Calculus PYQ — Page 26

${\int }_{0}^{2}(|2{x}^{2}-3x|+[x-\frac{1}{2}])dx$, where $[t]$ is the greatest integer function, is equal to

Let $y={y}_{1}(x)$ and $y={y}_{2}(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with ${y}_{1}(0)=0$ and ${y}_{2}(0)=1$ respectively. Then, the number of points of intersection of $y={y}_{1}(x)$ and $y={y}_{2}(x)$ is

Let $f(x)=\mathrm{min}{1,1+x\mathrm{sin}x},0\leq x\leq 2\pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $(m,n)$ is equal to

Let $f$ be a differentiable function in $(0,\frac{\pi }{2})$. If ${\int }_{\mathrm{cos}x}^{1}{t}^{2}f(t)dt={\mathrm{sin}}^{3}x+\mathrm{cos}x$, then $\frac{1}{\sqrt{3}}{f}^{'}(\frac{1}{\sqrt{3}})$ is equal to

Let $f$ be a twice differentiable function on $R$. If ${f}^{'}(0)=4$ and $f(x)+{\int }_{0}^{x}(x-t){f}^{'}(t)dt$ $=({e}^{2x}+{e}^{-2x})\mathrm{cos}2x+\frac{2}{a}x$, then ${(2a+1)}^{5}{a}^{2}$ is equal to _______.

The area enclosed by ${y}^{2}=8x$ and $y=\sqrt{2}x$ that lies outside the triangle formed by $y=\sqrt{2}x,x=1,y=2\sqrt{2}$, is equal to

Suppose $y=y(x)$ be the solution curve to the differential equation $\frac{dy}{dx}-y=2-{e}^{-x}$ such that $\underset{x\rightarrow \infty }{\mathrm{lim}}y(x)$ is finite. If $a$ and $b$ are respectively the $x-$ and $y-$intercept of the tangent to the curve at $x=0$, then the value of $a-4b$ is equal to _______.

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x)={\begin{matrix}x-[x], & \text{if}[x]\text{is odd} \\ 1+[x]-x, & \text{if}[x]\text{is even}\end{matrix}$ Then, the value of $\frac{{\pi }^{2}}{10}{\int }_{-10}^{10}f(x)cos\pi xdx$ is

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\frac{{(x+2\mathrm{cos}x)}^{3}+2{(x+2\mathrm{cos}x)}^{2}+3\mathrm{sin}(x+2\mathrm{cos}x)}{{(x+2)}^{3}+2{(x+2)}^{2}+3\mathrm{sin}(x+2)})}^{\frac{100}{x}}$ is equal to

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

For the curve $C:({x}^{2}+{y}^{2}-3)+{({x}^{2}-{y}^{2}-1)}^{5}=0$, the value of $3{y}^{'}-{y}^{3}{y}^{''}$, at the point $(\alpha ,\alpha ),\alpha >0$, on $C$, is equal to ________.

Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+{\int }_{0}^{1}(x-t)f(t)dt$. Then which of the following points $(x,y)$ lies on the curve $y=f(x)$?

If for $p\neq q\neq 0$, then function $f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then

If $[t]$ denotes the greatest integer $\leq t$, then the value of ${\int }_{0}^{1}[2x-|3{x}^{2}-5x+2|+1]dx$ is

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfy $f(x+y)={2}^{x}f(y)+{4}^{y}(f(x),\forall x$, $y\in \mathbb{R}$. If $f(2)=3$, then $14\cdot \frac{{f}^{'}(4)}{{f}^{'}(2)}$ is equal to _____.

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_{-3}^{101}([\mathrm{sin}(\pi x)]+{e}^{[\mathrm{cos}(2\pi x)]})dx$ is equal to

The minimum value of the twice differentiable function $f(x)={\int }_{0}^{x}{e}^{x-t}{f}^{'}(t)dt-({x}^{2}-x+1){e}^{x},x\in R$, is

Let $f:R\rightarrow R$ be defined as $f(x)={x}^{3}+x-5$. If $g(x)$ is a function such that $f(g(x))=x,\forall x\in R$, then ${g}^{'}(63)$ is equal to ______

Consider a cuboid of sides $2x,4x$ and $5x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is constant $k$, then the ratio $x:r$, for which the sum of their volumes is maximum, is

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is ${\mathrm{tan}}^{-1}\frac{3}{4}$. Water is poured in it at a constant rate of $6$ cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is $4$ meters, is _______.

Let $f(x)={3}^{{({x}^{2}-2)}^{3}+4},x\in R$. Then which of the following statements are true? $P:x=0$ is a point of local minima of $f$ $Q:x=\sqrt{2}$ is a point of inflection of $f$ $R:{f}^{'}$ is increasing for $x>\sqrt{2}$

If ${\mathrm{cos}}^{-1}(\frac{y}{2})={\mathrm{log}}_{e}{(\frac{x}{5})}^{5},|y|<2$, then

Water is being filled at the rate of $1{\mathrm{cm}}^{3}{\mathrm{sec}}^{-1}$ in a right circular conical vessel (vertex downwards) of height $35\mathrm{cm}$ and diameter $14\mathrm{cm}$. When the height of the water level is $10\mathrm{cm}$, the rate (in ${\mathrm{cm}}^{2}{\mathrm{sec}}^{-1}$) at which the wet conical surface area of the vessel increases is

${\int }_{0}^{5}\mathrm{cos}(\pi (x-[\frac{x}{2}]))dx$, where $[t]$ denotes greatest integer less than or equal to $t$, is equal to