Application of Derivatives · Mathematics · JEE Main

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MCQ (Single Correct Answer)

JEE Main 2024 (Online) 1st February Morning Shift
If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :
JEE Main 2024 (Online) 31st January Evening Shift
Let $$f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$$ be strictly increasing function such that $$\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}...
JEE Main 2024 (Online) 31st January Evening Shift
If the function $$f:(-\infty,-1] \rightarrow(a, b]$$ defined by $$f(x)=e^{x^3-3 x+1}$$ is one - one and onto, then the distance of the point $$P(2 b+4...
JEE Main 2024 (Online) 31st January Morning Shift
$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } ...
JEE Main 2024 (Online) 30th January Evening Shift
Let $$f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value o...
JEE Main 2024 (Online) 30th January Morning Shift
The maximum area of a triangle whose one vertex is at $$(0,0)$$ and the other two vertices lie on the curve $$y=-2 x^2+54$$ at points $$(x, y)$$ and $...
JEE Main 2024 (Online) 29th January Evening Shift
The function $$f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$$
JEE Main 2024 (Online) 29th January Evening Shift
The function $$f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$$, has
JEE Main 2024 (Online) 29th January Morning Shift
Consider the function $$f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$$ defined by $$f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$$. Consider the stateme...
JEE Main 2024 (Online) 27th January Evening Shift
Let $$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime}(x)>0$$ for all $$x \in(0,3)$$. If $$g$$ is decreasing in $$(0, \alpha)$$ and i...
JEE Main 2023 (Online) 13th April Morning Shift
$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$
JEE Main 2023 (Online) 12th April Morning Shift
If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $$...
JEE Main 2023 (Online) 11th April Morning Shift
Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)...
JEE Main 2023 (Online) 10th April Evening Shift
Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime}(x) > 0, x \in(0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0, a)$$ and increas...
JEE Main 2023 (Online) 10th April Morning Shift
The slope of tangent at any point (x, y) on a curve $$y=y(x)$$ is $${{{x^2} + {y^2}} \over {2xy}},x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ i...
JEE Main 2023 (Online) 10th April Morning Shift
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. ...
JEE Main 2023 (Online) 1st February Evening Shift
The sum of the absolute maximum and minimum values of the function $$f(x)=\left|x^{2}-5 x+6\right|-3 x+2$$ in the interval $$[-1,3]$$ is equal to :...
JEE Main 2023 (Online) 31st January Morning Shift
A wire of length $$20 \mathrm{~m}$$ is to be cut into two pieces. A piece of length $$l_{1}$$ is bent to make a square of area $$A_{1}$$ and the other...
JEE Main 2023 (Online) 30th January Evening Shift
If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7...
JEE Main 2023 (Online) 30th January Morning Shift
The number of points on the curve $$y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$$ at which the normal lines are parallel to $$x+90 y+2=0$$ is :...
JEE Main 2023 (Online) 25th January Evening Shift
Let the function $$f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$$ have a maxima for some value of $$x 0$$. Then, the set of all values of p is...
JEE Main 2023 (Online) 25th January Morning Shift
Let $$x=2$$ be a local minima of the function $$f(x)=2x^4-18x^2+8x+12,x\in(-4,4)$$. If M is local maximum value of the function $$f$$ in ($$-4,4)$$, t...
JEE Main 2023 (Online) 25th January Morning Shift
Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two...
JEE Main 2022 (Online) 29th July Morning Shift
Let $$f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathrm{R}$$. Then which of the following statements are true? $$\mathrm{P}: x=0$$ is a point of loca...
JEE Main 2022 (Online) 28th July Evening Shift
The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :
JEE Main 2022 (Online) 28th July Morning Shift
If the minimum value of $$f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0$$, is 14 , then the value of $$\alpha$$ is equal to :
JEE Main 2022 (Online) 26th July Evening Shift
If the maximum value of $$a$$, for which the function $$f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$$ is non-decreasing in $$\left(-\frac{\pi}{6}, \frac{\pi}{6}\r...
JEE Main 2022 (Online) 25th July Morning Shift
If the absolute maximum value of the function $$f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$$ in the interval $$...
