Calculus PYQ — Page 16

Let $I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}$, is continuous at $x=0$, then $f(0)$ is equal to :

Three points $O(0,0),P(a,{a}^{2}),Q(-b,{b}^{2}),a>0,b>0,$ are on the parabola $y={x}^{2}$. Let ${S}_{1}$ be the area of the region bounded by the line $PQ$ and the parabola, and ${S}_{2}$ be the area of the triangle $OPQ$. If the minimum value of $\frac{{S}_{1}}{{S}_{2}}$ is $\frac{m}{n},\mathrm{gcd}(m,n)=1,$ then $m+n$ is equal to:

The area (in square units) of the region bounded by the parabola ${y}^{2}=4(x-2)$ and the line $y=2x-8$.

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0, y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0$. Then $y(\sqrt{3})$ is equal to :

If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}$, then $y\left(\frac{\pi}{8}\right)$ is equal to:

Let $\lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$ $\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________

If the solution curve $y=y(x)$ of the differential equation $(1+{y}^{2})(1+{\mathrm{log}}_{e}x)dx+xdy=0,x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha -\mathrm{tan}(\frac{3}{2})}{\beta +\mathrm{tan}(\frac{3}{2})}$, then $\alpha +2\beta$ is

Let $f(x)={x}^{3}+{x}^{2}{f}^{'}(1)+x{f}^{"}(2)+{f}^{'''}(3),x\in R$. Then ${f}^{'}(10)$ is equal to

Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is________

Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^2 d x=d y, y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3 \alpha+\pi)^2+\beta^2=\gamma+\delta \sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma+\delta$ equals ______

Consider the function $f:(0,2)\rightarrow R$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)={\begin{matrix}\text{min}{f(t)}, & 0<t\leq x\text{ and }0<x\leq 1 \\ \frac{3}{2}+x, & 1<x<2\end{matrix}$. Then

Let the area of the region ${(x,y):0\leq x\leq 3,0\leq y\leq$ $\mathrm{min}{{x}^{2}+2,2x+2}}$ be $A$. Then $12A$ is equal to ______.

If $\mathrm{sin}(\frac{y}{x})={\mathrm{log}}_{e}|x|+\frac{\alpha }{2}$ is the solution of the differential equation $x\mathrm{cos}(\frac{y}{x})\frac{dy}{dx}=y\mathrm{cos}(\frac{y}{x})+x$ and $y(1)=\frac{\pi }{3}$, then ${\alpha }^{2}$ is equal to

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$ is equal to

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to______

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$ consider the following two statements : (I) $\mathrm{f}$ is increasing in $\left(0, \frac{\pi}{2}\right)$. (II) $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$. Between the above two statements,

Let $y=y(x)$ be the solution of the differential equation ${\mathrm{sec}}^{2}xdx+({e}^{2y}{\mathrm{tan}}^{2}x+\mathrm{tan}x)dy=0,$ $0<x<\frac{\pi }{2},y(\frac{\pi }{4})=0$. If $y(\frac{\pi }{6})=\alpha$, then ${e}^{8\alpha }$ is equal to ______.

Let $a$ be the sum of all coefficients in the expansion of $(1–2x+2{x}^{2}{)}^{2023}(3-4{x}^{2}+2{x}^{3}{)}^{2024}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}(\frac{{\int }_{0}^{x}\frac{\mathrm{log}(1+t)}{{t}^{2024}+1}dt}{{x}^{2}})$. If the equations $c{x}^{2}+dx+e=0$ and $2b{x}^{2}+ax+4=0$ have a common root, where $c,d,e\in R$, then $d:c:e$ equals

The value of the integral $\int _{0}^{\frac{\pi }{4}}\frac{xdx}{{\mathrm{sin}}^{4}(2x)+{\mathrm{cos}}^{4}(2x)}$ equals:

The function $f(x)=\frac{x}{{x}^{2}-6x-16},x\in \mathbb{R}-{-2,8}$

Suppose for a differentiable function $h, h(0)=0, h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $\mathrm{g}(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$, then $g^{\prime}(0)$ is equal to:

Let ${x}$ denote the fractional part of $x$ and $f(x)=\frac{{\mathrm{cos}}^{-1}(1-{{x}}^{2}){\mathrm{sin}}^{-1}(1-{x})}{{x}-{{x}}^{3}},x\neq 0$. If $L\text{and}R$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\frac{32}{{\pi }^{2}}({L}^{2}+{R}^{2})$ is equal to __________.