## Application of Derivatives

### Mathematics

#### (Past Years Questions)

More
The number of real roots of the equation $${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :
JEE Main 2021 (Online) 31st August Morning Shift
A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from ...
JEE Main 2021 (Online) 27th August Evening Shift
A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a r...
JEE Main 2021 (Online) 27th August Morning Shift
Let 'a' be a real number such that the function f(x) = ax2 + 6x $$-$$ 15, x $$\in$$ R is increasing in $$\left( { - \inf... JEE Main 2021 (Online) 20th July Morning Shift The maximum value of$$f(x) = \left| {\matrix{ {{{\sin }^2}x} &amp; {1 + {{\cos }^2}x} &amp; {\cos 2x} \cr {1 + ...
JEE Main 2021 (Online) 16th March Evening Shift
Let slope of the tangent line to a curve at any point P(x, y) be given by $${{x{y^2} + y} \over x}$$. If the curve inter...
JEE Main 2021 (Online) 26th February Evening Shift
Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2...
JEE Main 2021 (Online) 26th February Evening Shift
The maximum slope of the curve $$y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$$ occurs at the point :
JEE Main 2021 (Online) 26th February Morning Shift
The minimum value of $$f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$$, where a, $$x \in R$$ and a &gt; 0, is equal to :
JEE Main 2021 (Online) 25th February Evening Shift
If the curves, $${{{x^2}} \over a} + {{{y^2}} \over b} = 1$$ and $${{{x^2}} \over c} + {{{y^2}} \over d} = 1$$ intersect...
JEE Main 2021 (Online) 25th February Morning Slot
If the curve y = ax2 + bx + c, x$$\in$$R, passes through the point (1, 2) and the tangent line to this curve at origin...
JEE Main 2021 (Online) 24th February Evening Slot
Let $$f:R \to R$$ be defined as$$f(x) = \left\{ {\matrix{ { - 55x,} &amp; {if\,x &lt; - 5} \cr {2{x^3} - 3{x^2}... JEE Main 2021 (Online) 24th February Evening Slot For which of the following curves, the line$$x + \sqrt 3 y = 2\sqrt 3 $$is the tangent at the point$$\left( {{{3\sqrt...
JEE Main 2021 (Online) 24th February Evening Slot
If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point whic...
JEE Main 2021 (Online) 24th February Morning Slot
The function f(x) = $${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$$ :
JEE Main 2021 (Online) 24th February Morning Slot
If the tangent to the curve, y = f (x) = xloge x, (x &gt; 0) at a point (c, f(c)) is parallel to the line-segment joinin...
JEE Main 2020 (Online) 6th September Evening Slot
The set of all real values of $$\lambda$$ for which the function $$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lam... JEE Main 2020 (Online) 6th September Evening Slot The position of a moving car at time t is given by f(t) = at2 + bt + c, t &gt; 0, where a, b and c are real numbers grea... JEE Main 2020 (Online) 6th September Morning Slot If x = 1 is a critical point of the function f(x) = (3x2 + ax – 2 – a)ex , then : JEE Main 2020 (Online) 5th September Evening Slot Which of the following points lies on the tangent to the curve x4ey + 2$$\sqrt {y + 1} $$= 3 at the point (1, 0)?... JEE Main 2020 (Online) 5th September Evening Slot If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to: ... JEE Main 2020 (Online) 5th September Morning Slot The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie ... JEE Main 2020 (Online) 4th September Evening Slot The minimum value of 2sinx + 2cosx is: JEE Main 2020 (Online) 4th September Evening Slot If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of it... JEE Main 2020 (Online) 3rd September Evening Slot The function, f(x) = (3x – 7)x2/3, x$$ \in $$R, is increasing for all x lying in : JEE Main 2020 (Online) 3rd September Morning Slot The equation of the normal to the curve y = (1+x)2y + cos 2(sin–1x) at x = 0 is : JEE Main 2020 (Online) 2nd September Evening Slot Let f : (–1,$$\infty $$)$$ \to $$R be defined by f(0) = 1 and f(x) =$${1 \over x}{\log _e}\left( {1 + x} \right)$$, ... JEE Main 2020 (Online) 2nd September Evening Slot If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; ... JEE Main 2020 (Online) 2nd September Morning Slot If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining$$\left( {0,{3 \over 2}} \ri...
JEE Main 2020 (Online) 2nd September Morning Slot
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal t...
JEE Main 2020 (Online) 2nd September Morning Slot
A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/m...
JEE Main 2020 (Online) 9th January Morning Slot
A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec...