JEE Main 2022 (Online) 25th July Morning Shift
The curve $$y(x)=a x^{3}+b x^{2}+c x+5$$ touches the $$x$$-axis at the point $$\mathrm{P}(-2,0)$$ and cuts the $$y$$-axis at the point $$Q$$, where $$...
JEE Main 2022 (Online) 30th June Morning Shift
If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $$\alpha$$, 0) and (0, 50 + $$\alpha$$), $$\alpha$$ > 0,...
JEE Main 2022 (Online) 30th June Morning Shift
Let $$f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R$$. Then f :
JEE Main 2022 (Online) 29th June Evening Shift
Let f : R $$\to$$ R be a function defined by f(x) = (x $$-$$ 3)n1 (x $$-$$ 5)n2, n1, n2 $$\in$$ N. Then, which of the following is NOT true?...
JEE Main 2022 (Online) 29th June Morning Shift
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, th...
JEE Main 2022 (Online) 28th June Morning Shift
The number of real solutions of $${x^7} + 5{x^3} + 3x + 1 = 0$$ is equal to ____________.
JEE Main 2022 (Online) 26th June Evening Shift
Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r,...
JEE Main 2022 (Online) 26th June Morning Shift
The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x $$-$$ x2 + 2| $$-$$ x in the interval [$$-$$1, 2] is :...
JEE Main 2022 (Online) 26th June Morning Shift
Let S be the set of all the natural numbers, for which the line $${x \over a} + {y \over b} = 2$$ is a tangent to the curve $${\left( {{x \over a}} \r...
JEE Main 2022 (Online) 26th June Morning Shift
Let $$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$$, $$x \in [ - 1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b i...
JEE Main 2022 (Online) 25th June Evening Shift
Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the he...
JEE Main 2022 (Online) 25th June Evening Shift
If the angle made by the tangent at the point (x0, y0) on the curve $$x = 12(t + \sin t\cos t)$$, $$y = 12{(1 + \sin t)^2}$$, $$0 0 is equal to:...
JEE Main 2022 (Online) 24th June Evening Shift
The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by $${{{x^2}} \over {xy - {x^2}{y^2} - 1}}$$. If the curve passes...
JEE Main 2022 (Online) 24th June Evening Shift
Let $$\lambda$$$$^ * $$ be the largest value of $$\lambda$$ for which the function $${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$$ i...
JEE Main 2022 (Online) 24th June Morning Shift
The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and afte...
JEE Main 2022 (Online) 24th June Morning Shift
For the function $$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$$, which one of the following is NOT correct?
JEE Main 2022 (Online) 24th June Morning Shift
If the tangent at the point (x1, y1) on the curve $$y = {x^3} + 3{x^2} + 5$$ passes through the origin, then (x1, y1) does NOT lie on the curve :...
JEE Main 2022 (Online) 24th June Morning Shift
The sum of absolute maximum and absolute minimum values of the function $$f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$$ in the interval [0, 1] is :
JEE Main 2022 (Online) 24th June Morning Shift
Let $$\lambda x - 2y = \mu $$ be a tangent to the hyperbola $${a^2}{x^2} - {y^2} = {b^2}$$. Then $${\left( {{\lambda \over a}} \right)^2} - {\left( {...
JEE Main 2021 (Online) 1st September Evening Shift
The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :Statement 1 : there exists x1, x2 $$\in$$(2...
JEE Main 2021 (Online) 31st August Morning Shift
The number of real roots of the equation $${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :
JEE Main 2021 (Online) 27th August Evening Shift
A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and f...
JEE Main 2021 (Online) 27th August Morning Shift
A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the lengt...
JEE Main 2021 (Online) 26th August Evening Shift
The local maximum value of the function $$f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$$, x > 0, is
JEE Main 2021 (Online) 25th July Morning Shift
Let $$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$$, $$x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$. Then, f is :
JEE Main 2021 (Online) 22th July Evening Shift
Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le ...
JEE Main 2021 (Online) 20th July Evening Shift
The sum of all the local minimum values of the twice differentiable function f : R $$\to$$ R defined by $$f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x...
JEE Main 2021 (Online) 20th July Morning Shift
Let $$A = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {...
JEE Main 2021 (Online) 20th July Morning Shift
Let 'a' be a real number such that the function f(x) = ax2 + 6x $$-$$ 15, x $$\in$$ R is increasing in $$\left( { - \infty ,{3 \over 4}} \right)$$ and...
