $f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.$ then:

A

$f(x)$ is increasing on $\left[ { - 1,2} \right]$

B

$f(x)$ is continues on $\left[ { - 1,3} \right]$

C

$f'(2)$ does not exist

D

$f(x)$ has the maximum value at $x=2$

Question 62: Palette 62/71

JEE Advanced 1987 MCQ

Let $f$ and $g$ be increasing and decreasing functions, respectively from $\left[ {0,\infty } \right)$ to $\left[ {0,\infty } \right)$. Let $h\left( x \right) = f\left( {g\left( x \right)} \right).$ If $h\left( 0 \right) = 0,$ then $h\left( x \right) - h\left( 1 \right)$ is

A

always zero

B

always negative

C

always positive

D

strictly increasing

Question 63: Palette 63/71

JEE Advanced 1987 MCQ

The smallest positive root of the equation, $\tan x - x = 0$ lies in

A

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

B

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

C

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

D

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

Question 64: Palette 64/71

JEE Advanced 1986 MCQ

Let $P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}$ be a polynomial in a real variable $x$ with
$0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$ The function $P(x)$ has

A

neither a maximum nor a minimum

B

only one maximum

C

only one minimum

D

only one maximum and only one minimum

Question 65: Palette 65/71

JEE Advanced 1986 MSQ

If the line $ax+by+c=0$ is a normal to the curve $xy=1$, then

A

$a > 0,b > 0$

B

$a > 0,b < 0$

C

$a < 0,b > 0$

D

$a < 0,b < 0$

Question 66: Palette 66/71

JEE Advanced 1984 MCQ

For $0 < a < x,$ the minimum value of the function $lo{g_a}x + {\log _x}a$ is $2$.

A

TRUE

B

FALSE

Question 67: Palette 67/71

JEE Advanced 1983 MCQ

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

A

it makes a constant angle with the $x$-axis

B

it passes through the origin

C

it is at a constant distance from the origin

D

none of these

Question 68: Palette 68/71

JEE Advanced 1983 MCQ

If $a+b+c=0$, then the quadratic equation $3a{x^2} + 2bx + c = 0$ has

A

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

B

one root in $\left[ {2,3} \right]$ and the other in $\left[ {-2,-1} \right]$

C

imaginary roots

D

none of these

Question 69: Palette 69/71

JEE Advanced 1983 MCQ

$AB$ is a diameter of a circle and $C$ is any point on the circumference of the circle. Then

A

the area of $\Delta ABC$ is maximum when it is isosceles

B

the area of $\Delta ABC$ is minimum when it is isosceles

C

the perimeter of $\Delta ABC$ is minimum when it is isosceles

D

none of these

Question 70: Palette 70/71

JEE Advanced 1983 MCQ

If $y = a\,\,In\,x + b{x^2} + x$ has its extreamum values at $x=-1$ and $x=2$, then

A

$a = 2,b = - 1$

B

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

C

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

D

none of these

Question 71: Palette 71/71

JEE Advanced 1983 MCQ

If $x-r$ is a factor of the polynomial $f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$ repeated $m$ times $\left( {1 < m \le n} \right)$, then $r$ is a root of $\left( x \right) = 0$ repeated $m$ times.

A

TRUE

B

FALSE