JEE Main Mathematics — Calculus previous year questions with solutions.
If ${\mathrm{cos}}^{-1}(\frac{y}{2})={\mathrm{log}}_{e}{(\frac{x}{5})}^{5},|y|<2$, then
The value of $b>3$ for which $12{\int }_{3}^{b}\frac{1}{({x}^{2}-1)({x}^{2}-4)}dx={\mathrm{log}}_{e}(\frac{49}{40})$, is equal to _____.
${\int }_{0}^{2}(|2{x}^{2}-3x|+[x-\frac{1}{2}])dx$, where $[t]$ is the greatest integer function, is equal to
Let $f(x)=\mathrm{min}{1,1+x\mathrm{sin}x},0\leq x\leq 2\pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $(m,n)$ is equal to
Let $f$ be a differentiable function in $(0,\frac{\pi }{2})$. If ${\int }_{\mathrm{cos}x}^{1}{t}^{2}f(t)dt={\mathrm{sin}}^{3}x+\mathrm{cos}x$, then $\frac{1}{\sqrt{3}}{f}^{'}(\frac{1}{\sqrt{3}})$ is equal to
Let $f$ be a twice differentiable function on $R$. If ${f}^{'}(0)=4$ and $f(x)+{\int }_{0}^{x}(x-t){f}^{'}(t)dt$ $=({e}^{2x}+{e}^{-2x})\mathrm{cos}2x+\frac{2}{a}x$, then ${(2a+1)}^{5}{a}^{2}$ is equal to _______.
The value of the integral${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{(1+{e}^{x})({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x)}$ is equal to
The value of $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{({x}^{2}-1){\mathrm{sin}}^{2}(\pi x)}{{x}^{4}-2{x}^{3}+2x-1}$ is equal to:
Suppose $y=y(x)$ be the solution curve to the differential equation $\frac{dy}{dx}-y=2-{e}^{-x}$ such that $\underset{x\rightarrow \infty }{\mathrm{lim}}y(x)$ is finite. If $a$ and $b$ are respectively the $x-$ and $y-$intercept of the tangent to the curve at $x=0$, then the value of $a-4b$ is equal to _______.
If $y=y(x)$ is the solution of the differential equation $2{x}^{2}\frac{dy}{dx}-2xy+3{y}^{2}=0$ such that $y(e)=\frac{e}{3}$, then $y(1)$ is equal to
A wire of length $22m$ 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, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is
For the curve $C:({x}^{2}+{y}^{2}-3)+{({x}^{2}-{y}^{2}-1)}^{5}=0$, the value of $3{y}^{'}-{y}^{3}{y}^{''}$, at the point $(\alpha ,\alpha ),\alpha >0$, on $C$, is equal to ________.
For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x)={\begin{matrix}x-[x], & \text{if}[x]\text{is odd} \\ 1+[x]-x, & \text{if}[x]\text{is even}\end{matrix}$ Then, the value of $\frac{{\pi }^{2}}{10}{\int }_{-10}^{10}f(x)cos\pi xdx$ is
The area (in sq. units) of the region enclosed between the parabola ${y}^{2}=2x$ and the line $x+y=4$ is ______.
If for $p\neq q\neq 0$, then function $f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then
If $[t]$ denotes the greatest integer $\leq t$, then the value of ${\int }_{0}^{1}[2x-|3{x}^{2}-5x+2|+1]dx$ is
Let $S=(0,2\pi )-{\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\frac{dy}{dx}=\frac{1}{1+\mathrm{sin}2x},y(\frac{\pi }{4})=\frac{1}{2}$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt{2}\mathrm{sin}x$ is $\frac{k\pi }{12}$, then $k$ is equal to _____.
The integral ${\int }_{0}^{1}\frac{1}{{7}^{[\frac{1}{x}]}}dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to
Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_{-3}^{101}([\mathrm{sin}(\pi x)]+{e}^{[\mathrm{cos}(2\pi x)]})dx$ is equal to
The minimum value of the twice differentiable function $f(x)={\int }_{0}^{x}{e}^{x-t}{f}^{'}(t)dt-({x}^{2}-x+1){e}^{x},x\in R$, is
Let ${A}_{1}={(x,y):|x|\leq {y}^{2},|x|+2y\leq 8}$ and ${A}_{2}={(x,y):|x|+|y|\leq k}.$ If $27$ (Area ${A}_{1}$) $=5$(Area ${A}_{2}$), then $k$ is equal to
Let $f:R\rightarrow R$ 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 ______
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is ${\mathrm{tan}}^{-1}\frac{3}{4}$. Water is poured in it at a constant rate of $6$ cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is $4$ meters, is _______.
Water is being filled at the rate of $1{\mathrm{cm}}^{3}{\mathrm{sec}}^{-1}$ in a right circular conical vessel (vertex downwards) of height $35\mathrm{cm}$ and diameter $14\mathrm{cm}$. When the height of the water level is $10\mathrm{cm}$, the rate (in ${\mathrm{cm}}^{2}{\mathrm{sec}}^{-1}$) at which the wet conical surface area of the vessel increases is