Application of Derivatives · Mathematics · JEE Advanced

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MCQ (Single Correct Answer)

JEE Advanced 2023 Paper 1 Online
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 se...
JEE Advanced 2020 Paper 1 Offline
Consider the rectangles lying the region $$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le...
JEE Advanced 2017 Paper 1 Offline
Which of the following options is the only INCORRECT combination?
JEE Advanced 2017 Paper 1 Offline
Which of the following options is the only CORRECT combination?
JEE Advanced 2017 Paper 1 Offline
Which of the following options is the only CORRECT combination?
JEE Advanced 2016 Paper 1 Offline
The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is
JEE Advanced 2013 Paper 2 Offline
Let $$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$...
JEE Advanced 2013 Paper 2 Offline
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable, $$f(0) = f(1)=0$...
IIT-JEE 2012 Paper 2 Offline
Let $$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 {...
IIT-JEE 2012 Paper 2 Offline
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let $$g\left( x \right) = \int\limits_1^x {...
IIT-JEE 2008 Paper 1 Offline
The total number of local maxima and local minima of the function $$f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 ...
IIT-JEE 2007
The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c -...
IIT-JEE 2007
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i...
IIT-JEE 2007
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i...
IIT-JEE 2007
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root i...
IIT-JEE 2005 Screening
If $$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...
IIT-JEE 2004 Screening
If $$f\left( x \right) = {x^3} + b{x^2} + cx + d$$ and $$0 < {b^2} < c,$$ then in $$\left( { - \infty ,\infty } \right)$$
IIT-JEE 2004 Screening
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 $$\lef...
IIT-JEE 2003 Screening
In $$\left[ {0,1} \right]$$ Languages Mean Value theorem is NOT applicable to
IIT-JEE 2003 Screening
Tangent is drawn to ellipse $${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \...
IIT-JEE 2002 Screening
The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is
IIT-JEE 2002 Screening
The point(s) in the curve $${y^3} + 3{x^2} = 12y$$ where the tangent is vertical, is (are)
IIT-JEE 2001 Screening
If $$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$ then $$f(x)$$ is
IIT-JEE 2001 Screening
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 fi...
IIT-JEE 2001 Screening
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 o...
IIT-JEE 2000 Screening
Consider the following statements in $$S$$ and $$R$$ $$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interv...
IIT-JEE 2000 Screening
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, the...
IIT-JEE 2000 Screening
Let $$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $$ Then $$f$$ decreases in the interval
IIT-JEE 2000 Screening
Let $$f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {...
IIT-JEE 2000 Screening
For all $$x \in \left( {0,1} \right)$$
IIT-JEE 1999
The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if
IIT-JEE 1998
The number of values of $$x$$ where the function $$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is
IIT-JEE 1998
If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$
IIT-JEE 1997
If $$f\left( x \right) = {x \over {\sin x}}$$ and $$g\left( x \right) = {x \over {\tan x}}$$, where $$0 < x \le 1$$, then in this interval
IIT-JEE 1995 Screening
The function $$f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$$ is
IIT-JEE 1995 Screening
On the interval $$\left[ {0,1} \right]$$ the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value at the point
IIT-JEE 1995 Screening
The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$. If the curve passes through t...
IIT-JEE 1994
The function defined by $$f\left( x \right) = \left( {x + 2} \right){e^{ - x}}$$
IIT-JEE 1994
Which one of the following curves cut the parabola $${y^2} = 4ax$$ at right angles?
IIT-JEE 1987
The smallest positive root of the equation, $$\tan x - x = 0$$ lies in
IIT-JEE 1987
Let $$f$$ and $$g$$ be increasing and decreasing functions, respectively from $$\left[ {0,\infty } \right)$$ to $$\left[ {0,\infty } \right)$$. Let $$...
IIT-JEE 1986
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 $$0 < {a_0} &l...
IIT-JEE 1983
If $$a+b+c=0$$, then the quadratic equation $$3a{x^2} + 2bx + c = 0$$ has
IIT-JEE 1983
$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then
IIT-JEE 1983
The normal to the curve $$\,x = a\left( {\cos \theta + \theta \sin \theta } \right)$$, $$y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ a...
IIT-JEE 1983
If $$y = a\,\,In\,x + b{x^2} + x$$ has its extreamum values at $$x=-1$$ and $$x=2$$, then

MCQ (More than One Correct Answer)

