Calculus PYQ — Page 16

If $\int \frac{{\mathrm{sin}}^{\frac{3}{2}}x+{\mathrm{cos}}^{\frac{3}{2}}x}{\sqrt{{\mathrm{sin}}^{3}x{\mathrm{cos}}^{3}x\mathrm{sin}(x-\theta )}}dx=A\sqrt{\mathrm{cos}\theta \mathrm{tan}x-\mathrm{sin}\theta }+B\sqrt{\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}x}+C,$ where $C$ is the integration constant, then $AB$ 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 :

If the function $f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ 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 :

Let $f,g:(0,\infty )\rightarrow R$ be two functions defined by $f(x)={\int }_{-x}^{x}(|t|-{t}^{2}){e}^{-{t}^{2}}dt$ and $g(x)={\int }_{0}^{{x}^{2}}{t}^{\frac{1}{2}}{e}^{-{t}^{2}}dt$. Then the value of $9(f(\sqrt{{\mathrm{log}}_{e}9}+g(\sqrt{{\mathrm{log}}_{e}9}))$ is equal to

The temperature $T(t)$ of a body at time $t=0$ is ${160}^{^{\circ}}F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$, where $K$ is positive constant. If $T(15)={120}^{^{\circ}}F$, then $T(45)$ 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 $y=y(x)$ 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 ______

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 ______.

The value of the integral $\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{a-b\mathrm{cos}2x}{{x}^{2}};x<0 \\ {x}^{2}+cx+2;0\leq x\leq 1 \\ 2x+1;x>1\end{matrix}$ If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m+a+b+c$ equals:

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 ______.

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 __________.

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 ______.

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2{y}^{2}=kx$ and $k{y}^{2}=2(y-x)$ is maximum, is equal to:

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4 a^2+a-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

If the area of the region $\left\{(x, y): \frac{\mathrm{a}}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2,0 < \mathrm{a} < 1\right\}$ is $\left(\log _{\mathrm{e}} 2\right)-\frac{1}{7}$ then the value of $7 \mathrm{a}-3$ is equal to:

If the function $f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$ attains the maximum value at $x=\frac{1}{\mathrm{e}}$ then :

Let the area of the region enclosed by the curves $y=3 x, 2 y=27-3 x$ and $y=3 x-x \sqrt{x}$ be $A$. Then $10 A$ is equal to