Calculus PYQ — Page 13

The value $9{\int }_{0}^{9}[\sqrt{\frac{10x}{x+1}}]\mathrm{dx}$, where $t$ denotes the greatest integer less than or equal to $t$, is _____.

Let $f(x)={(x+3)}^{2}{(x-2)}^{3},x\in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is :

If the value of the integral${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{{x}^{2}\mathrm{cos}x}{1+{\pi }^{x}}+\frac{1+{\mathrm{sin}}^{2}x}{1+{e}^{{(\mathrm{sin}x)}^{2023}}})dx=\frac{\pi }{4}(\pi +a)-2,$ then the value of $a$ is

The area (in square units) of the region enclosed by the ellipse $x^2+3 y^2=18$ in the first quadrant below the line $y=x$ is

Let the area of the region enclosed by the curve $y=\min \{\sin x, \cos x\}$ and the $x$ axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^2$ is equal to ___________

The area of the region enclosed by the parabola $(y-2{)}^{2}=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is __________.

The solution of the differential equation $\left(x^2+y^2\right) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$, is :

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2x{(x+y)}^{3}-x(x+y)-1,y(0)=1$. Then, ${(\frac{1}{\sqrt{2}}+y(\frac{1}{\sqrt{2}}))}^{2}$ equals:

If the solution curve, of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is ${\mathrm{tan}}^{-1}(\frac{y-1}{x-1})-\frac{1}{\beta }{\mathrm{log}}_{e}(\alpha +{(\frac{y-1}{x-1})}^{2})={\mathrm{log}}_{e}|x-1|$, then $5\beta +\alpha$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\mathrm{sec}xdy+{2(1-x)\mathrm{tan}x+x(2-x)}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

The area of the region ${(x,y):{y}^{2}\leq 4x,x<4,\frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,x\neq 3}$ is

If $y=\frac{(\sqrt{x}+1)({x}^{2}-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3{\mathrm{cos}}^{2}x-5){\mathrm{cos}}^{3}x,$ then $96{y}^{'}(\frac{\pi }{6})$ is equal to:

Let $f:[-1,2] \rightarrow \mathbf{R}$ be given by $f(x)=2 x^2+x+\left[x^2\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

If $\int \frac{{\mathrm{sin}}^{\frac{3}{2}}x+{\mathrm{cos}}^{\frac{3}{2}}x}{\sqrt{{\mathrm{sin}}^{3}x{\mathrm{cos}}^{3}x\mathrm{sin}(x-\theta )}}dx=A\sqrt{\mathrm{cos}\theta \mathrm{tan}x-\mathrm{sin}\theta }+B\sqrt{\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}x}+C,$ where $C$ is the integration constant, then $AB$ is equal to

Let $g(x)=3f(\frac{x}{3})+f(3-x)$ and ${f}^{"}(x)>0$ for all $x\in (0,3)$. If $g$ is decreasing in $(0,\alpha )$ and increasing in $(\alpha ,3)$, then $8\alpha$ is

If $y=y(x)$ is the solution curve of the differential equation $({x}^{2}-4)\mathrm{dy}-({y}^{2}-3y)\mathrm{dx}=0$, $x>2,y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

If $f(t)=\int_0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0 < \mathrm{t} < \pi$, then the value of $\int_0^{\frac{\pi}{2}} \frac{\pi^2 \mathrm{dt}}{f(\mathrm{t})}$ equals_________

Let for a differentiable function $f:(0,\infty )\rightarrow R$, $f(x)-f(y)\geq {\mathrm{log}}_{e}(\frac{x}{y})+x-y,\forall x,y\in (0,\infty )$. Then $\sum _{n=1}^{20}{f}^{'}(\frac{1}{{n}^{2}})$ is equal to

If the function $f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to

$\lim _{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to

Let $y=y(x)$ be the solution of the differential equation $(1-{x}^{2})dy=[xy+({x}^{3}+2)\sqrt{3(1-{x}^{2})}]dx$, $-1<x<1,y(0)=0$. If $y(\frac{1}{2})=\frac{m}{n},m$ and $n$ are coprime numbers, then $m+n$ is equal to __________.

Let $\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}$ be the solution of the differential equation $x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0$, $y(1)=2$. Then $\alpha+\beta$ is equal to ________

The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2 x$ is equal to :

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cossc}^2 x+\frac{3}{2}\right)+\beta \log _\epsilon\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and $\mathrm{C}$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals _______