JEE Main 2021 — Calculus

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:

A function $f$ is defined on $[-3,3]$ as $f(x)={\begin{matrix}\mathrm{min}{|x|,2-{x}^{2}},-2\leq x\leq 2 \\ [|x|],2<|x|\leq 3\end{matrix}$ where $[x]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $(-3,3)$ is ___ .

A wire of length $36m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $(\frac{4}{\pi }+1)k$ is equal to

A wire of length $20m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

Consider the function $f(x)=\frac{P(x)}{\mathrm{sin}(x-2)},x\neq 2$, and $f(x)=7,x=2$where $P(x)$ is a polynomial such that ${P}^{"}(x)$ is always a constant and $P(3)=9.$ If $f(x)$ is continuous at $x=2,$ then $P(5)$ is equal to __________.

Consider the function $f:R\rightarrow R$ defined by $f(x)={\begin{matrix}(2-\mathrm{sin}(\frac{1}{x}))|x|, & x\neq 0 \\ 0, & x=0\end{matrix}.$ Then $f$ is:

Consider the integral $I={\int }_{0}^{10}\frac{[x]{e}^{[x]}}{{e}^{x-1}}dx$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to :

For $x>0$, if $f(x)={\int }_{1}^{x}\frac{{\mathrm{log}}_{e}t}{(1+t)}dt$, then $f(e)+f(\frac{1}{e})$ is equal to

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 ______ .

If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^{4},$ then for what value of $b,{\int }_{1}^{2}f(x)dx=\frac{62}{5}?$

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 $\alpha =\underset{x\rightarrow \pi /4}{\mathrm{lim}}\frac{{\mathrm{tan}}^{3}x-\mathrm{tan}x}{\mathrm{cos}(x+\frac{\pi }{4})}$ 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 :

If $f(x)=\mathrm{sin}({\mathrm{cos}}^{-1}(\frac{1-{2}^{2x}}{1+{2}^{2x}}))$ and its first derivative with respect to $x$ is $-\frac{b}{a}{\mathrm{log}}_{e}2$ when $x=1,$ where $a$ and $b$ are integers, then the minimum value of $|{a}^{2}-{b}^{2}|$ is _______.

If $y\frac{dy}{dx}=x[\frac{{y}^{2}}{{x}^{2}}+\frac{\phi (\frac{{y}^{2}}{{x}^{2}})}{{\phi }^{'}(\frac{{y}^{2}}{{x}^{2}})}],x>0,\phi >0,$ and $y(1)=-1,$ then $\phi (\frac{{y}^{2}}{4})$ is equal to:

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 _______.

If $f(x)=\int \frac{5{x}^{8}+7{x}^{6}}{{({x}^{2}+1+2{x}^{7})}^{2}}dx,(x\geq 0),f(0)=0$ and $f(1)=\frac{1}{K},$ then the value of $K$ is

If $\alpha ,\beta$ are the distinct roots of ${x}^{2}+bx+c=0,$ then $\underset{x\rightarrow \beta }{\mathrm{lim}}\frac{{e}^{2({x}^{2}+bx+c)}-1-2({x}^{2}+bx+c)}{(x-\beta {)}^{2}}$ is equal to

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral ${\int }_{-\pi /2}^{\pi /2}[[x]-\mathrm{sin}x]dx$ is equal to:

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{ax-({e}^{4x}-1)}{ax({e}^{4x}-1)}$ exists and is equal to $b$, then the value of $a-2b$ is ___ .

If ${I}_{m,n}={\int }_{0}^{1}{x}^{m-1}{(1-x)}^{n-1}dx$, for $m,n\geqslant 1$, and ${\int }_{0}^{1}\frac{{x}^{m-1}+{x}^{n-1}}{{(1+x)}^{m+n}}dx=\alpha {I}_{m,n},\alpha \in R$, then $\alpha$ equals ________.

If $f:R\rightarrow R$ is a function defined by $f(x)=[x-1]\mathrm{cos}(\frac{2x-1}{2})\pi ,$ where $[\cdot ]$ denotes the greatest integer function, then $f$ is:

If $y=y(x)$ is an implicit function of $x$ such that ${\mathrm{log}}_{e}(x+y)=4xy$, then $\frac{{d}^{2}y}{d{x}^{2}}$ at $x=0$ 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:

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{-1}x-{\mathrm{tan}}^{-1}x}{3{x}^{3}}$ is equal to $L,$ then the value of $(6L+1)$ is

If $[x]$ is the greatest integer $\leq x,$ then ${\pi }^{2}{\int }_{0}^{2}(\mathrm{sin}\frac{\pi x}{2}){(x-[x])}^{[x]}dx$ is equal to :

