Calculus PYQ — Page 29

The sum of the absolute maximum and absolute minimum values of the function $f(x)={\mathrm{tan}}^{-1}(\mathrm{sin}x-\mathrm{cos}x)$ in the interval $[0,\pi ]$ is

Let $f(x)={\begin{matrix}\frac{\mathrm{sin}(x-[x])}{x-[x]}, & x\in (-2,-1) \\ \mathrm{max}(2x,3[|x|]), & |x|<1 \\ 1, & \mathrm{otherwise}\end{matrix}$ where $[t]$ denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $(m,n)$ is:

Let $y=y(x),x>1$, be the solution of the differential equation $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y(2)=\frac{1+{e}^{4}}{2{e}^{4}}$. If $y(3)=\frac{{e}^{\alpha }+1}{\beta {e}^{\alpha }}$. then the value of $\alpha +\beta$ is equal to ______.

Let the solution curve of the differential equation $xdy=(\sqrt{{x}^{2}+{y}^{2}}+y)dx,x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is

If $y(x)={({x}^{x})}^{x},x>0$ then $\frac{{d}^{2}x}{d{y}^{2}}+20$ at $x=1$ is equal to

The number of distinct real roots of the equation ${x}^{7}-7x-2=0$ is

The number of real solutions of ${x}^{7}+5{x}^{3}+3x+1=0$ is equal to _____.

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae $3$ and $x(>3)$ be ${(\frac{y}{x})}^{3}$. If this curve also passes through the point $(\alpha ,6\sqrt{10})$ in the first quadrant, then $\alpha$ is equal to _______.

If $[t]$ denotes the greatest integer $\leq t$, then number of points, at which the function $f(x)=4|2x+3|+$ $9[x+\frac{1}{2}]-12[x+20]$ is not differentiable in the open interval $(-20,20)$, is ______.

Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral ${\int }_{0}^{1}[-8{x}^{2}+6x-1]dx$ is equal to

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)={\int }_{0}^{{x}^{2}}\frac{{t}^{2}-5t+4}{2+{e}^{t}}dt$, then the ordered pair $(m,n)$ is equal to

Let ${\lambda }^{*}$ be the largest value of $\lambda$ for which the function ${f}_{\lambda }(x)=4\lambda {x}^{3}-36\lambda {x}^{2}+36x+48$ is increasing for all $x\in \mathbb{R}$. Then ${f}_{{\lambda }^{*}}(1)+{f}_{\lambda ,*}(-1)$ is equal to:

Let $f:R\rightarrow R$ be a differentiable function such that $f(\frac{\pi }{4})=\sqrt{2},f(\frac{\pi }{2})=0$ and ${f}^{'}(\frac{\pi }{2})=1$ and let $g(x)={\int }_{x}^{\frac{\pi }{4}}({f}^{'}(t)\mathrm{sec}t+\mathrm{tan}t\mathrm{sec}tf(t))dt$ for $x\in [\frac{\pi }{4},\frac{\pi }{2}).$ Then $\underset{x\rightarrow {(\frac{\pi }{2})}^{-}}{\mathrm{lim}}g(x)$ is equal to

The function $f(x)=x{e}^{x(1-x)},x\in R$, is

The value of ${\int }_{0}^{\pi }\frac{{e}^{\mathrm{cos}x}\mathrm{sin}x}{(1+{\mathrm{cos}}^{2}x)({e}^{\mathrm{cos}x}+{e}^{-\mathrm{cos}x})}dx$ is equal to

The number of points where the function $f(x)={\begin{matrix}|2{x}^{2}-3x-7| & \mathrm{if}x\leqslant -1 \\ [4{x}^{2}-1] & \mathrm{if}-1<x<1 \\ |x+1|+|x-2| & \mathrm{if}x\geqslant 1\end{matrix}$, where $[t]$ denotes the greatest integer $\leqslant t$, is discontinuous is ______

The value of the integral ${\int }_{0}^{\frac{\pi }{2}}60\frac{\mathrm{sin}(6x)}{\mathrm{sin}x}dx$ is equal to

If the absolute maximum value of the function $f(x)=({x}^{2}-2x+7){e}^{(4{x}^{3}-12{x}^{2}-180x+31)}$in the interval $[-3,0]$ is $f(\alpha )$, then

The value of $b>3$ for which $12{\int }_{3}^{b}\frac{1}{({x}^{2}-1)({x}^{2}-4)}dx={\mathrm{log}}_{e}(\frac{49}{40})$, is equal to _____.

Let $x=x(y)$ be the solution of the differential equation $2y{e}^{\frac{x}{{y}^{2}}}dx+({y}^{2}-4x{e}^{\frac{x}{{y}^{2}}})dy=0$ such that $x(1)=0$. Then, $x(e)$ is equal to

Let ${I}_{n}(x)={\int }_{0}^{x}\frac{1}{{({t}^{2}+5)}^{n}}dt,n=1,2,3,\ldots .$ Then

Let $f:R\rightarrow R$ be continuous function satisfying $f(x)+f(x+k)=n$, for all $x\in R$ where $k>0$ and $n$ is a positive integer. If ${I}_{1}={\int }_{0}^{4nk}f(x)dx$ and ${I}_{2}={\int }_{-k}^{3k}f(x)dx$, then

Let $f(x)$ be a cubic polynomial with $f(1)=-10,f(-1)=6,$ and has a local minima at $x=1,$ and ${f}^{'}(x)$ has a local minima at $x=-1.$ Then $f(3)$ is equal to .

Let $a$ be a real number such that the function $f(x)=a{x}^{2}+6x-15,x\in R$ is increasing in $(-\infty ,\frac{3}{4})$ and decreasing in $(\frac{3}{4},\infty )$. Then the function $g(x)=a{x}^{2}-6x+15,x\in R$ has a