Calculus PYQ — Page 35

A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to:

If $\frac{dy}{dx}=\frac{{2}^{x}y+{2}^{y}\cdot {2}^{x}}{{2}^{x}+{2}^{x+y}{\mathrm{log}}_{e}2},y(0)=0,$ then for $y=1,$ the value of $x$ lies in the interval :

${\int }_{6}^{16}\frac{{\mathrm{log}}_{e}{x}^{2}}{{\mathrm{log}}_{e}{x}^{2}+{\mathrm{log}}_{e}({x}^{2}-44x+484)}dx$ is equal to

Which of the following statement is correct for the function $g(\alpha )$ for $\alpha \in R$ such that $g(\alpha )={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{{\mathrm{sin}}^{\alpha }x}{{\mathrm{cos}}^{\alpha }x+{\mathrm{sin}}^{\alpha }x}dx$

The value of the integral $\int \frac{\mathrm{sin}\theta \cdot \mathrm{sin}2\theta ({\mathrm{sin}}^{6}\theta +{\mathrm{sin}}^{4}\theta +{\mathrm{sin}}^{2}\theta )\sqrt{2{\mathrm{sin}}^{4}\theta +3{\mathrm{sin}}^{2}\theta +6}}{1-\mathrm{cos}2\theta }d\theta$ is (where $c$ is a constant of integration)

Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x,f(x))$ is given by $f(x)+x{f}^{'}(x)={x}^{2}.$ Then

If $\alpha =\underset{x\rightarrow \pi /4}{\mathrm{lim}}\frac{{\mathrm{tan}}^{3}x-\mathrm{tan}x}{\mathrm{cos}(x+\frac{\pi }{4})}$ and $\beta =\underset{x\rightarrow 0}{\mathrm{lim}}{(\mathrm{cos}x)}^{\mathrm{cot}x}$ are the roots of the equation, $a{x}^{2}+bx-4=0,$ then the ordered pair $(a,b)$ is :

The value of $\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{[r]+[2r]+...+[nr]}{{n}^{2}},$ where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r,$ is equal to :

Let $f:[-1,1]\rightarrow R$ be defined as $f(x)=a{x}^{2}+bx+c$ for all $x\in [-1,1],$ where $a,b,c\in R$ such that $f(-1)=2,{f}^{'}(-1)=1$ and for $x\in (-1,1)$ the maximum value of ${f}^{"}(x)$ is $\frac{1}{2}.$ If $f(x)\leq \alpha ,x\in [-1,1],$ then the least value of $\alpha$ is equal to

Let $f(x)=3{\mathrm{sin}}^{4}x+10{\mathrm{sin}}^{3}x+6{\mathrm{sin}}^{2}x-3$, $x\in [-\frac{\pi }{6},\frac{\pi }{2}]$. Then, $f$ is :

Let $f:[0,\infty )\rightarrow [0,3]$ be a function defined by $f(x)={\begin{matrix}\mathrm{max}{\mathrm{sin}t:0\leq t\leq \pi },x\in [0,\pi ] \\ 2+\mathrm{cos}x,x>\pi \end{matrix}$. Then which of the following is true ?

Let $f(x)$ be a polynomial of degree $6$ in $x,$ in which the coefficient of ${x}^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{f(x)}{{x}^{3}}=1,$ then $5\cdot f(2)$ is equal to

If $f(x)={\begin{matrix}\frac{1}{|x|} & ; & |x|\geq 1 \\ a{x}^{2}+b & ; & |x|<1\end{matrix}$ is differentiable at every point of the domain, then the values of $a$ and $b$ are respectively:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)((y+1){e}^{{x}^{2}/2}-x),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to

Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{x{y}^{2}+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of $y$, for which the point $(3,y)$ lies on the curve, is :

The population $P=P(t)$ at time $t$ of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450$. If $P(0)=850$, then the time at which population becomes zero is:

If $\int \frac{\mathrm{cos}x-\mathrm{sin}x}{\sqrt{8-\mathrm{sin}2x}}dx=a{\mathrm{sin}}^{-1}(\frac{\mathrm{sin}x+\mathrm{cos}x}{b})+c,$ where $c$ is a constant of integration, then the ordered pair $(a,b)$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}-55x, & \mathrm{if}x<-5 \\ 2{x}^{3}-3{x}^{2}-120x, & \mathrm{if}-5\leq x\leq 4 \\ 2{x}^{3}-3{x}^{2}-36x-336, & \mathrm{if}x>4\end{matrix}$ Let $A={x\in R:f$ is increasing}. Then $A$ is equal to:

If $y=y(x)$ is the solution of the equation ${e}^{\mathrm{sin}y}\mathrm{cos}y\frac{dy}{dx}+{e}^{\mathrm{sin}y}\mathrm{cos}x=\mathrm{cos}x,y(0)=0$; then $1+y(\frac{\pi }{6})+\frac{\sqrt{3}}{2}y(\frac{\pi }{3})+\frac{1}{\sqrt{2}}y(\frac{\pi }{4})$ is equal to _______.

The value of the integral ${\int }_{-1}^{1}\mathrm{log}(x+\sqrt{{x}^{2}+1})dx$ is:

Let $y=y(x)$ satisfies the equation $\frac{dy}{dx}-|A|=0,$ for all $x>0,$ where $A=[\begin{matrix}y & \mathrm{sin}x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{matrix}]$. If $y(\pi )=\pi +2,$ then the value of $y(\frac{\pi }{2})$ is:

The value of the limit $\underset{\theta \rightarrow 0}{\mathrm{lim}}\frac{\mathrm{tan}(\pi {\mathrm{cos}}^{2}\theta )}{\mathrm{sin}(2\pi {\mathrm{sin}}^{2}\theta )}$ is equal to :

Let ${J}_{n,m}={\int }_{0}^{1/2}\frac{{x}^{n}}{{x}^{m}-1}dx,\forall n>m$ and $n,m\in N.$ Consider a matrix $A={[{a}_{ij}]}_{3\times 3}$ where ${a}_{ij}={\begin{matrix}{J}_{6+i,3}-{J}_{i+3,3} & ,i\leq j \\ 0 & ,i>j\end{matrix}$. Then $|adj{A}^{-1}|$ is :

If $\int \frac{\mathrm{sin}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}dx=\alpha {\mathrm{log}}_{e}|1+\mathrm{tan}x|+\beta {\mathrm{log}}_{e}|1-\mathrm{tan}x+{\mathrm{tan}}^{2}x|+\gamma {\mathrm{tan}}^{-1}(\frac{2\mathrm{tan}x-1}{\sqrt{3}})+C,$ when $C$ is constant of integration, then the value of $18(\alpha +\beta +{\gamma }^{2})$ is