Calculus PYQ — Page 36

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{a{e}^{x}-b\mathrm{cos}x+c{e}^{-x}}{x\mathrm{sin}x}=2$, then $a+b+c$ is equal to ________.

If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^{4},$ then for what value of $b,{\int }_{1}^{2}f(x)dx=\frac{62}{5}?$

Let $f:[-3,1]\rightarrow R$ be given as $f(x)={\begin{matrix}min{(x+6),{x}^{2}}, & -3\leq x\leq 0 \\ max{\sqrt{x},{x}^{2}}, & 0\leq x\leq 1\end{matrix}.$If the area bounded by $y=f(x)$ and $x$-axis is $A$ sq units, then the value of $6A$ is equal to

If the function $f(x)=\frac{\mathrm{cos}(\mathrm{sin}x)-\mathrm{cos}x}{{x}^{4}}$ is continuous at each point in its domain and $f(0)=\frac{1}{k},$ then $k$ is _________.

Let $y=y(x)$ be the solution of the differential equation $dy={e}^{\alpha x+y}dx;\alpha \in N.$ If $y({\mathrm{log}}_{e}2)={\mathrm{log}}_{e}2$ and $y(0)={\mathrm{log}}_{e}(\frac{1}{2}),$ then the value of $\alpha$ is equal to ___.

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{1+{\mathrm{sin}}^{2}x}{1+{\pi }^{\mathrm{sin}x}})dx$ is :

If $y\frac{dy}{dx}=x[\frac{{y}^{2}}{{x}^{2}}+\frac{\phi (\frac{{y}^{2}}{{x}^{2}})}{{\phi }^{'}(\frac{{y}^{2}}{{x}^{2}})}],x>0,\phi >0,$ and $y(1)=-1,$ then $\phi (\frac{{y}^{2}}{4})$ is equal to:

If $[x]$ is the greatest integer $\leq x,$ then ${\pi }^{2}{\int }_{0}^{2}(\mathrm{sin}\frac{\pi x}{2}){(x-[x])}^{[x]}dx$ is equal to :

The local maximum value of the function, $f(x)={(\frac{2}{x})}^{{x}^{2}},x>0,$ is

$\int \frac{2{e}^{x}+3{e}^{-x}}{4{e}^{x}+7{e}^{-x}}dx=\frac{1}{14}(ux+v{\mathrm{log}}_{e}(4{e}^{x}+7{e}^{-x}))+C$, where $C$ is a constant of integration, then $u+v$ is equal to

If $y(x)={\mathrm{cot}}^{-1}(\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1-\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}x}-\sqrt{1-\mathrm{sin}x}}),x\in (\frac{\pi }{2},\pi )$, then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is:

The area of the region bounded by the parabola $(y-2{)}^{2}=(x-1)$, the tangent to it at the point whose ordinate is $3$ and the $x$-axis, is:

The function $f(x)=\frac{4{x}^{3}-3{x}^{2}}{6}-2\mathrm{sin}x+(2x-1)\mathrm{cos}x$:

A wire of length $20m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

The value of the integral, ${\int }_{1}^{3}[{x}^{2}-2x-2]dx,$ where $[x]$ denotes the greatest integer less than or equal to $x,$ is

If $y=y(x),y\in [0,\frac{\pi }{2})$ is the solution of the differential equation $\mathrm{sec}y\frac{dy}{dx}-\mathrm{sin}(x+y)-\mathrm{sin}(x-y)=0,$ with $y(0)=0$, then $5{y}^{'}(\frac{\pi }{2})$ is equal to _____.

The value of ${\int }_{-2}^{2}|3{x}^{2}-3x-6|dx$ is

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis is $\frac{4\sqrt{8}}{3}$, then the value of $y(1)$ is equal to ________.

Let $f(x)$ be a differentiable function defined on $[0,2]$ such that ${f}^{'}(x)={f}^{'}(2-x)$ for all $x\in (0,2),f(0)=1$ and $f(2)={e}^{2}.$ Then the value of ${\int }_{0}^{2}f(x)dx$ is

Let $y=y(x)$ be the solution of the differential equation $xdy-ydx=\sqrt{({x}^{2}-{y}^{2})}dx,x\geq 1$, with $y(1)=0$. If the area bounded by the line $x=1,x={e}^{\pi },y=0$ and $y=y(x)$ is $\alpha {e}^{2\pi }+\beta ,$ then the value of $10(\alpha +\beta )$ is equal to ___ .

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1,{f}^{'}(0)=2$ and ${f}^{'}(x)\neq 0$ for all $x\in R.$ If $|\begin{matrix}f(x) & {f}^{'}(x) \\ {f}^{'}(x) & {f}^{''}(x)\end{matrix}|=0,$ for all $x\in R,$ then the value of $f(1)$ lies in the interval

Let $f:R\rightarrow R$ be a continuous function such that $f(x)+f(x+1)=2$ for all $x\in R$. If ${I}_{1}={\int }_{0}^{8}f(x)dx$ and ${I}_{2}={\int }_{-1}^{3}f(x)dx$, then the value of ${I}_{1}+2{I}_{2}$ is equal to ________.

Let a curve $y=y(x)$ be given by the solution of the differential equation $\mathrm{cos}(\frac{1}{2}{\mathrm{cos}}^{-1}({e}^{-x}))dx=(\sqrt{{e}^{2x}-1})dy$. If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x-$axis is $(\alpha ,0)$, then ${e}^{\alpha }$ is equal to

Let $T$ be the tangent to the ellipse $E:{x}^{2}+4{y}^{2}=5$ at the point $P(1,1)$. If the area of the region bounded by the tangent $T$, ellipse $E$, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\mathrm{cos}}^{-1}(\frac{1}{\sqrt{5}})$, then $|\alpha +\beta +\gamma |$ is equal to______.