JEE Main 2021 (Online) 17th March Evening Shift
Consider the function f : R $$ \to $$ R defined by $$f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfi...
JEE Main 2021 (Online) 16th March Evening Shift
Let f be a real valued function, defined on R $$-$$ {$$-$$1, 1} and given by f(x) = 3 loge $$\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - ...
JEE Main 2021 (Online) 16th March Evening Shift
The maximum value of $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^...
JEE Main 2021 (Online) 26th February Evening Shift
Let slope of the tangent line to a curve at any point P(x, y) be given by $${{x{y^2} + y} \over x}$$. If the curve intersects the line x + 2y = 4 at x...
JEE Main 2021 (Online) 26th February Morning Shift
The maximum slope of the curve $$y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$$ occurs at the point :
JEE Main 2021 (Online) 26th February Morning Shift
Let f be any function defined on R and let it satisfy the condition : $$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$$If f(0) = 1, then :...
JEE Main 2021 (Online) 25th February Morning Shift
If the curves, $${{{x^2}} \over a} + {{{y^2}} \over b} = 1$$ and $${{{x^2}} \over c} + {{{y^2}} \over d} = 1$$ intersect each other at an angle of 90$...
JEE Main 2021 (Online) 25th February Morning Shift
If Rolle's theorem holds for the function $$f(x) = {x^3} - a{x^2} + bx - 4$$, $$x \in [1,2]$$ with $$f'\left( {{4 \over 3}} \right) = 0$$, then ordere...
JEE Main 2021 (Online) 24th February Evening Shift
For which of the following curves, the line $$x + \sqrt 3 y = 2\sqrt 3 $$ is the tangent at the point $$\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \ri...
JEE Main 2021 (Online) 24th February Evening Shift
Let $$f:R \to R$$ be defined as$$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le ...
JEE Main 2021 (Online) 24th February Evening Shift
If the curve y = ax2 + bx + c, x$$ \in $$R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible v...
JEE Main 2021 (Online) 24th February Morning Shift
The function f(x) = $${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$$ :
JEE Main 2021 (Online) 24th February Morning Shift
If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the...
JEE Main 2020 (Online) 6th September Evening Slot
The set of all real values of $$\lambda $$ for which the function $$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \...
JEE Main 2020 (Online) 6th September Evening Slot
If the tangent to the curve, y = f (x) = xloge x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1, 0) and (e, e)...
JEE Main 2020 (Online) 6th September Morning Slot
The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average s...
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 x = 1 is a critical point of the function f(x) = (3x2 + ax – 2 – a)ex , then :
JEE Main 2020 (Online) 5th September Morning 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) 4th September Evening 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 on the parabola, y = x2–1 belo...
JEE Main 2020 (Online) 4th September Morning Slot
Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $$ \ge $$ 1 and f''(x) $$ \ge $$ 4, for all x $$ \in $$ (1, 6), then...
JEE Main 2020 (Online) 3rd 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 its volume (in cm3/sec), when th...
JEE Main 2020 (Online) 3rd September Morning Slot
The function, f(x) = (3x – 7)x2/3, x $$ \in $$ R, is increasing for all x lying in :
JEE Main 2020 (Online) 2nd September Evening Slot
Let f : (–1, $$\infty $$) $$ \to $$ R be defined by f(0) = 1 and f(x) = $${1 \over x}{\log _e}\left( {1 + x} \right)$$, x $$ \ne $$ 0. Then the functi...
JEE Main 2020 (Online) 2nd September Evening 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 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 to the curve at P is :...
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 $$\left( {0,{3 \over 2}} \right)$$ and $$\left( {{1 \over ...
JEE Main 2020 (Online) 2nd September Morning Slot
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :
JEE Main 2020 (Online) 9th January Morning Slot
A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/min. When the thickness of ice ...
JEE Main 2020 (Online) 8th January Evening Slot
The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2,2) is
JEE Main 2020 (Online) 8th January Morning Slot
If c is a point at which Rolle's theorem holds for the function, f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$$ in the interval [...
JEE Main 2020 (Online) 8th January Morning Slot
Let ƒ(x) = xcos–1(–sin|x|), $$x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, then which of the following is true?
