Calculus PYQ — Page 35

The triangle of maximum area that can be inscribed in a given circle of radius '$r$' is :

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then ${A}^{4}$ is equal to

Let $y=y(x)$ be the solution of the differential equation $xdy=(y+{x}^{3}\mathrm{cos}x)dx$ with $y(\pi )=0,$ then $y(\frac{\pi }{2})$ is equal to:

If $[\cdot ]$ represents the greatest integer function, then the value of $|{\int }_{0}^{\sqrt{\frac{\pi }{2}}}[[{x}^{2}]-\mathrm{cos}x]dx|$ is ___________.

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

If $y=y(x)$ is the solution curve of the differential equation ${x}^{2}dy+(y-\frac{1}{x})dx=0;x>0$ and $y(1)=1,$ then $y(\frac{1}{2})$ is equal to :

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 ________.

The number of real roots of the equation ${e}^{6x}-{e}^{4x}-2{e}^{3x}-12{e}^{2x}+{e}^{x}+1=0$ is:

Let $f:S\rightarrow S$ where $S=(0,\infty )$ be a twice differentiable function such that $f(x+1)=xf(x)$. If $g:S\rightarrow R$ be defined as $g(x)={\mathrm{log}}_{e}f(x)$, then the value of $|{g}^{''}(5)-{g}^{''}(1)|$ is equal to :

If $f(x)=\int \frac{5{x}^{8}+7{x}^{6}}{{({x}^{2}+1+2{x}^{7})}^{2}}dx,(x\geq 0),f(0)=0$ and $f(1)=\frac{1}{K},$ then the value of $K$ is

Let ${C}_{1}$ be the curve obtained by the solution of differential equation $2xy\frac{dy}{dx}={y}^{2}-{x}^{2},x>0$. Let the curve ${C}_{2}$ be the solution of $\frac{2xy}{{x}^{2}-{y}^{2}}=\frac{dy}{dx}$. If both the curves pass through $(1,1),$ then the area (in sq. units) enclosed by the curves ${C}_{1}$ and ${C}_{2}$ is equal to :

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y;y(0)=0$

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix}3(1-\frac{|x|}{2}) & \text{if} & |x|\leq 2 \\ 0 & \text{if} & |x|>2\end{matrix}.$ Let $g:R\rightarrow R$ be given by $g(x)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

The area bounded by the lines $y=||x-1|-2|$ and $y=2$ is _____.

The value of $\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{[r]+[2r]+...+[nr]}{{n}^{2}},$ where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r,$ is equal to :

The value of $\underset{x\rightarrow 0}{\mathrm{lim}}(\frac{x}{\sqrt[8]{1-\mathrm{sin}x}-\sqrt[8]{1+\mathrm{sin}x}})$ is equal to :

Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{x{y}^{2}+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of $y$, for which the point $(3,y)$ lies on the curve, 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 _____.

Let $y(x)$ be the solution of the differential equation $2{x}^{2}dy+({e}^{y}-2x)dx=0,x>0.$ If $y(e)=1,$ then $y(1)$ is equal to:

The population $P=P(t)$ at time $t$ of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450$. If $P(0)=850$, then the time at which population becomes zero is:

The area of the region: $R={(x,y):5{x}^{2}\leq y\leq 2{x}^{2}+9}$ is

Let the functions $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as : $f(x)={\begin{matrix}x+2, & x<0 \\ {x}^{2}, & x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}{x}^{3}, & x<1 \\ 3x-2, & x\geq 1\end{matrix}$ Then, the number of points in $R$ where $(fog)(x)$ is NOT differentiable is equal to :

If $\frac{dy}{dx}=\frac{{2}^{x}y+{2}^{y}\cdot {2}^{x}}{{2}^{x}+{2}^{x+y}{\mathrm{log}}_{e}2},y(0)=0,$ then for $y=1,$ the value of $x$ lies in the interval :

The value of the limit $\underset{\theta \rightarrow 0}{\mathrm{lim}}\frac{\mathrm{tan}(\pi {\mathrm{cos}}^{2}\theta )}{\mathrm{sin}(2\pi {\mathrm{sin}}^{2}\theta )}$ is equal to :