Application of Derivatives
584 Questions
Start JEE Mains Test
2004
Q501
JEE Mains
MCQ
14 Mar 2026
A point on the parabola ${y^2} = 18x$ at which the ordinate increases at twice the rate of the abscissa is
A.
$\left( {{9 \over 8},{9 \over 2}} \right)$
B.
$(2, -4)$
C.
$\left( {{-9 \over 8},{9 \over 2}} \right)$
D.
$(2, 4)$
2004
Q502
JEE Advanced
MCQ
14 Mar 2026
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.
$1/2$
2004
Q503
JEE Advanced
MCQ
14 Mar 2026
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
2004
Q504
JEE Advanced
Numerical
14 Mar 2026
Prove that for $x \in \left[ {0,{\pi \over 2}} \right],$ $\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$. Explain
the identity if any used in the proof.
the identity if any used in the proof.
Correct Answer: Solve it.
2004
Q505
JEE Advanced
Numerical
14 Mar 2026
Using Rolle's theorem, prove that there is at least one root
in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial
$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$.
in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial
$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$.
Correct Answer: Solve it.
2003
Q506
JEE Mains
MCQ
14 Mar 2026
The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :
A.
-2
B.
2
C.
1
D.
-1
2003
Q507
JEE Mains
MCQ
14 Mar 2026
If the function $f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$ where $a>0,$ attains its maximum and minimum at $p$ and $q$ respectively such that ${p^2} = q$ , then $a$ equals
A.
${1 \over 2}$
B.
$3$
C.
$1$
D.
$2$
2003
Q508
JEE Advanced
MCQ
14 Mar 2026
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.
$f\left( x \right) = \left| x \right|$
2003
Q509
JEE Advanced
MCQ
14 Mar 2026
Tangent is drawn to ellipse
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.
${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$.
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.
$\pi /4$
2003
Q510
JEE Advanced
Numerical
14 Mar 2026
Find a point on the curve ${x^2} + 2{y^2} = 6$ whose distance from
the line $x+y=7$, is minimum.
the line $x+y=7$, is minimum.
Correct Answer: $$(2, 1)$$
2003
Q511
JEE Advanced
Numerical
14 Mar 2026
Using the relation $2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$ or otherwise,
prove that $\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$
prove that $\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$
Correct Answer: Solve it.
2003
Q512
JEE Advanced
Numerical
14 Mar 2026
If the function $f:\left[ {0,4} \right] \to R$ is differentiable then show that
(i)$\,\,\,\,\,$ For $a, b$$\,\,$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$
(ii)$\,\,\,\,\,$ $\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $
(i)$\,\,\,\,\,$ For $a, b$$\,\,$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$
(ii)$\,\,\,\,\,$ $\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $
Correct Answer: Solve it.
2003
Q513
JEE Advanced
Numerical
14 Mar 2026
If $P(1)=0$ and ${{dp\left( x \right)} \over {dx}} > P\left( x \right)$ for all $x \ge 1$ then prove that
$P(x)>0$ for all $x>1$.
$P(x)>0$ for all $x>1$.
Correct Answer: Solve it.
2002
Q514
JEE Mains
MCQ
14 Mar 2026
The maximum distance from origin of a point on the curve
$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$
$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$ both $a,b > 0$ is
$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$
$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$ both $a,b > 0$ is
A.
$a-b$
B.
$a+b$
C.
$\sqrt {{a^2} + {b^2}} $
D.
$\sqrt {{a^2} - {b^2}} $
2002
Q515
JEE Mains
MCQ
14 Mar 2026
If $2a+3b+6c=0,$ $\left( {a,b,c \in R} \right)$ then the quadratic equation $a{x^2} + bx + c = 0$ has
A.
at least one root in $\left[ {0,1} \right]$
B.
at least one root in $\left[ {2,3} \right]$
C.
at least one root in $\left[ {4,5} \right]$
D.
none of these
2002
Q516
JEE Advanced
MCQ
14 Mar 2026
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 $
2002
Q517
JEE Advanced
MCQ
14 Mar 2026
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.
$\left( { \pm {4 \over {\sqrt 3 }}, 2} \right)$
2001
Q518
JEE Advanced
MCQ
14 Mar 2026
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.
$\left( {0,\,1} \right]$
2001
Q519
JEE Advanced
MCQ
14 Mar 2026
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.
$1$
2001
Q520
JEE Advanced
MCQ
14 Mar 2026
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 $\left[ { - 1/2,1} \right]$
2001
Q521
JEE Advanced
Numerical
14 Mar 2026
Let $ - 1 \le p \le 1$. Show that the equation $4{x^3} - 3x - p = 0$
has a unique root in the interval $\left[ {1/2,\,1} \right]$ and identify it.
has a unique root in the interval $\left[ {1/2,\,1} \right]$ and identify it.
Correct Answer: Solve it.
2000
Q522
JEE Advanced
MCQ
14 Mar 2026
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 ?
$S:$ $\,\,\,$$ Both $\sin \,\,x$ and $\cos \,\,x$ are decreasing functions in the interval $\left( {{\pi \over 2},\pi } \right)$
$R:$$\,\,\,$ 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
2000
Q523
JEE Advanced
MCQ
14 Mar 2026
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.
$\left( {2, + \infty } \right)$
2000
Q524
JEE Advanced
MCQ
14 Mar 2026
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
2000
Q525
JEE Advanced
MCQ
14 Mar 2026
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.
$1$
2000
Q526
JEE Advanced
MCQ
14 Mar 2026
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.
${\log _e}x > x$
2000
Q527
JEE Advanced
Numerical
14 Mar 2026
Suppose $p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$ If
$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$ for all $x \ge 0$, prove that
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$ for all $x \ge 0$, prove that
$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$.
Correct Answer: Solve it.
1999
Q528
JEE Advanced
MCQ
14 Mar 2026
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.
$5\pi /8 < x < 3\pi /4$
1999
Q529
JEE Advanced
MSQ
14 Mar 2026
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.
$3$
1998
Q530
JEE Advanced
MCQ
14 Mar 2026
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
1998
Q531
JEE Advanced
MCQ
14 Mar 2026
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
$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$ attains its maximum is
A.
$0$
B.
$1$
C.
$2$
D.
infinite
1998
Q532
JEE Advanced
MSQ
14 Mar 2026
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.
1998
Q533
JEE Advanced
Numerical
14 Mar 2026
Suppose $f(x)$ is a function satisfying the following conditions
(a) $f(0)=2,f(1)=1$,
(b) $f$has a minimum value at $x=5/2$, and
(c) for all $x$, $$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$
where $a,b$ are some constants. Determine the constants $a, b$ and the function $f(x)$.
(a) $f(0)=2,f(1)=1$,
(b) $f$has a minimum value at $x=5/2$, and
(c) for all $x$, $$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$
where $a,b$ are some constants. Determine the constants $a, b$ and the function $f(x)$.
Correct Answer: $$a = {1 \over 4},\,\,b = - {5 \over 4}$$
<br>$$f\left( x \right) = {1 \over 4}{x^2} - {5 \over 4}x + 2$$
1998
Q534
JEE Advanced
Numerical
14 Mar 2026
A curve $C$ has the property that if the tangent drawn at any point $P$ on $C$ meets the co-ordinate axes at $A$ and $B$, then $P$ is the mid-point of $AB$. The curve passes through the point $(1, 1)$. Determine the equation of the curve.
Correct Answer: $$xy=1$$
1997
Q535
JEE Advanced
MCQ
14 Mar 2026
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
1997
Q536
JEE Advanced
Numerical
14 Mar 2026
Let $a+b=4$, where $a<2,$ and let $g(x)$ be a differentiable function.