If $R$ is the least value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is increasing on $[1,2]$ and $S$ is the greatest value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is decreasing on $[1,2]$, then the value of $|R-S|$ is

If $y=y(x)$ is the solution curve of the differential equation ${x}^{2}dy+(y-\frac{1}{x})dx=0;x>0$ and $y(1)=1,$ then $y(\frac{1}{2})$ is equal to :

If $y=y(x),y\in [0,\frac{\pi }{2})$ is the solution of the differential equation $\mathrm{sec}y\frac{dy}{dx}-\mathrm{sin}(x+y)-\mathrm{sin}(x-y)=0,$ with $y(0)=0$, then $5{y}^{'}(\frac{\pi }{2})$ is equal to _____.

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\mathrm{tan}x)y=\mathrm{sin}x,0\leq x\leq \frac{\pi }{3},$ with $y(0)=0,$ then $y(\frac{\pi }{4})$ is equal to

If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\mathrm{tan}x=\mathrm{sin}x,y(\frac{\pi }{3})=0$, then the maximum value of the function $y(x)$ over $R$ 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 _______.

If $[\cdot ]$ represents the greatest integer function, then the value of $|{\int }_{0}^{\sqrt{\frac{\pi }{2}}}[[{x}^{2}]-\mathrm{cos}x]dx|$ is ___________.

If the area of the bounded region $R={(x,y):\mathrm{max}{0,{\mathrm{log}}_{e}x}\leq y\leq {2}^{x},\frac{1}{2}\leq x\leq 2}$ is, $\alpha {({\mathrm{log}}_{e}2)}^{-1}+\beta ({\mathrm{log}}_{e}2)+\gamma$ then the value of ${(\alpha +\beta -2\gamma )}^{2}$ is equal to:

If the curve $y=y(x)$ is the solution of the differential equation $2({x}^{2}+{x}^{5/4})dy-y(x+{x}^{1/4})dx=2{x}^{9/4}dx,x>0$ which passes through the point $(1,1-\frac{4}{3}{\mathrm{log}}_{e}2),$ then the value of $y(16)$ is equal to

If the curve, $y=y(x)$ represented by the solution of the differential equation $(2x{y}^{2}-y)dx+xdy=0$, passes through the intersection of the lines, $2x-3y=1$ and $3x+2y=8$, then $|y(1)|$ is equal to ___ .

If the function $f(x)=\frac{\mathrm{cos}(\mathrm{sin}x)-\mathrm{cos}x}{{x}^{4}}$ is continuous at each point in its domain and $f(0)=\frac{1}{k},$ then $k$ is _________.

If the function $f(x)={\begin{matrix}\frac{1}{x}{\mathrm{log}}_{e}(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}),x<0 \\ k,x=0 \\ \frac{{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1}{\sqrt{{x}^{2}+1}-1},x>0\end{matrix}$ is continuous at $x=0,$ then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :

If the integral ${\int }_{0}^{10}\frac{[\mathrm{sin}2\pi x]}{{e}^{x-[x]}}dx=\alpha {e}^{-1}+\beta {e}^{-\frac{1}{2}}+\gamma ,$ where $\alpha ,\beta ,\gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

If the line $y=mx$ bisects the area enclosed by the lines $x=0,y=0,x=\frac{3}{2}$ and the curve $y=1+4x-{x}^{2},$ then $12m$ is equal to .

If the normal to the curve $y(x)={\int }_{0}^{x}(2{t}^{2}-15t+10)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5,a>1$, then the value of $|a+6b|$ is equal to ________.

If the solution curve of the differential equation $(2x-10{y}^{3})dy+ydx=0$, passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation?

If the value of $\underset{x\rightarrow 0}{\mathrm{lim}}{(2-\mathrm{cos}x\sqrt{\mathrm{cos}2x})}^{(\frac{x+2}{{x}^{2}})}$ is equal to ${e}^{a}$, then $a$ is equal to_____.