JEE Main 2019 (Online) 12th April Morning Slot
If m is the minimum value of k for which the function f(x) = x$$\sqrt {kx - {x^2}}$$ is increasing in the interval [0,3...
JEE Main 2019 (Online) 12th April Morning Slot
If the tangent to the curve $$y = {x \over {{x^2} - 3}}$$ , $$x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)$$, at a p...
JEE Main 2019 (Online) 10th April Evening Slot
A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3 ...
JEE Main 2019 (Online) 10th April Evening Slot
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $${\tan ^{ - 1}}\left( {{1 \... JEE Main 2019 (Online) 9th April Evening Slot If the tangent to the curve, y = x3 + ax – b at the point (1, –5) is perpendicular to the line, –x + y + 4 = 0, then whi... JEE Main 2019 (Online) 9th April Morning Slot Let S be the set of all values of x for which the tangent to the curve y = ƒ(x) = x3 – x2 – 2x at (x, y) is parallel to... JEE Main 2019 (Online) 9th April Morning Slot If ƒ(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x$$ \i...
JEE Main 2019 (Online) 9th April Morning Slot
The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is
JEE Main 2019 (Online) 8th April Evening Slot
Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $$2y \over x^2$$. If the curve passes th...
JEE Main 2019 (Online) 8th April Evening Slot
Let ƒ : [0, 2] $$\to$$ R be a twice differentiable function such that ƒ''(x) &gt; 0, for all x $$\in$$ (0, 2). If $$... JEE Main 2019 (Online) 8th April Morning Slot If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, ƒ(x) = 9x4 + 12x3 – 3... JEE Main 2019 (Online) 8th April Morning Slot The tangent to the curve y = x2 – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point : JEE Main 2019 (Online) 12th January Evening Slot Let f(x) =$${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$$... JEE Main 2019 (Online) 11th January Evening Slot If y(x) is the solution of the differential equation&nbsp;$${{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {...
JEE Main 2019 (Online) 11th January Morning Slot
The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = $$\left\{ {x\, \in R:{x^2} + 30 \le 11x} ... JEE Main 2019 (Online) 11th January Morning Slot The tangent to the curve, y = xex2 passing through the point (1, e) also passes through the point JEE Main 2019 (Online) 10th January Evening Slot The shortest distance between the point&nbsp;&nbsp;$$\left( {{3 \over 2},0} \right)$$&nbsp;&nbsp;and the curve y =$$\s...
JEE Main 2019 (Online) 10th January Morning Slot
If &nbsp;$${{dy} \over {dx}} + {3 \over {{{\cos }^2}x}}y = {1 \over {{{\cos }^2}x}},\,\,x \in \left( {{{ - \pi } \over 3... JEE Main 2019 (Online) 10th January Morning Slot Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3$$-$$9x2 ... JEE Main 2018 (Online) 16th April Morning Slot If$$\beta $$is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cos$$\theta $$,$$...
JEE Main 2018 (Online) 15th April Morning Slot
If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (...
JEE Main 2018 (Online) 15th April Morning Slot
The function f defined by f(x) = x3 $$-$$ 3x2 + 5x + 7 , is :
JEE Main 2017 (Online) 9th April Morning Slot
A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the...
JEE Main 2017 (Online) 9th April Morning Slot
The tangent at the point (2, $$-$$2) to the curve, x2y2 $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point : ...
JEE Main 2017 (Online) 8th April Morning Slot
Let C be a curve given by y(x) = 1 + $$\sqrt {4x - 3} ,x &gt; {3 \over 4}.$$ If P is a point on C, such that the tangent...
JEE Main 2016 (Online) 10th April Morning Slot
The minimum distance of a point on the curve y = x2−4 from the origin is :
JEE Main 2016 (Online) 9th April Morning Slot
If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3−1, t $$\in$$ R, meets the curve again...
JEE Main 2016 (Online) 9th April Morning Slot
Consider : f $$\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left(... JEE Main 2016 (Offline) A wire of length$$2$$units is cut into two parts which are bent respectively to form a square of side$$=x$$units and... JEE Main 2016 (Offline) Let$$f(x)$$be a polynomial of degree four having extreme values at$$x=1$$and$$x=2$$. If$$\mathop {\lim }\limits_{...
