Calculus PYQ — Page 34

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

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

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{ax-({e}^{4x}-1)}{ax({e}^{4x}-1)}$ exists and is equal to $b$, then the value of $a-2b$ 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 $f:S\rightarrow S$ 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 :

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 ___________.

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 :

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

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

If $x\phi (x)={\int }_{5}^{x}(3{t}^{2}-2{\phi }^{'}(t))dt,x>-2,$ $\phi (0)=4,$ then $\phi (2)$ 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 ________.

If $R$ is the least value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is increasing on $[1,2]$ and $S$ is the greatest value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is decreasing on $[1,2]$, then the value of $|R-S|$ is

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2{x}^{3}-3{x}^{2}-12x$. If $A$ is the total area of the region bounded by $y=f(x),$ the $x$-axis and the lines $x=a$ and $x=b,$ then $4A$ is equal to ______.

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

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 $y(x)$ 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:

Let $[t]$ denote the greatest integer less than or equal to $t.$ Let $f(x)=x-[x],g(x)=1-x+[x],$ and $h(x)=\mathrm{min}{f(x),g(x)},x\in [-2,2].$ Then $h$ is :

If $\frac{dy}{dx}=\frac{{2}^{x+y}-{2}^{x}}{{2}^{y}},y(0)=1,$ then $y(1)$ is equal to :

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

The difference between degree and order of a differential equation that represents the family of curves given by ${y}^{2}=a(x+\frac{\sqrt{a}}{2}),a>0$ 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 :