If the value of the integral ${\int }_{0}^{5}\frac{x+[x]}{{e}^{x-[x]}}dx=\alpha {e}^{-1}+\beta ,$ where $\alpha ,\beta \in R,5\alpha +6\beta =0,$ and $[x]$ denotes the greatest integer less than or equal to $x;$ then the value of $(\alpha +\beta {)}^{2}$ is equal to :

If $y(x)={\mathrm{cot}}^{-1}(\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1-\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}x}-\sqrt{1-\mathrm{sin}x}}),x\in (\frac{\pi }{2},\pi )$, then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is:

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 :

If ${I}_{n}={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\mathrm{cot}}^{n}xdx$, then

If $x\phi (x)={\int }_{5}^{x}(3{t}^{2}-2{\phi }^{'}(t))dt,x>-2,$ $\phi (0)=4,$ then $\phi (2)$ is

If $\frac{dy}{dx}=\frac{{2}^{x+y}-{2}^{x}}{{2}^{y}},y(0)=1,$ then $y(1)$ is equal to :

If ${\int }_{0}^{\pi }({\mathrm{sin}}^{3}x){e}^{-{\mathrm{sin}}^{2}x}dx=\alpha -\frac{\beta }{e}{\int }_{0}^{1}\sqrt{t}{e}^{t}dt,$ then $\alpha +\beta$ is equal to

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{a{e}^{x}-b\mathrm{cos}x+c{e}^{-x}}{x\mathrm{sin}x}=2$, then $a+b+c$ 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

If $\underset{x\rightarrow \infty }{\mathrm{lim}}(\sqrt{{x}^{2}-x+1}-ax)=b$, then the ordered pair $(a,b)$ is:

If $\underset{x\rightarrow 0}{\mathrm{lim}}[\frac{\alpha x{e}^{x}-\beta {\mathrm{log}}_{e}(1+x)+\gamma {x}^{2}{e}^{-x}}{x{\mathrm{sin}}^{2}x}]=10,\alpha ,\beta ,\gamma \in R$, then the value of $\alpha +\beta +\gamma$ 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

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:

If $\int \frac{dx}{{({x}^{2}+x+1)}^{2}}=a{\mathrm{tan}}^{-1}(\frac{2x+1}{\sqrt{3}})+b(\frac{2x+1}{{x}^{2}+x+1})+C,x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3}a+b)$ is equal to _________.

If ${\int }_{0}^{100\pi }\frac{{\mathrm{sin}}^{2}x}{{e}^{(\frac{x}{\pi }-[\frac{x}{\pi }])}}dx=\frac{\alpha {\pi }^{3}}{1+4{\pi }^{2}},\alpha \in R$ where $[x]$ is the greatest integer less than or equal to $x,$ then the value of $\alpha$ is:

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}(\pi {\mathrm{cos}}^{4}x)}{{x}^{4}}$ is equal to :

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\int }_{0}^{{x}^{2}}(\mathrm{sin}\sqrt{t})dt}{{x}^{3}}$ is equal to:

$\underset{n\rightarrow \infty }{\mathrm{lim}}{(1+\frac{1+\frac{1}{2}+\ldots \ldots +\frac{1}{n}}{{n}^{2}})}^{n}$ is equal to

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

$\underset{x\rightarrow 2}{\mathrm{lim}}(\sum _{n=1}^{9}\frac{x}{n(n+1){x}^{2}+2(2n+1)x+4})$ is equal to :

Let a curve $y=y(x)$ be given by the solution of the differential equation $\mathrm{cos}(\frac{1}{2}{\mathrm{cos}}^{-1}({e}^{-x}))dx=(\sqrt{{e}^{2x}-1})dy$. If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x-$axis is $(\alpha ,0)$, then ${e}^{\alpha }$ is equal to

Let a function $g:[0,4]\rightarrow R$ be defined as $g(x)={\begin{matrix}\underset{0\leq t\leq x}{\mathrm{max}{{t}^{3}-6{t}^{2}+9t-3}}, & 0\leq x\leq 3 \\ 4-x, & 3<x\leq 4\end{matrix}$ then the number of points in the interval $(0,4)$ where $g(x)$ is NOT differentiable, is _________.

Let a function $f:R\rightarrow R$ be defined as, $f(x)={\begin{matrix}\mathrm{sin}x-{e}^{x} & \mathrm{if}x\leq 0 \\ a+[-x] & \mathrm{if}0<x<1 \\ 2x-b & \mathrm{if}x\geq 1\end{matrix}$ Where $[x]$ is the greatest integer less than or equal to $x.$ If $f$ is continuous on $R$, then $(a+b)$ is equal to:

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 $f(x)$ and $g(x)$ be two functions satisfying $f({x}^{2})+g(4-x)=4{x}^{3}$ and $g(4-x)+g(x)=0,$ then the value of ${\int }_{-4}^{4}f({x}^{2})dx$ is

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 :

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:

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2{x}^{3}-3{x}^{2}-12x$. If $A$ is the total area of the region bounded by $y=f(x),$ the $x$-axis and the lines $x=a$ and $x=b,$ then $4A$ is equal to ______.

