Calculus PYQ — Page 27

Let $f,g:R\rightarrow R$ be functions defined by $f(x)={\begin{matrix}[x] & ,x<0 \\ |1-x| & ,x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}{e}^{x}-x, & x<0 \\ {(x-1)}^{2}-1, & x\geq 0\end{matrix}$ where $[x]$ denote the greatest integer less than or equal to $x$. Then, the function fog is discontinuous at exactly

The value of the integral ${\int }_{-2}^{2}\frac{|{x}^{3}+x|}{({e}^{x|x|}+1)}dx$ is equal to

The area of the region ${(x,y):|x-1|\leq y\leq \sqrt{5-{x}^{2}}}$ is equal to

If $\frac{dy}{dx}+{e}^{x}({x}^{2}-2)y=({x}^{2}-2x)({x}^{2}-2){e}^{2x}$ and $y(0)=0$, then the value of $y(2)$ is

The area (in sq. units) of the region enclosed between the parabola ${y}^{2}=2x$ and the line $x+y=4$ is ______.

The area bounded by the curves $y=|{x}^{2}-1|$ and $y=1$ is

A wire of length $22m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is

$\int \frac{({x}^{2}+1){e}^{x}}{{(x+1)}^{2}}dx=f(x){e}^{x}+C$, where $C$ is a constant, then $\frac{{d}^{3}f}{d{x}^{3}}$ at $x=1$ is equal to

Let $\beta =\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\alpha x-({e}^{3x}-1)}{\alpha x({e}^{3x}-1)}$ for some $\alpha \in \mathbb{R}$. Then the value of $\alpha +\beta$ is:

The number of distinct real roots of the equation ${x}^{5}({x}^{3}-{x}^{2}-x+1)+x(3{x}^{3}-4{x}^{2}-2x+4)-1=0$ is

If $\underset{n\rightarrow \infty }{\mathrm{lim}}(\sqrt{{n}^{2}-n-1}+n\alpha +\beta )=0$ then $8(\alpha +\beta )$ is equal to

Let $f(x)=\mathrm{max}{|x+1|,|x+2|,\ldots ,|x+5|}$. Then ${\int }_{-6}^{0}f(x)dx$ is equal to ______.

The lengths of the sides of a triangle are $10+{x}^{2}$, $10+{x}^{2}$ and $20-2{x}^{2}$. If for $x=k$, the area of the triangle is maximum, then $3{k}^{2}$ is equal to

For real numbers $a,b(a>b>0)$, let Area ${(x,y):{x}^{2}+{y}^{2}\leq {a}^{2}$ and $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\geq 1}=30\pi$ and Area ${(x,y):{x}^{2}+{y}^{2}\geq {b}^{2}$ and $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\leq 1}=18\pi$ Then the value of $(a-b){}^{2}$ is equal to _____.

Let $S=(0,2\pi )-{\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}}$. Let $y=y(x)$, $x\in S$, be the solution curve of the differential equation $\frac{dy}{dx}=\frac{1}{1+\mathrm{sin}2x},y(\frac{\pi }{4})=\frac{1}{2}$. If the sum of abscissas of all the points of intersection of the curve $y=y(x)$ with the curve $y=\sqrt{2}\mathrm{sin}x$ is $\frac{k\pi }{12}$, then $k$ is equal to _____.

If the function $f(x)={\begin{matrix}\frac{{\mathrm{log}}_{e}(1-x+{x}^{2})+{\mathrm{log}}_{e}(1+x+{x}^{2})}{secx-\mathrm{cos}x}, & x\in (\frac{-\pi }{2},\frac{\pi }{2})-{0} \\ k & ,x=0\end{matrix}$ is continuous at $x=0$, then $k$ is equal to:

Let the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+\frac{xy}{{x}^{2}-1}=\frac{{x}^{4}+2x}{\sqrt{1-{x}^{2}}},x\in (-1,1)$ pass through the origin. Then ${\int }_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f(x)dx$ is equal to

The sum of the absolute minimum and the absolute maximum values of the function $f(x)=|3x-{x}^{2}+2|-x$ in the interval $[-1,2]$ is

The value of the integral${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{(1+{e}^{x})({\mathrm{sin}}^{6}x+{\mathrm{cos}}^{6}x)}$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $\mathrm{sin}(2{x}^{2}){\mathrm{log}}_{e}(\mathrm{tan}{x}^{2})dy+(4xy-4\sqrt{2}x\mathrm{sin}({x}^{2}-\frac{\pi }{4}))dx=0,0<x<\sqrt{\frac{\pi }{2}}$ , which passes through the point $(\sqrt{\frac{\pi }{6}},1)$. Then $|y(\sqrt{\frac{\pi }{3}})|$ is equal to _______.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined as $f(x)=a\mathrm{sin}(\frac{\pi [x]}{2})+[2-x],a\in \mathbb{R}$, where $[t]$ is the greatest integer less than or equal to $t$. If $\underset{x\rightarrow -1}{\mathrm{lim}}f(x)$ exists, then the value of ${\int }_{0}^{4}f(x)dx$ is equal to

If $x=x(y)$ is the solution of the differential equation $y\frac{dx}{dy}=2x+{y}^{3}(y+1){e}^{y},x(1)=0$; then $x(e)$ is equal to

Let the solution curve $y=y(x)$ of the differential equation, $[\frac{x}{\sqrt{{x}^{2}-{y}^{2}}}+{e}^{\frac{y}{x}}]x\frac{dy}{dx}=x+[\frac{x}{\sqrt{{x}^{2}-{y}^{2}}}+{e}^{\frac{y}{x}}]y$ pass through the points $(1,0)$ and $(2\alpha ,\alpha ),\alpha >0$. Then $\alpha$ is equal to

Let $f(x)=2{\mathrm{cos}}^{-1}x+4{\mathrm{cot}}^{-1}x-3{x}^{2}-2x+10,x\in [-1,1]$. If $[a,b]$ is the range of the function, then $4a-b$ is equal to