JEE Main 2020 (Online) 7th January Evening Slot
The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x $$ \in $$ [0, 1] is: ...
JEE Main 2020 (Online) 7th January Evening Slot
Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If $$\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} ...
JEE Main 2020 (Online) 7th January Morning Slot
Let the function, ƒ:[-7, 0]$$ \to $$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $$ \le $$ 2, for all x $$ \in $$...
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] and M is the maximum value o...
JEE Main 2019 (Online) 12th April 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, then the rate (in cm/sec.) a...
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 /min. When the thickness of th...
JEE Main 2019 (Online) 10th April Evening 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 point ($$\alpha $$, $$\beta $$)...
JEE Main 2019 (Online) 9th April Evening Slot
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$. Water is p...
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 $$ \in $$ R : ƒ(x) = ƒ(0)} Contains...
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 the line segment joining the ...
JEE Main 2019 (Online) 9th April Morning 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 which one of the following points...
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 through the centre of the circle...
JEE Main 2019 (Online) 8th April Evening 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 Morning Slot
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, ƒ(x) = 9x4 + 12x3 – 36x2 + 25, x $$ \in $$ R, then ...
JEE Main 2019 (Online) 8th April Morning Slot
Let ƒ : [0, 2] $$ \to $$ R be a twice differentiable function such that ƒ''(x) > 0, for all x $$ \in $$ (0, 2). If $$\phi $$(x) = ƒ(x) + ƒ(2 – x), ...
JEE Main 2019 (Online) 12th January Evening Slot
If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a$$ \in $$R is increasing in (0, 1] and decreasing in [1, 5), then a root of th...
JEE Main 2019 (Online) 12th January Evening 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) 11th January Evening Slot
Let f(x) = $${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$$ x $$\, \in $$ R, where a, b a...
JEE Main 2019 (Online) 11th January Morning Slot
The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = $$\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}$$ is :...
JEE Main 2019 (Online) 10th January Evening 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
A helicopter is flying along the curve given by y – x3/2 = 7, (x $$ \ge $$ 0). A soldier positioned at the point $$\left( {{1 \over 2},7} \right)$$ wa...
JEE Main 2019 (Online) 10th January Morning Slot
The shortest distance between the point  $$\left( {{3 \over 2},0} \right)$$   and the curve y = $$\sqrt x $$, (x > 0), is -
JEE Main 2019 (Online) 9th January Morning Slot
The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :
JEE Main 2018 (Online) 16th April Morning Slot
Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 $$-$$ 9x2 + 12x + 5 in the interval [0, ...
JEE Main 2018 (Offline)
If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :
JEE Main 2018 (Offline)
Let $$f\left( x \right) = {x^2} + {1 \over {{x^2}}}$$ and $$g\left( x \right) = x - {1 \over x}$$, $$x \in R - \left\{ { - 1,0,1} \right\}$$. If $$h\...
JEE Main 2018 (Online) 15th April Morning Slot
If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is : ...
JEE Main 2018 (Online) 15th April Morning Slot
If $$\beta $$ is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cos $$\theta $$, $$\sqrt 3 \sin \theta $$) and (...
JEE Main 2017 (Online) 9th April Morning Slot
A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the...
JEE Main 2017 (Online) 9th April Morning Slot
The function f defined by f(x) = x3 $$-$$ 3x2 + 5x + 7 , is :
JEE Main 2017 (Online) 8th April Morning Slot
The tangent at the point (2, $$-$$2) to the curve, x2y2 $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point :
JEE Main 2017 (Offline)
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, ...
JEE Main 2017 (Offline)
The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point :
JEE Main 2016 (Online) 10th April Morning Slot
Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :
JEE Main 2016 (Online) 10th April Morning Slot
Let C be a curve given by y(x) = 1 + $$\sqrt {4x - 3} ,x > {3 \over 4}.$$ If P is a point on C, such that the tangent at P has slope $${2 \over 3}$...
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 at a point Q, then the coordi...
JEE Main 2016 (Online) 9th April Morning Slot
The minimum distance of a point on the curve y = x2−4 from the origin is :
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 a circle of radius $$=r$$ uni...
JEE Main 2016 (Offline)
Consider : f $$\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right).$$...
