Calculus PYQ — Page 20

Let $A={(x,y)\in {\mathbb{R}}^{2}:y\geq 0,2x\leq y\leq \sqrt{4-{(x-1)}^{2}}}$ and $B={(x,y)\in \mathbb{R}\times \mathbb{R}:0\leq y\leq \mathrm{min}{2x,\sqrt{4-{(x-1)}^{2}}}}$. Then the ratio of the area of $A$ to the area of $B$ is

Let $y=y(x)$ be a solution of the differential equation $(x\mathrm{cos}x)dy+(xy\mathrm{sin}x+y\mathrm{cos}x-1)dx=0,0<x<\frac{\pi }{2}.$ If $\frac{\pi }{3}y(\frac{\pi }{3})=\sqrt{3},$ then $|\frac{\pi }{6}{y}^{"}(\frac{\pi }{6})+2{y}^{'}(\frac{\pi }{6})|$ is equal to $_______.$

Let the solution curve $y=y(x)$ of the differential equation $\frac{dy}{dx}-\frac{3{x}^{5}{\mathrm{tan}}^{-1}({x}^{3})}{{(1+{x}^{6})}^{\frac{3}{2}}}y=2x$ $\mathrm{exp}\frac{{x}^{3}-{\mathrm{tan}}^{-1}{x}^{3}}{\sqrt{(1+x{)}^{6}}}$ pass through the origin. Then $y(1)$ is equal to:

${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4{x}^{2}}}dx$ is equal to

If the area of the region ${(x,y):|{x}^{2}-2|\leq y\leq x}$ is $A$, then $6A+16\sqrt{2}$ is equal to ______________

If ${\int }_{\frac{1}{3}}^{3}|{\mathrm{log}}_{e}x|\mathrm{dx}=\frac{m}{n}{\mathrm{log}}_{e}(\frac{{n}^{2}}{e})$, where $m$ and $n$ are coprime natural numbers, then ${m}^{2}+{n}^{2}-5$ is equal to _____ .

The set of values of $a$ for which $\underset{x\rightarrow a}{\mathrm{lim}}([x-5]-[2x+2])=0$, where, $[\zeta ]$ denotes the greatest integer less than or equal to $\zeta$ is equal to

The value of $\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{1+2-3+4+5-6+\ldots +(3n-2)+(3n-1)-3n}{\sqrt{2{n}^{4}+4n+3-}\sqrt{{n}^{4}+5n+4}}$ is

Let $f(x)=[{x}^{2}-x]+|-x+[x]|$, where $x\in \mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}=\frac{y}{x}(1+{x}^{2}(1+{\mathrm{log}}_{e}x))$, $x>0,y(1)=3$. Then $\frac{{y}^{2}(x)}{9}$ is equal to :

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{({x}^{2}-1)}y=\frac{x+2}{{({x}^{2}-1)}^{\frac{5}{2}}},x>1$ such that $y(2)=\frac{2}{9}{\mathrm{log}}_{e}(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha {\mathrm{log}}_{e}(\sqrt{\alpha }+\beta )+\beta -\sqrt{\gamma },\alpha ,\beta ,\gamma \in \mathbb{N}$, then $\alpha \beta \gamma$ is equal to

The value of ${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{(2+3\mathrm{sin}x)}{\mathrm{sin}x(1+\mathrm{cos}x)}dx$ is equal to

Let $f:[2,4]\rightarrow \mathbb{R}$ be a differentiable function such that $(x{\mathrm{log}}_{e}x){f}^{'}(x)+({\mathrm{log}}_{e}x)f(x)+f(x)\geq 1,x\in [2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{2}$. Consider the following two statements: (A) $f(x)\leq 1,\text{for all}x\in [2,4]$ (B) $f(x)\geq 1/8,\text{for all}x\in [2,4]$ Then,

If $I(x)=\int {e}^{{\mathrm{sin}}^{2}x}(\mathrm{cos}x\mathrm{sin}2x-\mathrm{sin}x)dx$ and $I(0)=1$, then $I(\frac{\pi }{3})$ is equal to

A wire of length $20m$ is to be cut into two pieces. A piece of length ${\ell }_{1}$ is bent to make a square of area ${A}_{1}$ and the other piece of length ${\ell }_{2}$ is made into a circle of area ${A}_{2}$. If $2{A}_{1}+3{A}_{2}$ is minimum then $(\pi {\ell }_{1}):{\ell }_{2}$ is equal to:

If $\int \sqrt{\mathrm{sec}2x-1}dx=\alpha {\mathrm{log}}_{e}|\mathrm{cos}2x+\beta +\sqrt{\mathrm{cos}2x(1+\mathrm{cos}\frac{1}{\beta }x)}|$ $+$ constant, then $\beta -\alpha$ is equal to ______.

Let $A$ be the area of the region ${(x,y):y\geq {x}^{2},y\geq (1-x{)}^{2},y\leq 2x(1-x)}$. Then $540A$ is equal to

Let $y=f(x)={\mathrm{sin}}^{3}(\frac{\pi }{3}(\mathrm{cos}(\frac{\pi }{3\sqrt{2}}{(-4{x}^{3}+5{x}^{2}+1)}^{\frac{3}{2}})))$. Then, at $x=1$,

Let ${f}_{n}={\int }_{0}^{\frac{\pi }{2}}(\sum _{k=1}^{n}{\mathrm{sin}}^{k-1}x)(\sum _{k=1}^{n}(2k-1){\mathrm{sin}}^{k-1}x)\mathrm{cos}xdx,n\in \mathbb{N}.$ Then ${f}_{21}-{f}_{20}$ is equal to

Suppose f is a function satisfying $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{N}$ and $f(1)=\frac{1}{5}$. If $\sum _{n=1}^{m}\frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$ then $m$ is equal to ______.

Let $[t]$ denote the greatest integer function. If ${\int }_{0}^{2.4}[{x}^{2}]dx=\alpha +\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}\text{, then }\alpha +\beta +\gamma +\delta$ is equal to

Let $y=y(x),y>0,$ be a solution curve of the differential equation $(1+{x}^{2})dy=y(x-y)dx$. If $y(0)=1$ and $y(2\sqrt{2})=\beta ,$ then

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{e}^{ax}-\mathrm{cos}(bx)-\frac{cx{e}^{-cx}}{2}}{1-\mathrm{cos}(2x)}=17$, then $5{a}^{2}+{b}^{2}$ is equal to

Let $[x]$ denote the greatest integer function and $f(x)=\mathrm{max}{1+x+[x],2+x,x+2[x]},0\leq x\leq 2$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not differentiable. Then ${(m+n)}^{2}+2$ is equal to