JEE MAIN Differentiation Previous Year Question Paper and Solution with PDF

JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Differentiation PYQ

Qus : 1 JEE MAIN PYQ

Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_{e}\!\big(\sqrt{1+x^{2}}+x\big),\ x\in[0,3]$. Then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Morning Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

If $y(x)=x^{x},\ x>0$, then $y''(2)-2y'(2)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

The derivative of ${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$, with respect to ${x \over 2}$ , where $\left( {x \in \left( {0,{\pi \over 2}} \right)} \right)$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

If $y=\sec(\tan^{-1}x)$, then $\dfrac{dy}{dx}$ at $x=1$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 5 JEE MAIN PYQ

Let $f(x)=x^{5}+2x^{3}+3x+1,; x\in\mathbb{R}$, and let $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{R}$. Then $\dfrac{g'(7)}{g'(7)}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $ \ne $ 0 for all x $ \in $ R. If $\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right|$ = 0, for all x$ \in $R, then the value of f(1) lies in the interval :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

If $2x^{y}+3y^{x}=20$, then $\dfrac{dy}{dx}$ at $(2,2)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0$ then, the minimum value of $y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

The local maximum value of the function $f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$, x > 0, is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

If $f(x)=\sin^{-1}\left(\dfrac{2x^{3}}{1+9x^{2}}\right)$, then $f'!\left(-\dfrac{1}{2}\right)$ equals

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

Let $x(t) = 2\sqrt{2}\cos t \sqrt{\sin 2t}$ and $y(t) = 2\sqrt{2}\sin t \sqrt{\sin 2t}, \; t \in (0,\tfrac{\pi}{2}).$ Then $\dfrac{1+\left(\tfrac{dy}{dx}\right)^2}{\tfrac{d^2y}{dx^2}}$ at $t=\tfrac{\pi}{4}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

et y = y(x) be the solution of the differential equation ${{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x$ such that y(0) = 7. Then y($\pi$) is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

Let us consider a curve, y = f(x) passing through the point ($-$2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x². Then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

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 length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

Suppose $f(x)=\dfrac{(2^{x}+2^{-x})\tan x\,\sqrt{\tan^{-1}(x^{2}-x+1)}}{(7x^{2}+3x+1)^{3}}$. Then the value of $f'(0)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

If the solution curve of the differential equation (2x $-$ 10y³)dy + ydx = 0, passes through the points (0, 1) and (2, $\beta$), then $\beta$ is a root of the equation :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

A box open from top is made from a rectangular sheet of dimension a x

b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

The maximum slope of the curve $y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$ occurs at the point :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

If $f(x) = \displaystyle\int_{0}^{x} t(\sin x - \sin t),dt$ then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after ${k \over {{{\log }_e}\left( {{6 \over 5}} \right)}}$ hours, then ${\left( {{k \over {{{\log }_e}2}}} \right)^2}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

If $y(x)=\cot^{-1}\!\left(\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right),\; x\in\left(\tfrac{\pi}{2},\pi\right)$, then $\dfrac{dy}{dx}$ at $x=\tfrac{5\pi}{6}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Evening Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Evening Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

Let $f$ be a differentiable function such that $f(1)=2$ and $f'(x)=f(x)$ for all $x\in\mathbb{R}$. If $h(x)=f(f(x))$, then $h'(1)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

Suppose for a differentiable function $h$, $h(0)=0$, $h(1)=1$ and $h'(0)=h'(1)=2$. If $g(x)=h(e^{x}),e^{h(x)}$, then $g'(0)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

Let $g:\mathbb{R}\to\mathbb{R}$ be a non-constant twice-differentiable function such that $g'\!\left(\tfrac12\right)=g'\!\left(\tfrac32\right)$. If a real-valued function $f$ is defined as $f(x)=\dfrac12\,[\,g(x)+g(2-x)\,]$, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

Let f : S $ \to $ S where S = (0, $\infty $) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $ \to $ R be defined as g(x) = log_e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