JEE Main 2015 (Offline)
The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$
JEE Main 2015 (Offline)
Let $$f(x)$$ be a polynomial of degree four having extreme values at $$x=1$$ and $$x=2$$. If $$\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left(...
JEE Main 2014 (Offline)
If $$x=-1$$ and $$x=2$$ are extreme points of $$f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x$$ then
JEE Main 2014 (Offline)
If $$f$$ and $$g$$ are differentiable functions in $$\left[ {0,1} \right]$$ satisfying $$f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) ...
JEE Main 2013 (Offline)
The real number $$k$$ for which the equation, $$2{x^3} + 3x + k = 0$$ has two distinct real roots in $$\left[ {0,\,1} \right]$$
JEE Main 2013 (Offline)
The intercepts on $$x$$-axis made by tangents to the curve, $$y = \int\limits_0^x {\left| t \right|dt,x \in R,} $$ which are parallel to the line $$y...
AIEEE 2012
A spherical balloon is filled with $$4500\pi $$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $$72\pi $...
AIEEE 2012
Let $$a,b \in R$$ be such that the function $$f$$ given by $$f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0$$ has extreme values at $$...
AIEEE 2012
A line is drawn through the point $$(1, 2)$$ to meet the coordinate axes at $$P$$ and $$Q$$ such that it forms a triangle $$OPQ,$$ where $$O$$ is the...
AIEEE 2011
For $$x \in \left( {0,{{5\pi } \over 2}} \right),$$ define $$f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.} $$ Then $$f$$ has
AIEEE 2011
The shortest distance between line $$y-x=1$$ and curve $$x = {y^2}$$ is
AIEEE 2010
Let $$f:R \to R$$ be defined by $$$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x >...
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 a continuous function defined by $$$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$$ Statement - 1 : $$f\left( c \right) =...
AIEEE 2009
Given $$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( x \right) = 0.$$ If $$P\left( ...
AIEEE 2008
How many real solutions does the equation $${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?
AIEEE 2008
Suppose the cubic $${x^3} - px + q$$ has three distinct real roots where $$p>0$$ and $$q>0$$. Then which one of the following holds?
AIEEE 2007
The function $$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$ is an incresing function in
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 the interval $$\left[ {1,3} \...
AIEEE 2007
If $$p$$ and $$q$$ are positive real numbers such that $${p^2} + {q^2} = 1$$, then the maximum value of $$(p+q)$$ is
AIEEE 2006
Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is
AIEEE 2006
The function $$f\left( x \right) = {x \over 2} + {2 \over x}$$ has a local minimum at
AIEEE 2006
A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$...
AIEEE 2005
Area of the greatest rectangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$
AIEEE 2005
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 ...
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 $$50$$ cm$$^3$$ /min. When the th...
AIEEE 2005
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 positive root $$x = \alpha $$, then ...
AIEEE 2005
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
AIEEE 2005
Let f be differentiable for all x. If f(1) = -2 and f'(x) $$ \ge $$ 2 for x $$ \in \left[ {1,6} \right]$$, then
AIEEE 2005
A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the ins...
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
A function $$y=f(x)$$ has a second order derivative $$f''\left( x \right) = 6\left( {x - 1} \right).$$ If its graph passes through the point $$(2, 1)$...
AIEEE 2004
If $$2a+3b+6c=0$$, then at least one root of the equation $$a{x^2} + bx + c = 0$$ lies in the interval
AIEEE 2004
The normal to the curve x = a(1 + cos $$\theta $$), $$y = a\sin \theta $$ at $$'\theta '$$ always passes through the fixed point
AIEEE 2003
The real number $$x$$ when added to its inverse gives the minimum sum at $$x$$ equal :
AIEEE 2003
If the function $$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$$ respecti...
AIEEE 2002
The maximum distance from origin of a point on the curve $$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$ $$y = a\cos t - b\cos \left( {{{at} ...
AIEEE 2002
If $$2a+3b+6c=0,$$ $$\left( {a,b,c \in R} \right)$$ then the quadratic equation $$a{x^2} + bx + c = 0$$ has

Numerical

JEE Main 2024 (Online) 29th January Morning Shift
Let $$f(x)=2^x-x^2, x \in \mathbb{R}$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y=f(x)$$ and $$y=f^{\prime}(x)$...