JEE Main 2015 (Offline)
If $$f$$ and $$g$$ are differentiable functions in $$\left[ {0,1} \right]$$ satisfying $$f\left( 0 \right) = 2 = g\left... JEE Main 2014 (Offline) The intercepts on$$x$$-axis made by tangents to the curve,$$y = \int\limits_0^x {\left| t \right|dt,x \in R,} $$whic... JEE Main 2013 (Offline) A line is drawn through the point$$(1, 2)$$to meet the coordinate axes at$$P$$and$$Q$$such that it forms a triang... AIEEE 2012 Let$$a,b \in R$$be such that the function$$f$$given by$$f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \n...
AIEEE 2012
A spherical balloon is filled with $$4500\pi$$ cubic meters of helium gas. If a leak in the balloon causes the gas to e...
AIEEE 2012
The shortest distance between line $$y-x=1$$ and curve $$x = {y^2}$$ is
AIEEE 2011
For $$x \in \left( {0,{{5\pi } \over 2}} \right),$$ define $$f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.} ... AIEEE 2011 Let$$f:R \to R$$be a continuous function defined by$$$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$Stateme... AIEEE 2010 The equation of the tangent to the curve$$y = x + {4 \over {{x^2}}}$$, that is parallel to the$$x$$-axis, is AIEEE 2010 Let$$f:R \to R$$be defined by$$$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} &amp; {x \le - 1} \cr ...
AIEEE 2010
Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only real root of $$P'\,\left( ... AIEEE 2009 Let$$f\left( x \right) = x\left| x \right|$$and$$g\left( x \right) = \sin x.$$Statement-1: gof is differentiable at ... AIEEE 2009 Suppose the cubic$${x^3} - px + q$$has three distinct real roots where$$p&gt;0$$and$$q&gt;0$$. Then which one of t... AIEEE 2008 How many real solutions does the equation$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$have? AIEEE 2008 If$$p$$and$$q$$are positive real numbers such that$${p^2} + {q^2} = 1$$, then the maximum value of$$(p+q)$$is AIEEE 2007 A value of$$c$$for which conclusion of Mean Value Theorem holds for the function$$f\left( x \right) = {\log _e}x$$on... AIEEE 2007 The function$$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$is an incresing function in AIEEE 2007 A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides havi... AIEEE 2006 The function$$f\left( x \right) = {x \over 2} + {2 \over x}$$has a local minimum at AIEEE 2006 If the equation$${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0{a_1} \ne 0,n \ge 2,$$has a positi... AIEEE 2005 A spherical iron ball$$10$$cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of$$5...
AIEEE 2005
The normal to the curve $$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \... AIEEE 2005 Area of the greatest rectangle that can be inscribed in the ellipse$${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}}...
AIEEE 2005
The normal to the curve x = a(1 + cos $$\theta$$), $$y = a\sin \theta$$ at $$'\theta '$$ always passes through the fix...
AIEEE 2004
If $$2a+3b+6c=0$$, then at least one root of the equation $$a{x^2} + bx + c = 0$$ lies in the interval
AIEEE 2004
A function $$y=f(x)$$ has a second order derivative $$f''\left( x \right) = 6\left( {x - 1} \right).$$ If its graph pass...
AIEEE 2004
A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
AIEEE 2004
If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a&gt;0,$$ attains its maximum and minim...
AIEEE 2003
If $$2a+3b+6c=0,$$ $$\left( {a,b,c \in R} \right)$$ then the quadratic equation $$a{x^2} + bx + c = 0$$ has
AIEEE 2002
The maximum distance from origin of a point on the curve $$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$ $$y = ... AIEEE 2002 ## Numerical More The number of distinct real roots of the equation 3x4 + 4x3$$-$$12x2 + 4 = 0 is _____________.... JEE Main 2021 (Online) 27th August Morning Shift A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a... JEE Main 2021 (Online) 26th August Morning Shift Let f : [$$-$$1, 1]$$ \to $$R be defined as f(x) = ax2 + bx + c for all x$$\in$$[$$-$$1, 1], where a, b, c$$\in$$R suc... JEE Main 2021 (Online) 17th March Evening Shift Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interva... JEE Main 2021 (Online) 26th February Evening Shift If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________. JEE Main 2021 (Online) 25th February Evening Shift The minimum value of$$\alpha $$for which the equation$${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha $$has a... JEE Main 2021 (Online) 24th February Morning Slot Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m... JEE Main 2020 (Online) 6th September Morning Slot If the lines x + y = a and x – y = b touch the curve y = x2 – 3x + 2 at the points where the curve intersects the x-axi... JEE Main 2020 (Online) 5th September Evening Slot Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at$$\left( {0,{3 \over 2}} \right)\$...
JEE Main 2020 (Online) 8th January Morning Slot