Suppose$ \displaystyle f(x)=\frac{(x^{2}+2-x),\tan x;\sqrt{\tan^{-1}!\left(\frac{x^{2}-x+1}{x}\right)}}{(7x^{2}+3x+1)^{3}}. $ Then the value of $f'(0)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

If for $x\in\left(0,\dfrac14\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1-9x^{3}}\right)$ is $\sqrt{x}\cdot g(x)$, then $g(x)$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 31 JEE MAIN PYQ

If $\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C} $, where C is a constant, then ${{{d^3}f} \over {d{x^3}}}$ at x = 1 is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Morning Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

If $f(x)=x^{3}-x^{2}f'(1)+x f''(2)-f'''(3),\ x\in\mathbb{R}$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

If $y^{2} + \log_{e}(\cos^{2}x) = y,\; x \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right),$ then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 34 JEE MAIN PYQ

Let $y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$. Then $y' - y''$ at $x=-1$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

If $f(1)=1,\ f'(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^{2}$ at $x=1$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

If y = ${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$ then (x² $-$ 1) ${{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

For the differentiable function $f:\mathbb{R}\setminus{0}\to\mathbb{R}$, let $3f(x)+2f!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10$. Then $\left|,f(3)+f'!\left(\dfrac{1}{4}\right)\right|$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

Let $f$ be a polynomial function such that $f(3x) = f'(x)\cdot f''(x)$ for all $x \in \mathbb{R}$. Then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x \sin y)(f(2 x+2 y)+f(2 x-2 y))$, for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

The functions $f$ and $g$ are twice differentiable on $\mathbb{R}$ such that $f''(x) = g''(x) + 6x$ $f'(1) = 4g'(1) - 3 = 9$ $f(2) = 3g(2) = 12$ Then which of the following is NOT true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

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 decreasing in $\left( {{3 \over 4},\infty } \right)$. Then the function g(x) = ax² $-$ 6x + 15, x$\in$R has a :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Morning) PYQ

Solution

Qus : 44 JEE MAIN PYQ

If $f(x)$ is a differentiable function in the interval $(0,\infty)$ such that $f(1) = 1$ and $\displaystyle \lim_{t \to x} \frac{t^{2}f(x) - x^{2}f(t)}{t - x} = 1$, for each $x > 0$, then $f\left(\dfrac{3}{2}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_\limits{n=1}^{100} \log _e f(n)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Morning Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

$ \text { If } y(x)=\left|\begin{array}{ccc} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array}\right|, x \in \mathbb{R} \text {, then } \frac{d^2 y}{d x^2}+y \text { is equal to } $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

If $x=3\tan t$ and $y=3\sec t$, then the value of $\dfrac{d^{2}y}{dx^{2}}$ at $t=\dfrac{\pi}{4}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

Let f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le 0} \cr } } \right.$. Then f is increasing function in the interval

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

The solution of the differential equation${{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0$ is:(where c is a constant of integration)

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 50 JEE MAIN PYQ

The slope of the tangent to a curve $C: y=y(x)$ at any point $(x, y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If $C$ passes through the points $\left(0, \tfrac{1}{2}+\tfrac{\pi}{2\sqrt{2}}\right)$ and $\left(\alpha, \tfrac{1}{2}e^{2\alpha}\right)$, then $e^{\alpha}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Morning Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

The derivative of ${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with respect to ${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$ at x = ${1 \over 2}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Evening) PYQ

Solution

Qus : 52 JEE MAIN PYQ

If x = 1 is a critical point of the function f(x) = (3x² + ax – 2 – a)e^x, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Evening) PYQ

Solution

Qus : 53 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that \[ f(x)=x^{3}+x^{2}f'(1)+x f'(2)+f''(3),\qquad x\in\mathbb{R}. \] Then $f(2)$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

If $\displaystyle \int_{0}^{x} f(t)\,dt \;=\; x^{2} \;+\; \int_{x}^{1} t^{2} f(t)\,dt$, then $f'\!\left(\tfrac{1}{2}\right)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

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