JEE Main 2024 (Online) 27th January Morning Shift
Let for a differentiable function $f:(0, \infty) \rightarrow \mathbf{R}, f(x)-f(y) \geqslant \log _{\mathrm{e}}\left(\frac{x}{y}\right)+x-y, \forall x...
JEE Main 2023 (Online) 15th April Morning Shift
Consider the triangles with vertices $A(2,1), B(0,0)$ and $C(t, 4), t \in[0,4]$. If the maximum and the minimum perimeters of such triangles are obta...
JEE Main 2023 (Online) 10th April Evening Shift
Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the...
JEE Main 2023 (Online) 8th April Morning Shift
If $$a_{\alpha}$$ is the greatest term in the sequence $$\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots$$, then $$\alpha$$ is equal to __________...
JEE Main 2023 (Online) 6th April Evening Shift
Let a curve $$y=f(x), x \in(0, \infty)$$ pass through the points $$P\left(1, \frac{3}{2}\right)$$ and $$Q\left(a, \frac{1}{2}\right)$$. If the tangent...
JEE Main 2023 (Online) 6th April Evening Shift
The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.
JEE Main 2023 (Online) 29th January Evening Shift
If the equation of the normal to the curve $$y = {{x - a} \over {(x + b)(x - 2)}}$$ at the point (1, $$-$$3) is $$x - 4y = 13$$, then the value of $$a...
JEE Main 2022 (Online) 29th July Evening Shift
If the tangent to the curve $$y=x^{3}-x^{2}+x$$ at the point $$(a, b)$$ is also tangent to the curve $$y = 5{x^2} + 2x - 25$$ at the point (2, $$-$$1)...
JEE Main 2022 (Online) 27th July Evening Shift
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Wa...
JEE Main 2022 (Online) 27th July Morning Shift
Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-a...
JEE Main 2022 (Online) 26th July Morning Shift
Let the function $$f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$$, be decreasing in $$(0, \mathrm{a})$$ and increasing in $$(\mathrm{a}, 4)$$. A tangent to ...
JEE Main 2022 (Online) 25th July Evening Shift
The sum of the maximum and minimum values of the function $$f(x)=|5 x-7|+\left[x^{2}+2 x\right]$$ in the interval $$\left[\frac{5}{4}, 2\right]$$, whe...
JEE Main 2022 (Online) 30th June Morning Shift
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate a...
JEE Main 2022 (Online) 28th June Morning Shift
Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then ...
JEE Main 2022 (Online) 25th June Evening Shift
Let $$f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$$. If m and M are respectively the number of points of local minimum and local maximum of f i...
JEE Main 2021 (Online) 31st August Evening Shift
Let f(x) be a cubic polynomial with f(1) = $$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $$-$$1. Then f...
JEE Main 2021 (Online) 31st August Morning Shift
If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the ...
JEE Main 2021 (Online) 27th August Morning Shift
The number of distinct real roots of the equation 3x4 + 4x3 $$-$$ 12x2 + 4 = 0 is _____________.
JEE Main 2021 (Online) 26th 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 circle. If the sum of the are...
JEE Main 2021 (Online) 17th March Evening 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 such that f($$-$$1) = 2, f'($$-$$...
JEE Main 2021 (Online) 26th February Evening Shift
Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $$-$$3) and (4, $$-$$2$$\sqrt 2 ...
JEE Main 2021 (Online) 26th February Evening Shift
Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equ...
JEE Main 2021 (Online) 25th February Evening Shift
If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________.
JEE Main 2021 (Online) 25th February Morning Shift
Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $$-$$1 and x = 1. If $$\mathop {\lim }\li...
JEE Main 2021 (Online) 24th February Morning Shift
The minimum value of $$\alpha $$ for which the equation $${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha $$ has at least one solution in $$\lef...
JEE Main 2020 (Online) 5th September Evening Slot
If the lines x + y = a and x – y = b touch the curve y = x2 – 3x + 2 at the points where the curve intersects the x-axis, then $${a \over b}$$ is equ...
JEE Main 2020 (Online) 8th January Evening Slot
Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. The...
JEE Main 2020 (Online) 8th January Morning Slot
Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at $$\left( {0,{3 \over 2}} \right)$$ . If m is the slope of the t...
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