Calculus PYQ — Page 48

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^2 y}{d x^2}$ at the point $(-2,0)$ is

If a right circularcone having maximum volume, is inscribed in a sphere of radius $3 \mathrm{~cm}$, then the curved surface area (in $\mathrm{cm}^2$ ) of this cone is

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{(27+x)}^{\frac{1}{3}}-3}{9-{(27+x)}^{\frac{2}{3}}}$ equals

Let $S=\left\{(\lambda, \mu) \in R \times R: f(t)=\left(|\lambda| e^t-\mu\right) \cdot \sin (2|t|)\right.$, $t \in R$, is a differentiable function $\}$. Then $S$ is a subest of?

Let $S={(\lambda ,\mu )\in R\times R :f(t)=(|\lambda |{e}^{|t|}-\mu )\mathrm{sin}⁡(2|t|),t\in R$ is a differential function}. Then, $S$ is a subset of :

The area (in sq. units) of the region $\{x \in R: x \geq 0, y \geq 0, y \geq x-2$ and $y \leq \sqrt{x}\}$, is

Let $f(x)$ be a polynomial of degree 4 having extreme values at $x=1$ and $x=2$. If $\lim _{x \rightarrow 0}\left(\frac{f(x)}{x^2}+1\right)=3$ then $f(-1)$ is equal to

Let $f(x)=\left\{\begin{array}{cc}(x-1)^{\frac{1}{2-x}}, & x>1, x \neq 2 \\ k, & x=2\end{array}\right.$ The value of $k$ for which $f$ is continuous at $x=2$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+2y=f(x)$, where $f(x)= {\begin{matrix}1, & x\in [0, 1] \\ 0, & otherwise\end{matrix}$. If $y(0)=0$, then $y (\frac{3}{2})$ is

The area (in sq. units) of the region ${(x, y):x\geq 0, x+y\leq 3, {x}^{2}\leq 4y and y\leq 1+\sqrt{x}}$ is

The function $f$ defined by $f(x)={x}^{3}-3{x}^{2}+5x+7$ is:

$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}\frac{\mathrm{cot}x-\mathrm{cos}x}{{(\pi -2x)}^{3}}$ equals

The integral $\int _{\frac{\pi }{12}}^{\frac{\pi }{4}}\frac{8\mathrm{cos}2x}{{(\mathrm{tan}x+\mathrm{cot}x)}^{3}}dx$ equals

If $\int _{1}^{2}\frac{dx}{{({x}^{2}-2x+4)}^{\frac{3}{2}}}=\frac{k}{k+5}$, then $k$ is equal to

The integral $\int _{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{dx}{1+\mathrm{cos}x}$ is equal to

If for $x\in (0,\frac{1}{4}),$ the derivative of ${\mathrm{tan}}^{-1}⁡(\frac{6x\sqrt{x}}{1-9{x}^{3}})$ is $\sqrt{x} \cdot g(x)$ , then $g(x)$ equals:

If $(2+\mathrm{sin}x)\frac{dy}{dx}+(y+1)\mathrm{cos}x=0$ and $y(0)=1$, then $y(\frac{\pi }{2})$ is equal to

If $2x={y}^{\frac{1}{5}}+{y}^{-\frac{1}{5}}$ and $({x}^{2}-1)\frac{{d}^{2}y}{d{x}^{2}}+\lambda x\frac{dy}{dx}+ky=0$, then $\lambda +k$ is equal to

If $f(\frac{3x-4}{3x+4})=x+2, x\neq -\frac{4}{3}$, and $\int f(x)dx=A\mathrm{log}|1-x|+Bx+C$ , then the ordered pair $(A, B)$ is equal to

Let $f$ be a polynomial function such that $f(3x)={f}^{'}(x). {f}^{''}(x),$ for all $x\in R.$ Then :

The area (in sq. units) of the smaller portion enclosed between the curves, ${x}^{2}+{y}^{2}=4$ and ${y}^{2}=3x$, is:

The value of $k$ which the function $f(x)= {\begin{matrix}{(\frac{4}{5})}^{\frac{\mathrm{tan}⁡4x}{\mathrm{tan}⁡5x}}, & 0<x<\frac{\pi }{2} \\ k+\frac{2}{5}, & x=\frac{\pi }{2}\end{matrix}$ is continuous at $x=\frac{\pi }{2},$ is

The integral $\int \sqrt{1+2\mathrm{cot}x(\mathrm{cosec}x+\mathrm{cot}x)}dx, (0<x<\frac{\pi }{2})$is equal to

A tangent to the curve, $y=f(x)$ at $P(x, y)$ meets $x$-axis at $A$ and $y$-axis at $B$. If $AP:BP=1:3$ and $f(1)=1,$ then the curve also passes through the point