Calculus PYQ — Page 23

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 _______.

The area of the region $S={(x,y):{y}^{2}\leq 8x,y\geq \sqrt{2}x,x\geq 1}$ is

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 _____.

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

If the area of the region ${(x,y):{x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}\leq 1,x+y\geq 0,y\geq 0}$ is $A$, then $\frac{256A}{\pi }$ is

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 integral ${\int }_{0}^{\frac{\pi }{2}}\frac{1}{3+2\mathrm{sin}x+\mathrm{cos}x}dx$ is equal to:

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\alpha {e}^{x}+\beta {e}^{-x}+\gamma \mathrm{sin}x}{x{\mathrm{sin}}^{2}x}=\frac{2}{3}$, where $\alpha ,\beta ,\gamma \in R$, then which of the following is NOT correct?

${\int }_{0}^{20\pi }{(|\mathrm{sin}x|+|\mathrm{cos}x|)}^{2}dx$ 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

∫ eˣ dx is equal to:

The maximum value of sin x + cos x is:

$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}({\mathrm{tan}}^{2}x({(2{\mathrm{sin}}^{2}x+3\mathrm{sin}x+4)}^{\frac{1}{2}}-{({\mathrm{sin}}^{2}x+6\mathrm{sin}x+2)}^{\frac{1}{2}}))$ is equal to

The function $f:R\rightarrow R$ defined by $f(x)=\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{\mathrm{cos}(2\pi x)-{x}^{2n}\mathrm{sin}(x-1)}{1+{x}^{2n+1}-{x}^{2n}}$ is continuous for all $x$ in

Let $f(x)={\begin{matrix}|4{x}^{2}-8x+5|, & \mathrm{if}8{x}^{2}-6x+1\geq 0 \\ [4{x}^{2}-8x+5], & \mathrm{if}8{x}^{2}-6x+1<0\end{matrix}$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is _____ .

$f,g:R\rightarrow R$ be two real valued function defined as $f(x)={\begin{matrix}-|x+3| & , & x<0 \\ {e}^{x} & , & x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}{x}^{2}+{k}_{1}x & , & x<0 \\ 4x+{k}_{2} & , & x\geq 0\end{matrix}$, where ${k}_{1}$ and ${k}_{2}$ are real constants. If $gof$ is differentiable at $x=0$, then $gof(-4)+$$gof(4)$ is equal to

If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the point $(2,1)$ and $(k+1,2),k>0$, then

If ${b}_{n}={\int }_{0}^{\frac{\pi }{2}}\frac{{\mathrm{cos}}^{2}nx}{\mathrm{sin}x}dx,n\in \mathbb{N}$, then

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

The area enclosed by the curves $y={\mathrm{log}}_{e}(x+{e}^{2}),x={\mathrm{log}}_{e}(\frac{2}{y})$ and $x={\mathrm{log}}_{e}2$, above the line $y=1$ is

Let $y=y(x)$ be the solution of the differential equation $x(1-{x}^{2})\frac{dy}{dx}+(3{x}^{2}y-y-4{x}^{3})=0,x>1$ with $y(2)=-2$. Then $y(3)$ is equal to

Let $f(x)=[2{x}^{2}+1]$ and $g(x)={\begin{matrix}2x-3, & x<0 \\ 2x+3, & x\geq 0\end{matrix}$, where $[t]$ is the greatest integer $\leq t$. Then, in the open interval $(-1,1)$, the number of points where fog is discontinuous is equal to ______.

The number of distinct real roots of ${x}^{4}-4x+1=0$ is

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\mathrm{cos}(\mathrm{sin}x)-\mathrm{cos}x}{{x}^{4}}$ is equal to