Calculus PYQ — Page 12

The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $2 f(2)+3 f(3)$ is equal to _______

Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]\rightarrow R$ be a differentiable function such that $f(0)=\frac{1}{2}$, If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{x{\int }_{0}^{x}f(t)dt}{{e}^{{x}^{2}}-1}=\alpha$, then $8{\alpha }^{2}$ is equal to :

$\lim _{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}$ is equal to :

Let $f(x)=|2{x}^{2}+5|x|-3|,x\in R$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

Let $\mathrm{a}>0$ be a root of the equation $2 x^2+x-2=0$. If $\lim _{x \rightarrow \frac{1}{\mathrm{a}}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-\mathrm{a} x)^2}=\alpha+\beta \sqrt{17}$, where $\alpha, \beta \in Z$, then $\alpha+\beta$ is equal to_______

Let the area of the region ${(x,y):x-2y+4\geq 0$, $x+2{y}^{2}\geq 0,x+4{y}^{2}\leq 8,y\geq 0}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to ______.

If $5f(x)+4f(\frac{1}{x})={x}^{2}-2,\forall x\neq 0$ and $y=9{x}^{2}f(x),$ then $y$ is strictly increasing in:

If $\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^2+b x-\sqrt{\mathrm{e}}=0$, then $12 \log _{\mathrm{e}}(\mathrm{a}+\mathrm{b})$ is equal to__________

If $\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to_________

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is

Let $[\mathrm{t}]$ denote the greatest integer less than or equal to $\mathrm{t}$. Let $f:[0, \infty) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to _______

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by $f(x)=\left\{\begin{array}{ll} \frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0 \end{array}\right.$ where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to :

For $x\in (-\frac{\pi }{2},\frac{\pi }{2})$, if $y(x)=\int \frac{cosecx+\mathrm{sin}x}{cosecx\mathrm{sec}x+\mathrm{tan}x{\mathrm{sin}}^{2}x}dx$ and $\underset{x\rightarrow {(\frac{\pi }{2})}^{-}}{\mathrm{lim}}y(x)=0$ then $y(\frac{\pi }{4})$ is equal to

Suppose $f(x)=\frac{({2}^{x}+{2}^{-x})\mathrm{tan}x\sqrt{{\mathrm{tan}}^{-1}({x}^{2}-x+1)}}{{(7{x}^{2}+3x+1)}^{3}}$. Then the value of ${f}^{'}(0)$ is equal to

Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

Let the set of all values of $p$, for which $f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does not have any critical point, be the interval $(a, b)$. Then $16 a b$ is equal to _______

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$. (S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

If $\frac{dx}{dy}=\frac{1+x-{y}^{2}}{y},x(1)=1,$ then $5x(2)$ is equal to:

Let $\alpha$ be a non-zero real number. Suppose $f:R\rightarrow R$ is a differentiable function such that $f(0)=1$ and $\underset{x\rightarrow -\infty }{\mathrm{lim}}f(x)=1$. If ${f}^{'}(x)=\alpha f(x)+3$, for all $x\in R,$ then $f(-{\mathrm{log}}_{e}2)$ is equal to ________.

Let $\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

If $\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$, where $\mathrm{C}$ is the constant of integration, then the value of $\alpha+\beta+20 \mathrm{AB}$ is__________

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to_______