Let $f:R\rightarrow R$ be a continuous function such that $f(x)+f(x+1)=2$ for all $x\in R$. If ${I}_{1}={\int }_{0}^{8}f(x)dx$ and ${I}_{2}={\int }_{-1}^{3}f(x)dx$, then the value of ${I}_{1}+2{I}_{2}$ 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 $f(x)$ be a cubic polynomial with $f(1)=-10,f(-1)=6,$ and has a local minima at $x=1,$ and ${f}^{'}(x)$ has a local minima at $x=-1.$ Then $f(3)$ is equal to .

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 $f(x)$ be a differentiable function defined on $[0,2]$ such that ${f}^{'}(x)={f}^{'}(2-x)$ for all $x\in (0,2),f(0)=1$ and $f(2)={e}^{2}.$ Then the value of ${\int }_{0}^{2}f(x)dx$ is

Let $f(x)={\int }_{0}^{x}{e}^{t}f(t)dt+{e}^{x}$ be a differentiable function for all $x\in R$. Then $f(x)$ equals :

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix}\frac{\mathrm{sin}(a+1)x+\mathrm{sin}2x}{2x} & ,\mathrm{if}x<0 \\ b & ,\mathrm{if}x=0 \\ \frac{\sqrt{x+b{x}^{3}}-\sqrt{x}}{b{x}^{5/2}} & ,\mathrm{if}x>0\end{matrix}$ If $f$ is continuous at $x=0$, then the value of $a+b$ is equal to :

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix}3(1-\frac{|x|}{2}) & \text{if} & |x|\leq 2 \\ 0 & \text{if} & |x|>2\end{matrix}.$ Let $g:R\rightarrow R$ be given by $g(x)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

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 $f:R\rightarrow R$ be a function such that $f(2)=4$ and ${f}^{'}(2)=1.$ Then, the value of $\underset{x\rightarrow 2}{\mathrm{lim}}\frac{{x}^{2}f(2)-4f(x)}{x-2}$ is equal to:

Let $A=[{a}_{ij}]$ be a $3\times 3$ matrix, where ${a}_{ij}={\begin{matrix}1 & , & \mathrm{if}i=j \\ -x & , & \mathrm{if}|i-j|=1 \\ 2x+1 & , & \mathrm{otherwise}\end{matrix}$ Let a function $f:R\rightarrow R$ be defined as $f(x)=det(A)$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1).$ If ${\int }_{0}^{x}\sqrt{1-{({f}^{'}(t))}^{2}}\mathrm{dt}={\int }_{0}^{x}f(t)\mathrm{dt},0\leq x\leq 1$ and $f(0)=0,$ then $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{1}{{x}^{2}}{\int }_{0}^{x}f(t)\mathrm{dt}:$

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

Let $a$ be a positive real number such that ${\int }_{0}^{a}{e}^{x-[x]}dx=10e-9$ where, $[x]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to:

Let $P(x)={x}^{2}+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_{0}^{1}P(x)dx=1$ and $P(x)$ leaves remainder $5$ when it is divided by $(x-2)$ Then the value of $9(b+c)$ is equal to:

Let $a$ be a real number such that the function $f(x)=a{x}^{2}+6x-15,x\in R$ is increasing in $(-\infty ,\frac{3}{4})$ and decreasing in $(\frac{3}{4},\infty )$. Then the function $g(x)=a{x}^{2}-6x+15,x\in R$ has a

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 ___ .

Let $f$ be a real valued function, defined on$R-{-1,1}$ and given by $f(x)=3{\mathrm{log}}_{e}|\frac{x-1}{x+1}|-\frac{2}{x-1}$. Then in which of the following intervals, function $f(x)$ is increasing?

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:

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1,{f}^{'}(0)=2$ and ${f}^{'}(x)\neq 0$ for all $x\in R.$ If $|\begin{matrix}f(x) & {f}^{'}(x) \\ {f}^{'}(x) & {f}^{''}(x)\end{matrix}|=0,$ for all $x\in R,$ then the value of $f(1)$ lies in the interval

Let $F:[3,5]\rightarrow R$ be a twice differentiable function on $(3,5)$ such that $F(x)={e}^{-x}{\int }_{3}^{x}(3{t}^{2}+2t+4{F}^{'}(t))dt.$ If ${F}^{'}(4)=\frac{\alpha {e}^{\beta }-224}{{({e}^{\beta }-4)}^{2}}$, then $\alpha +\beta$ is equal to _____.