JEE Advanced 2022 Paper 2 Online
Let $$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$ Let $g:[0,1] \rightarrow \mathbb{R}$ be the function d...
JEE Advanced 2019 Paper 2 Offline
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 > 0Then which of the following opt...
JEE Advanced 2019 Paper 2 Offline
Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 ...
JEE Advanced 2017 Paper 2 Offline
f : R $$ \to $$ R is a differentiable function such that f'(x) > 2f(x) for all x$$ \in $$R, and f(0) = 1 then
JEE Advanced 2017 Paper 2 Offline
If $$f(x) = \left| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} &...
JEE Advanced 2016 Paper 2 Offline
Let f: R $$ \to \left( {0,\infty } \right)$$ and g : R $$ \to $$ R be twice differentiable functions such that f'' and g'' are continuous functions on...
JEE Advanced 2015 Paper 2 Offline
Let $$f, g :$$ $$\left[ { - 1,2} \right] \to R$$ be continuous functions which are twice differentiable on the interval $$(-1, 2)$$. Let the values of...
JEE Advanced 2013 Paper 2 Offline
The function $$f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|$$ has a local minimum o...
JEE Advanced 2013 Paper 1 Offline
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $$8:15$$ is converted into an open rectangular box by folding afte...
IIT-JEE 2012 Paper 2 Offline
If $$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),$$ then
IIT-JEE 2009 Paper 2 Offline
For the function $$$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,$$$
IIT-JEE 2006
$$f(x)$$ is cubic polynomial with $$f(2)=18$$ and $$f(1)=-1$$. Also $$f(x)$$ has local maxima at $$x=-1$$ and $$f'(x)$$ has local minima at $$x=0$$, t...
IIT-JEE 2006
Let $$f\left( x \right) = \left\{ {\matrix{ {{e^x},} & {0 \le x \le 1} \cr {2 - {e^{x - 1}},} & {1 < x \le 2} \cr {x - e,} &am...
IIT-JEE 1999
The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {...
IIT-JEE 1998
Let $$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$$ for every real number ...
IIT-JEE 1993
If $$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...
IIT-JEE 1986
If the line $$ax+by+c=0$$ is a normal to the curve $$xy=1$$, then

Numerical

JEE Advanced 2018 Paper 1 Offline
For each positive integer n, let $${y_n} = {1 \over n}(n + 1)(n + 2)...{(n + n)^{{1 \over n}}}$$. For x$$ \in $$R, let [x] be the greatest integer les...
JEE Advanced 2015 Paper 1 Offline
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V$$ $$m{m^3}$$, has...
JEE Advanced 2014 Paper 1 Offline
The slope of the tangent to the curve $${\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}$$ at the point $$(1, 3)$$ is
IIT-JEE 2012 Paper 1 Offline
Let $$f:IR \to IR$$ be defined as $$f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.$$ The total number of points at which $$f$$ att...
IIT-JEE 2012 Paper 1 Offline
Let $$p(x)$$ be a real polynomial of least degree which has a local maximum at $$x=1$$ and a local minimum at $$x=3$$. If $$p(1)=6$$ and $$p(3)=2$$, t...
IIT-JEE 2010 Paper 1 Offline
Let $$f$$ be a real-valued differentiable function on $$R$$ (the set of all real numbers) such that $$f(1)=1$$. If the $$y$$-intercept of the tangent ...
IIT-JEE 2010 Paper 2 Offline
Let $$f$$ be a function defined on $$R$$ (the set of all real numbers) such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 201...
IIT-JEE 2009 Paper 2 Offline
Let $$p(x)$$ be a polynomial of degree $$4$$ having extremum at $$x = 1,2$$ and $$\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \...
IIT-JEE 2009 Paper 2 Offline
The maximum value of the function $$f(x) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set $$A = \{ x|{x^2} + 20 \le 9x|\} $$ is __________.

