Calculus PYQ — Page 30

If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____. 

If ${y}^{1/4}+{y}^{-1/4}=2x,$ and $({x}^{2}-1)\frac{{d}^{2}y}{d{x}^{2}}+\alpha x\frac{dy}{dx}+\beta y=0,$ then $|\alpha -\beta |$ is equal to _______.

Let $y=y(x)$ be a solution curve of the differential equation $(y+1){\mathrm{tan}}^{2}xdx+\mathrm{tan}xdy+ydx=0$, $x\in (0,\frac{\pi }{2})$. If $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}xy(x)=1$, then the value of $y(\frac{\pi }{4})$ is:

The area (in sq. unit) bounded by the curve $4{y}^{2}={x}^{2}(4-x)(x-2)$ is equal to

Let $[t]$ denote the greatest integer $\leq t.$ Then the value of $8\cdot {\int }_{-\frac{1}{2}}^{1}([2x]+|x|)dx$ is

Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as $f(x)={\begin{matrix}x+a, & x<0 \\ |x-1|, & x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}x+1, & x<0 \\ (x-1{)}^{2}+b, & x\geq 0\end{matrix}$, where $a,b$ are non-negative real numbers. If $gof(x)$ is continuous for all $x\in R,$ then $a+b$ is equal to ______ .

Let $P(x)$ be a real polynomial of degree $3$ which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and ${\int }_{-1}^{1}P(x)dx=18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to ___ .

The maximum slope of the curve $y=\frac{1}{2}{x}^{4}-5{x}^{3}+18{x}^{2}-19x$ occurs at the point

For real numbers $\alpha ,\beta ,\gamma$ and $\delta ,$ if $\int \frac{({x}^{2}-1)+{\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})}{({x}^{4}+3{x}^{2}+1){\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})}dx=\alpha {\mathrm{log}}_{e}({\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x}))+\beta {\mathrm{tan}}^{-1}(\frac{\gamma ({x}^{2}-1)}{x})+\delta {\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})+C$ where $C$ is an arbitrary constant, then the value of $10(\alpha +\beta \gamma +\delta )$ is equal to ______ .

Let $f:R\rightarrow R$ be a continuous function. Then $\underset{x\rightarrow \pi /4}{\mathrm{lim}}\frac{\frac{\pi }{4}{\int }_{2}^{{\mathrm{sec}}^{2}x}f(x)dx}{{x}^{2}-\frac{{\pi }^{2}}{16}}$ is equal to:

Let $g(x)={\int }_{0}^{x}f(t)dt$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\leq f(t)\leq 1$ for all $t\in [0,1]$ and $0\leq f(t)\leq \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which $g(3)$ lies is :

lim(x→0) (sin x)/x is equal to:

If f(x) = |x - 2|, then f(x) is not differentiable at x =:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=1+x{e}^{y-x},-\sqrt{2}<x<\sqrt{2},y(0)=0$, then the minimum value of $y(x),x\in (-\sqrt{2},\sqrt{2})$ is equal to :

Let $f(x)={x}^{6}+2{x}^{4}+{x}^{3}+2x+3,x\in R$. Then the natural number $n$ for which $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{{x}^{n}f(1)-f(x)}{x-1}=44$ is _____ .

Let $f(x)$ be a differentiable function at $x=a$ with ${f}^{'}(a)=2$ and $f(a)=4$. Then $\underset{x\rightarrow a}{\mathrm{lim}}\frac{xf(a)-af(x)}{x-a}$ equals:

Let $y=y(x)$ be solution of the differential equation ${\mathrm{log}}_{e}(\frac{dy}{dx})=3x+4y,$ with $y(0)=0$. If $y(-\frac{2}{3}{\mathrm{log}}_{e}2)=\alpha {\mathrm{log}}_{e}2$, then the value of $\alpha$ is equal to:

Let $f$ be any function defined on $R$ and let it satisfy the condition: $|f(x)-f(y)|\leq |{(x-y)}^{2}|,\forall (x,y)\in R$. If $f(0)=1,$ then :

The integral $\int \frac{(2x-1)\mathrm{cos}\sqrt{(2x-1{)}^{2}+5}}{\sqrt{4{x}^{2}-4x+6}}dx$ is equal to (where $c$ is a constant of integration)

The function $f(x),$ that satisfies the condition $f(x)=x+{\int }_{0}^{\pi /2}\mathrm{sin}x\mathrm{cos}yf(y)dy,$ is :

Let $y=y(x)$ be the solution of the differential equation ${e}^{x}\sqrt{1-{y}^{2}}dx+(\frac{y}{x})dy=0,y(1)=-1$ Then the value of ${(y(3))}^{2}$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}-\frac{4}{3}{x}^{3}+2{x}^{2}+3x, & x>0 \\ 3x{e}^{x}, & x\leq 0\end{matrix}.$ Then $f$ is increasing function in the interval

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)={\mathrm{tan}}^{-1}(\mathrm{sin}x+\mathrm{cos}x)$ in $[0,\frac{\pi }{2}]$. Then the value of $\mathrm{tan}(M-m)$ is equal to:

If $f(x)={\begin{matrix}{\int }_{0}^{x}(5+|1-t|)dt, & x>2 \\ 5x+1, & x\leq 2\end{matrix},$ then