Calculus PYQ — Page 20

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

If $f(x)={x}^{2}+{g}^{'}(1)x+{g}^{"}(2)$ and $g(x)=f(1){x}^{2}+x{f}^{'}(x)+{f}^{"}(x)$, then the value of $f(4)-g(4)$ is equal to _____ .

Let $\alpha$ be the area of the larger region bounded by the curve ${y}^{2}=8x$ and the lines $y=x$ and $x=2$, which lies in the first quadrant. Then the value of $3\alpha$ 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 ______.

If the total maximum value of the function $f(x)={(\frac{\sqrt{3e}}{2\mathrm{sin}x})}^{{\mathrm{sin}}^{2}x},x\in (0,\frac{\pi }{2}),$ is $\frac{k}{e},$ then ${(\frac{k}{e})}^{8}+\frac{{k}^{8}}{{e}^{5}}+{k}^{8}$ 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:

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 $f$ be a differentiable function defined on $[0,\frac{\pi }{2}]$ such that $f(x)>0$ and $f(x)+{\int }_{0}^{x}f(t)\sqrt{1-{({\mathrm{log}}_{e}(f(t)))}^{2}}dt=e$ $\forall x\in [0,\frac{\pi }{2}]$, then ${{6{\mathrm{log}}_{e}(f(\frac{\pi }{6}))}}^{2}$ 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

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

Let $T$ and $C$ respectively, be the transverse and conjugate axes of the hyperbola $16{x}^{2}-{y}^{2}+64x+4y+44=0$. Then the area of the region above the parabola ${x}^{2}=y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is:

Let $f:(-2,2)\rightarrow \mathbb{R}$ be defined by $f(x)={\begin{matrix}x[x] & , & -2<x<0 \\ (x-1)[x] & , & 0\leq x<2\end{matrix}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(–2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to ________.

The area of the region enclosed by the curve $f(x)=\mathrm{max}{\mathrm{sin}x,\mathrm{cos}x},-\pi \leq x\leq \pi$ and the $x-$axis is

$\underset{0\leq x\leq \pi }{\mathrm{max}}{x-2\mathrm{sin}x\mathrm{cos}x+\frac{1}{3}\mathrm{sin}3x}=$

If the area bounded by the curve $2{y}^{2}=3x$, lines $x+y=3,y=0$ and outside the circle ${(x-3)}^{2}+{y}^{2}=2$ is A, then $4(\pi +4A)$ is equal to __________.