If ${{dg} \over {dx}} > 0$ for all $x$, prove that $\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $
increases as $(b-a)$ increases.
Correct Answer: Solve it.
1996
Q537
JEE Advanced
Numerical
14 Mar 2026
Let $f\left( x \right) = \left\{ {\matrix{
{x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr
{x + a{x^2} - {x^3},\,x > 0} \cr
} } \right.$
Where a is a positive constant. Find the interval in which $f'(x)$ is increasing.
Correct Answer: $$\left( { - {2 \over a},{a \over 3}} \right)$$
1996
Q538
JEE Advanced
Numerical
14 Mar 2026
Determine the points of maxima and minima of the function
$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$ where $b \ge 0$ is a constant.
$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$ where $b \ge 0$ is a constant.
Correct Answer: min at $$x = {1 \over 4}\left( {b + \sqrt {{b^2} - 1} } \right)$$
<br>max at $$x = {1 \over 4}\left( {b - \sqrt {{b^2} - 1} } \right)$$
1996
Q539
JEE Advanced
Numerical
14 Mar 2026
A curve $y=f(x)$ passes through the point $P(1, 1)$. The normal to the curve at $P$ is $a(y-1)+(x-1)=0$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $y$-axis, the curve and the normal to the curve at $P$.
Correct Answer: $$y = {e^{a\left( {x - 1} \right)}}$$
<br> Area $$=$$ $$1$$ sq. unit.
1995
Q540
JEE Advanced
MCQ
14 Mar 2026
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 $\left( {\pi /e,\infty } \right)$
1995
Q541
JEE Advanced
MCQ
14 Mar 2026
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.
$6$
1995
Q542
JEE Advanced
MCQ
14 Mar 2026
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.
${1 \over 3}$
1995
Q543
JEE Advanced
Numerical
14 Mar 2026
Let $(h, k)$ be a fixed point, where $h > 0,k > 0.$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $P$ and $Q$. Find the minimum area of the triangle $OPQ$, $O$ being the origin.
Correct Answer: $$2$$ $$kh$$
1994
Q544
JEE Advanced
MCQ
14 Mar 2026
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.
${x^2} = 4ay$
1994
Q545
JEE Advanced
MCQ
14 Mar 2026
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 $\left( { - \infty , - 1} \right)$
1994
Q546
JEE Advanced
Numerical
14 Mar 2026
The circle ${x^2} + {y^2} = 1$ cuts the $x$-axis at $P$ and $Q$. Another circle with centre at $Q$ and variable radius intersects the first circle at $R$ above the $x$-axis and the line segment $PQ$ at $S$. Find the maximum area of the triangle $QSR$.
Correct Answer: $${{4\sqrt 3 } \over 9}$$ sq. units
1994
Q547
JEE Advanced
Numerical
14 Mar 2026
The curve $y = a{x^3} + b{x^2} + cx + 5$, touches the $x$-axis at $P(-2, 0)$ and cuts the $y$ axis at a point $Q$, where its gradient is $3$. Find $a, b, c$.
Correct Answer: $$a = - {1 \over 2},b = - {3 \over 4},c = 3$$
1994
Q548
JEE Advanced
Numerical
14 Mar 2026
Let $P$ be a variable point on the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ with foci ${F_1}$ and ${F_2}$. If $A$ is the area of the triangle $P{F_1}{F_2}$ then the maximum value of $A$ is ..........
Correct Answer: $$abc$$
1994
Q549
JEE Advanced
Numerical
14 Mar 2026
Let $C$ be the curve ${y^3} - 3xy + 2 = 0$. If $H$ is the set of points on the curve $C$ where the tangent is horizontal and $V$ is the set of the point on the curve $C$ where the tangent is vertical then $H=$.............. and $V=$ .................
Correct Answer: $$H = \phi $$
<br>$$V = \left\{ {\left( {1,1} \right)} \right\}$$
1993
Q550
JEE Advanced
MSQ
14 Mar 2026
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 $x=2$