Calculus PYQ — Page 14

Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ______

If the function $f(x)=2 x^3-9 x^2+12 \mathrm{a}^2 x+1, \mathrm{a}>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :

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 $\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$ constant, then the maximum value of $\mathrm{a} \sin x+\mathrm{b} \cos x$, is :

The integral $\int \frac{({x}^{8}-{x}^{2})\mathrm{dx}}{({x}^{12}+3{x}^{6}+1){\mathrm{tan}}^{-1}({x}^{3}+\frac{1}{{x}^{3}})}$ is equal to :

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

Suppose the solution of the differential equation $\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}$ represents a circle passing through origin. Then the radius of this circle is :

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $a,b\in R$. Given that $x(0)=2;y(0)=1$ and $3y(1)=2x(1)$, the value of $t$, for which $x(t)=y(t)$, is :

The function $f(x)=2x+3{x}^{\frac{2}{3}},x\in R$, has

The area of the region enclosed by the parabola $y=4x-{x}^{2}$ and $3y={(x-4)}^{2}$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0 .$ Then the area enclosed by the curve $f(x)=y(x) \mathrm{e}^{-\frac{1}{\left(1+x^2\right)}}$ and the line $y-x=4$ is__________

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\frac{\pi }{6}$ and $\frac{\pi }{4}$, respectively with positive $x$-axis. If $27{\int }_{1}^{3}({({f}^{'}(t))}^{2}+1){f}^{"}(t)dt=\alpha +\beta \sqrt{3}$ where $\alpha ,\beta$ are integers, then the value of $\alpha +\beta$ equals

The area of the region enclosed by the parabolas $y=x^2-5 x$ and $y=7 x-x^2$ is

Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2+2 \beta^2$ is equal to

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

If $\int _{0}^{\frac{\pi }{3}}{\mathrm{cos}}^{4}xdx=a\pi +b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to:

Let $f:(0,\infty )\rightarrow R$ and $F(x)=\int _{0}^{x}tf(t)dt$. If $F({x}^{2})={x}^{4}+{x}^{5},$ then $\sum _{r=1}^{12}f({r}^{2})$ is equal to:

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and ${I}_{1}={\int }_{a}^{b}\mathrm{xsin}(4x-{x}^{2})\mathrm{dx},{I}_{2}={\int }_{a}^{b}\mathrm{sin}(4x-{x}^{2})\mathrm{dx}$ , then $36\frac{{I}_{1}}{{I}_{2}}$ 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 $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\mathrm{tan}x)+y}{\mathrm{sin}x(\mathrm{sec}x-\mathrm{sin}x\mathrm{tan}x)},x\in (0,\frac{\pi }{2})$ satisfying the condition $y(\frac{\pi }{4})=2$. Then, $y(\frac{\pi }{3})$ is

Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in interval $(0,2 \pi)$ is

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to:

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{{4}^{x}}{{4}^{x}+2}$ and $M=\int _{f(a)}^{f(1-a)}x{\mathrm{sin}}^{4}(x(1-x))dx$, $N=\int _{f(a)}^{f(1-a)}{\mathrm{sin}}^{4}(x(1-x))dx;a\neq \frac{1}{2}$. If $\alpha M=\beta N,\alpha ,\beta \in \mathbb{N}$, then the least value of ${\alpha }^{2}+{\beta }^{2}$ is equal to ______