Subjective

IIT-JEE 2006
For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \righ...
IIT-JEE 2005
If $$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$$ for all $${x_1},{x_2} \in R$$. Fin...
IIT-JEE 2005
If $$p(x)$$ be a polynomial of degree $$3$$ satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maxima at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. ...
IIT-JEE 2004
Using Rolle's theorem, prove that there is at least one root in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial $$P\left( x \right) = 51{x^...
IIT-JEE 2004
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 ...
IIT-JEE 2003
Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise, prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \i...
IIT-JEE 2003
Find a point on the curve $${x^2} + 2{y^2} = 6$$ whose distance from the line $$x+y=7$$, is minimum.
IIT-JEE 2003
If the function $$f:\left[ {0,4} \right] \to R$$ is differentiable then show that (i)$$\,\,\,\,\,$$ For $$a, b$$$$\,\,$$$$ \in \left( {0,4} \right),{...
IIT-JEE 2003
If $$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$$....
IIT-JEE 2001
Let $$ - 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]$$ and identify ...
IIT-JEE 2000
Suppose $$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}...
IIT-JEE 1998
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 ...
IIT-JEE 1998
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...
IIT-JEE 1997
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 $$\int_0^a {...
IIT-JEE 1996
Let $$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.$$ Where...
IIT-JEE 1996
A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any po...
IIT-JEE 1996
Determine the points of maxima and minima of the function $$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a ...
IIT-JEE 1995
Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordina...
IIT-JEE 1994
The curve $$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...
IIT-JEE 1994
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 ...
IIT-JEE 1993
Find the equation of the normal to the curve $$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$
IIT-JEE 1993
Let $$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & ...
IIT-JEE 1992
A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14}...
IIT-JEE 1992
What normal to the curve $$y = {x^2}$$ forms the shortest chord?
IIT-JEE 1992
In this questions there are entries in columns $$I$$ and $$II$$. Each entry in column $$I$$ is related to exactly one entry in column $$II$$. Write th...
IIT-JEE 1991
A window of perimeter $$P$$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is f...
IIT-JEE 1990
Show that $$2\sin x + \tan x \ge 3x$$ where $$0 \le x < {\pi \over 2}$$.
IIT-JEE 1990
A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P$$. Determine the maximum possi...
IIT-JEE 1989
Find all maxima and minima of the function $$$y = x{\left( {x - 1} \right)^2},0 \le x \le 2$$$ Also determine the area bounded by the curve $$y = x{\...
IIT-JEE 1988
Investigate for maxima and minimum the function $$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} ...
IIT-JEE 1987
Find the point on the curve $$\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8$$ that is farthest from the point $$(0, -2)$$.
IIT-JEE 1985
Find all the tangents to the curve $$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.
IIT-JEE 1985
Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$ Find the intervals in which $$\lambda $$ sho...
IIT-JEE 1983
Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$ for all $$x \ge 0$$
IIT-JEE 1983
Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$ where the tangent to the curve has the greatest slope.
IIT-JEE 1982
If $$f(x)$$ and $$g(x)$$ are differentiable function for $$0 \le x \le 1$$ such that $$f(0)=2$$, $$g(0)=0$$, $$f(1)=6$$; $$g(1)=2$$, then show that th...
IIT-JEE 1982
If $$a{x^2} + {b \over x} \ge c$$ for all positive $$x$$ where $$a>0$$ and $$b>0$$ show that $$27a{b^2} \ge 4{c^3}$$.
IIT-JEE 1981
Use the function $$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers $${e^\pi }$$ and $${\pi ^e}$$
IIT-JEE 1981
Let $$x$$ and $$y$$ be two real variables such that $$x>0$$ and $$xy=1$$. Find the minimum value of $$x+y$$.
IIT-JEE 1981
For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)...
IIT-JEE 1979
Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$ $$a,b > c,x > - c$$ is $${...

Fill in the Blanks

IIT-JEE 1994
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 ...
IIT-JEE 1994
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 ...
IIT-JEE 1987
The set of all $$x$$ for which $$in\left( {1 + x} \right) \le x$$ is equal to ..........
IIT-JEE 1983
The larger of $$\cos \left( {In\,\,\theta } \right)$$ and $$In $$ $$\left( {\cos \,\,\theta } \right)$$ If $${e^{ - \pi /2}} < \theta < {\pi ...
IIT-JEE 1983
The function $$y = 2{x^2} - In\,\left| x \right|$$ is monotonically increasing for values of $$x\left( {x \ne 0} \right)$$ satisfying the inequalities...

True or False

IIT-JEE 1984
For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$.
IIT-JEE 1983
If $$x-r$$ is a factor of the polynomial $$f\left( x \right) = {a_n}{x^4} + ..... + {a_0},$$ repeated $$m$$ times $$\left( {1 < m \le n} \right)$$,...
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