JEE Main Mathematics — Calculus previous year questions with solutions.
The function $f(x)=x{e}^{x(1-x)},x\in R$, is
Let the function $f(x)=2{x}^{2}-{\mathrm{log}}_{e}x,x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola ${y}^{2}=4ax$ at a point $P$ on it passes through the point $(8a,8a-1)$ but does not pass through the point $(-\frac{1}{a},0)$. If the equation of the normal at $P$ is $\frac{x}{\alpha }+\frac{y}{\beta }=1$, then $\alpha +\beta$ is equal to
Let $P$ and $Q$ be any points on the curves ${(x-1)}^{2}+{(y+1)}^{2}=1$ and $y={x}^{2}$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval
Let $f:R\rightarrow R$ be a function defined by $f(x)={(x-3)}^{{n}_{1}}{(x-5)}^{{n}_{2}},{n}_{1},{n}_{2}\in N$. The, which of the following is NOT true?
The curve $y(x)=a{x}^{3}+b{x}^{2}+cx+5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $\mathrm{Q}$, where ${y}^{'}$ is equal to $3$. Then the local maximum value of $y(x)$ is
If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)={\int }_{0}^{{x}^{2}}\frac{{t}^{2}-5t+4}{2+{e}^{t}}dt$, then the ordered pair $(m,n)$ is equal to
If the sum of all the roots of the equation ${e}^{2x}-11{e}^{x}-45{e}^{-x}+\frac{81}{2}=0$ is ${\mathrm{log}}_{e}P$, then $P$ is equal to _____.
Let $f(x)=|(x-1)({x}^{2}-2x-3)|+x-3,x\in \mathbb{R}$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$, then $m+M$ is equal to _____.
Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be two functions defined by $f(x)={\mathrm{log}}_{e}({x}^{2}+1)-{e}^{-x}+1$ and $g(x)=\frac{1-2{e}^{2x}}{{e}^{x}}\cdot$ Then, for which of the following range of $\alpha$, the inequality $f(g(\frac{{(\alpha -1)}^{2}}{3}))>f(g(\alpha -\frac{5}{3}))$ holds?
The sum of absolute maximum and absolute minimum values of the function $f(x)=|2{x}^{2}+3x-2|+\mathrm{sin}x\mathrm{cos}x$ in the interval $[0,1]$ is
For the function $f(x)=4{\mathrm{log}}_{e}(x-1)-2{x}^{2}+4x+5,x>1$, which one of the following is NOT correct?
If $y=y(x)$ is the solution of the differential equation $x\frac{dy}{dx}+2y=x{e}^{x},y(1)=0$ then the local maximum value of the function $z(x)={x}^{2}y(x)-{e}^{x},x\in R$ is
If ${\int }_{0}^{\sqrt{3}}\frac{15{x}^{3}}{\sqrt{1+{x}^{2}+\sqrt{{(1+{x}^{2})}^{3}}}}dx=\alpha \sqrt{2}+\beta \sqrt{3}$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to
The slope of normal at any point $(x,y),x>0,y>0$ on the curve $y=y(x)$ is given by $\frac{{x}^{2}}{xy-{x}^{2}{y}^{2}-1}$. If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to
If $x=x(y)$ is the solution of the differential equation $y\frac{dx}{dy}=2x+{y}^{3}(y+1){e}^{y},x(1)=0$; then $x(e)$ is equal to
For $I(x)=\int \frac{{\mathrm{sec}}^{2}x-2022}{{\mathrm{sin}}^{2022}x}dx$, if $I(\frac{\pi }{4})={2}^{1011}$, then
The integral $\int \frac{(1-\frac{1}{\sqrt{3}})(\mathrm{cos}x-\mathrm{sin}x)}{(1+\frac{2}{\sqrt{3}}\mathrm{sin}2x)}dx$ is equal to
Let $g:(0,\infty )\rightarrow R$ be a differentiable function such that $\int (\frac{x(\mathrm{cos}x-\mathrm{sin}x)}{{e}^{x}+1}+\frac{g(x)({e}^{x}+1-x{e}^{x})}{{({e}^{x}+1)}^{2}})dx=\frac{xg(x)}{{e}^{x}+1}+C$, for all $x>0$, where $C$ is an arbitrary constant. Then
Let $f(x)=\mathrm{min}{[x-1],[x-2],\ldots ,[x-10]}$ where $[t]$ denotes the greatest integer $\leq t$. Then ${\int }_{0}^{10}f(x)dx+{\int }_{0}^{10}{(f(x))}^{2}dx+{\int }_{0}^{10}|f(x)|dx$ is equal _______. to
$I={\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}(\frac{8\mathrm{sin}x-\mathrm{sin}2x}{x})dx$. Then
${\int }_{0}^{5}\mathrm{cos}(\pi (x-[\frac{x}{2}]))dx$, where $[t]$ denotes greatest integer less than or equal to $t$, is equal to
If ${\int }_{0}^{2}(\sqrt{2x}-\sqrt{2x-{x}^{2}})dx=$ ${\int }_{0}^{1}(1-\sqrt{1-{y}^{2}}-\frac{{y}^{2}}{2})dy+{\int }_{1}^{2}(2-\frac{{y}^{2}}{2})dy+I$, then $I$ equal to
Let $f:R\rightarrow R$ be continuous function satisfying $f(x)+f(x+k)=n$, for all $x\in R$ where $k>0$ and $n$ is a positive integer. If ${I}_{1}={\int }_{0}^{4nk}f(x)dx$ and ${I}_{2}={\int }_{-k}^{3k}f(x)dx$, then
The value of the integral $\frac{48}{{\pi }^{4}}{\int }_{0}^{\pi }(\frac{3\pi {x}^{2}}{2}-{x}^{3})\frac{\mathrm{sin}x}{1+{\mathrm{cos}}^{2}x}dx$ is equal to ______.