JEE Main Mathematics — Calculus previous year questions with solutions.
Let a function $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as: $f(x)={\begin{matrix}{\int }_{0}^{x}(5-|t-3|)dt, & x>4 \\ {x}^{2}+bx, & x\leq 4\end{matrix}$ where $b\in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?
The area of the region bounded by ${y}^{2}=8x$ and ${y}^{2}=16(3-x)$ is equal to
If the solution curve of the differential equation $(({\mathrm{tan}}^{-1}y)-x)dy=(1+{y}^{2})dx$ passes through the point $(1,0)$ then the abscissa of the point on the curve whose ordinate is $\mathrm{tan}(1)$ is
The integral ${\int }_{0}^{\frac{\pi }{2}}\frac{1}{3+2\mathrm{sin}x+\mathrm{cos}x}dx$ is equal to:
The area enclosed by ${y}^{2}=8x$ and $y=\sqrt{2}x$ that lies outside the triangle formed by $y=\sqrt{2}x,x=1,y=2\sqrt{2}$, is equal to
Let $S$ be the region bounded by the curves $y={x}^{3}$ and ${y}^{2}=x$. The curve $y=2|x|$ divides $S$ into two regions of areas ${R}_{1}$ and ${R}_{2}$. If $\mathrm{max}|{R}_{1},{R}_{2}|={R}_{2}$, then $\frac{{R}_{2}}{{R}_{1}}$ is equal to ______
The sum of the absolute maximum and absolute minimum values of the function $f(x)={\mathrm{tan}}^{-1}(\mathrm{sin}x-\mathrm{cos}x)$ in the interval $[0,\pi ]$ is
Let $y=y(x),x>1$, be the solution of the differential equation $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y(2)=\frac{1+{e}^{4}}{2{e}^{4}}$. If $y(3)=\frac{{e}^{\alpha }+1}{\beta {e}^{\alpha }}$. then the value of $\alpha +\beta$ is equal to ______.
Let the solution curve of the differential equation $xdy=(\sqrt{{x}^{2}+{y}^{2}}+y)dx,x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is
The number of distinct real roots of the equation ${x}^{7}-7x-2=0$ is
The number of real solutions of ${x}^{7}+5{x}^{3}+3x+1=0$ is equal to _____.
Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae $3$ and $x(>3)$ be ${(\frac{y}{x})}^{3}$. If this curve also passes through the point $(\alpha ,6\sqrt{10})$ in the first quadrant, then $\alpha$ is equal to _______.
If $[t]$ denotes the greatest integer $\leq t$, then number of points, at which the function $f(x)=4|2x+3|+$ $9[x+\frac{1}{2}]-12[x+20]$ is not differentiable in the open interval $(-20,20)$, is ______.
Let $f(x)={3}^{{({x}^{2}-2)}^{3}+4},x\in R$. Then which of the following statements are true? $P:x=0$ is a point of local minima of $f$ $Q:x=\sqrt{2}$ is a point of inflection of $f$ $R:{f}^{'}$ is increasing for $x>\sqrt{2}$
Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral ${\int }_{0}^{1}[-8{x}^{2}+6x-1]dx$ is equal to
Let ${\lambda }^{*}$ be the largest value of $\lambda$ for which the function ${f}_{\lambda }(x)=4\lambda {x}^{3}-36\lambda {x}^{2}+36x+48$ is increasing for all $x\in \mathbb{R}$. Then ${f}_{{\lambda }^{*}}(1)+{f}_{\lambda ,*}(-1)$ is equal to:
Let $f:R\rightarrow R$ be a differentiable function such that $f(\frac{\pi }{4})=\sqrt{2},f(\frac{\pi }{2})=0$ and ${f}^{'}(\frac{\pi }{2})=1$ and let $g(x)={\int }_{x}^{\frac{\pi }{4}}({f}^{'}(t)\mathrm{sec}t+\mathrm{tan}t\mathrm{sec}tf(t))dt$ for $x\in [\frac{\pi }{4},\frac{\pi }{2}).$ Then $\underset{x\rightarrow {(\frac{\pi }{2})}^{-}}{\mathrm{lim}}g(x)$ is equal to
The value of ${\int }_{0}^{\pi }\frac{{e}^{\mathrm{cos}x}\mathrm{sin}x}{(1+{\mathrm{cos}}^{2}x)({e}^{\mathrm{cos}x}+{e}^{-\mathrm{cos}x})}dx$ is equal to
The value of the integral ${\int }_{0}^{\frac{\pi }{2}}60\frac{\mathrm{sin}(6x)}{\mathrm{sin}x}dx$ is equal to
Let $f$ be a differentiable function satisfying $f(x)=\frac{2}{\sqrt{3}}{\int }_{0}^{\sqrt{3}}f(\frac{{\lambda }^{2}x}{3})d\lambda ,x>0$ and $f(1)=\sqrt{3}$. If $y=f(x)$ passes through the point $(\alpha ,6)$, then $\alpha$ is equal to _______.
$\int \frac{({x}^{2}+1){e}^{x}}{{(x+1)}^{2}}dx=f(x){e}^{x}+C$, where $C$ is a constant, then $\frac{{d}^{3}f}{d{x}^{3}}$ at $x=1$ is equal to
The general solution of the differential equation $(x-{y}^{2})dx+y(5x+{y}^{2})dy=0$ is
Let $f:[0,3]\rightarrow R$ be defined by $f(x)=\mathrm{min}{x-[x],1+[x]-x}$ where $[x]$ is the greatest integer less than or equal to $x.$ Let $P$ denote the set containing all $x\in [0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in (0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to _____.
If $f:R\rightarrow R$ is a function defined by $f(x)=[x-1]\mathrm{cos}(\frac{2x-1}{2})\pi ,$ where $[\cdot ]$ denotes the greatest integer function, then $f$ is: