Calculus PYQ — Page 18

Let $a\in \mathbb{Z}$ and $[t]$ be the greatest integer $\leq t$, then the number of points, where the function $f(x)=[a+13\mathrm{sin}x],x\in (0,\pi )$ is not differentiable, is $____________$

If the function $f(x)={\begin{matrix}(1+|\mathrm{cos}x|)\frac{\lambda }{|\mathrm{cos}x|}, & 0<x<\frac{\pi }{2} \\ \mu , & x=\frac{\pi }{2} \\ {e}^{\frac{\mathrm{cot}6x}{\mathrm{cot}4x}}, & \frac{\pi }{2}<x<\pi \end{matrix}$is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6{\mathrm{log}}_{c}\mu +{\mu }^{6}-{e}^{6\lambda }$ is equal to

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 ________.

If $\phi (x)=\frac{1}{\sqrt{x}}{\int }_{\frac{\pi }{4}}^{x}(4\sqrt{2}\mathrm{sin}t-3{\phi }^{'}(t))dt,x>0$ then ${\phi }^{'}(\frac{\pi }{4})$ is equal to

Let $y=y(x)$ be the solution of the differential equation $({x}^{2}–3{y}^{2})\mathrm{dx}+3\mathrm{xy}\mathrm{dy}=0,y(1)=1$. Then $6{y}^{2}(e)$ is equal to

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

Let f be a continuous function satisfying ${\int }_{0}^{{t}^{2}}(f(x)+{x}^{2})\mathrm{dx}=\frac{4}{3}{t}^{3},\forall t>0$ . Then $f(\frac{{\pi }^{2}}{4})$ is equal to

Let $f,g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x)={\begin{matrix}\frac{x}{|x|}, & x\neq 0 \\ 1, & x=0\end{matrix},g(x)={\begin{matrix}\frac{\mathrm{sin}(x+1)}{(x+1)}, & x\neq -1 \\ 1, & x=-1\end{matrix}$and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\underset{x\rightarrow 1}{\mathrm{lim}}g(h(x-1))$ is

Let $f(x)={\begin{matrix}{x}^{2}\mathrm{sin}(\frac{1}{x});x\neq 0 \\ 0;x=0\end{matrix}$, then at $x=0$

The number of points, where the curve $y={x}^{5}-20{x}^{3}+50x+2$ crosses the $x$-axis, is _____.

Let the area enclosed by the lines $x+y=2,y=0$, $x=0$ and the curve $f(x)=\mathrm{min}{{x}^{2}+\frac{3}{4},1+[x]}$ where $[x]$ denotes the greatest integer $\leq x$, be $A$. Then the value of $12A$ is

Let$y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{5}{x({x}^{5}+1)}y=\frac{{({x}^{5}+1)}^{2}}{{x}^{7}},x>0$. If $y(1)=2$, then $y(2)$ is equal to

Let $f(x)=\int \frac{2x}{({x}^{2}+1)({x}^{2}+3)}dx$. If $f(3)=\frac{1}{2}({\mathrm{log}}_{e}5-{\mathrm{log}}_{e}6)$, then $f(4)$ is equal to

Let $y=y(x)$ be a solution curve of the differential equation, $(1-{x}^{2}{y}^{2})dx=ydx+xdy$, If the line $x=1$ intersects the curve $y=y(x)$ at $y=2$ and the line $x=2$ intersects the curve $y=y(x)$ at $y=\alpha$, then a value of $\alpha$ is

Let $y(x)=(1+x)(1+{x}^{2})(1+{x}^{4})(1+{x}^{8})(1+{x}^{16})$. Then ${y}^{'}-{y}^{"}$ at $x=-1$ is equal to

The area enclosed between the curves ${y}^{2}+4x=4$ and $y-2x=2$ is

If ${\int }_{-0.15}^{0.15}|100{x}^{2}-1|dx=\frac{k}{3000},$ then $k$ is equal to _____.

The value of the integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{x+\frac{\pi }{4}}{2-\mathrm{cos}2x}dx$ is :

Let $y=y(x)$ be the solution of the differential equation $x{\mathrm{log}}_{e}x\frac{dy}{dx}+y={x}^{2}{\mathrm{log}}_{e}x,(x>1)$. If $y(2)=2$, then $y(e)$ is equal to

The value of the integral ${\int }_{1/2}^{2}\frac{{\mathrm{tan}}^{-1}x}{x}dx$ is equal to

If the area enclosed by the parabolas ${P}_{1}:2y=5{x}^{2}$ and ${P}_{2}:{x}^{2}-y+6=0$ is equal to the area enclosed by ${P}_{1}$ and $y=\alpha x,\alpha >0$, then ${\alpha }^{3}$ is equal to _____ .

If [ $t$ denotes the greatest integer $\leq 1$, then the value of $\frac{3(e-1)}{e}{\int }_{1}^{2}{x}^{2}{e}^{[x]+[{x}^{3}]}dx$ is :

The area of the region $x,y:{x}^{2}\leq y\leq |{x}^{2}-4|,y\geq 1$ is

Let $x=2$ be a root of the equation ${x}^{2}+px+q=0$ and $f(x)={\begin{matrix}\frac{1-\mathrm{cos}({x}^{2}-4px+{q}^{2}+8q+16)}{{(x-2p)}^{4}}, & x\neq 2p \\ 0, & x=2p\end{matrix}$. Then $\underset{x\rightarrow 2{p}^{+}}{\mathrm{lim}}[f(x)]$ where $[\cdot ]$ denotes greatest integer function, is