Let $a$ be an integer such that all the real roots of the polynomial $2{x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}+10x+10$ lie in the interval $(a,a+1)$. Then, $|a|$ 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 :

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 $f:R\rightarrow R$ be defined as $f(x)={e}^{-x}\mathrm{sin}x.$ If $F:[0,1]\rightarrow R$ is a differentiable function such that $F(x)={\int }_{0}^{x}f(t)dt,$ then the value of ${\int }_{0}^{1}({F}^{'}(x)+f(x)){e}^{x}dx$ lies in the interval

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{{x}^{3}}{(1-\mathrm{cos}2x{)}^{2}}{\mathrm{log}}_{e}(\frac{1+2x{e}^{-2x}}{{(1-x{e}^{-x})}^{2}}) & ,x\neq 0 \\ \alpha & ,x=0\end{matrix}$ If $f$ is continuous at $x=0,$ then $\alpha$ is equal to:

Let $f:(-\frac{\pi }{4},\frac{\pi }{4})\rightarrow R$ be defined as, $f(x)={\begin{matrix}{(1+|\mathrm{sin}x|)}^{\frac{3a}{|\mathrm{sin}x|}} & ,-\frac{\pi }{4}<x<0 \\ b & ,x=0 \\ {e}^{\mathrm{cot}4x/\mathrm{cot}2x} & ,0<x<\frac{\pi }{4}\end{matrix}$ If $f$ is continuous at $x=0$ then the value of $6a+{b}^{2}$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}2\mathrm{sin}(-\frac{\pi x}{2}), & \mathrm{if}x<-1 \\ |a{x}^{2}+x+b|, & \mathrm{if}-1\leq x\leq 1 \\ \mathrm{sin}(\pi x), & \mathrm{if}x>1\end{matrix}$ If $f(x)$ is continuous on $R$, then $a+b$ equals :

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:

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 $f:[0,\infty )\rightarrow [0,\infty )$ be defined as $f(x)={\int }_{0}^{x}[y]dy$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{\lambda |{x}^{2}-5x+6|}{\mu (5x-{x}^{2}-6)} & x<2 \\ {e}^{\frac{\mathrm{tan}(x-2)}{x-[x]}} & x>2 \\ \mu & x=2\end{matrix}$ where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x=2$, then $\lambda +\mu$ is equal to :

Let $f:[0,3]\rightarrow R$ be defined by $f(x)=\mathrm{min}{x-[x],1+[x]-x}$ where $[x]$ is the greatest integer less than or equal to $x.$ Let $P$ denote the set containing all $x\in [0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in (0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to _____.

Let $f:[-3,1]\rightarrow R$ be given as $f(x)={\begin{matrix}min{(x+6),{x}^{2}}, & -3\leq x\leq 0 \\ max{\sqrt{x},{x}^{2}}, & 0\leq x\leq 1\end{matrix}.$If the area bounded by $y=f(x)$ and $x$-axis is $A$ sq units, then the value of $6A$ is equal to

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 $y=y(x)$ be solution of the following differential equation ${e}^{y}\frac{dy}{dx}-2{e}^{y}\mathrm{sin}x+\mathrm{sin}x{\mathrm{cos}}^{2}x=0,y(\frac{\pi }{2})=0$. If $y(0)={\mathrm{log}}_{e}(\alpha +\beta {e}^{-2})$, then $4(\alpha +\beta )$ is equal to .

Let $\alpha \in R$ be such that the function $f(x)={\begin{matrix}\frac{{\mathrm{cos}}^{-1}(1-{{x}}^{2}){\mathrm{sin}}^{-1}(1-{x})}{{x}-{{x}}^{3}}, & x\neq 0 \\ \alpha , & x=0\end{matrix}$ is continuous at $x=0,$ where ${x}=x-[x],[x]$ is the greatest integer less than or equal to $x$. Then :

Let ${A}_{1}$ be the area of the region bounded by the curves $y=\mathrm{sin}x,y=\mathrm{cos}x$ and $y$-axis in the first quadrant. Also, let ${A}_{2}$ be the area of the region bounded by the curves $y=\mathrm{sin}x,y=\mathrm{cos}x$, $x$-axis and $x=\frac{\pi }{2}$ in the first quadrant. Then,

Let ${C}_{1}$ be the curve obtained by the solution of differential equation $2xy\frac{dy}{dx}={y}^{2}-{x}^{2},x>0$. Let the curve ${C}_{2}$ be the solution of $\frac{2xy}{{x}^{2}-{y}^{2}}=\frac{dy}{dx}$. If both the curves pass through $(1,1),$ then the area (in sq. units) enclosed by the curves ${C}_{1}$ and ${C}_{2}$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $(x-{x}^{3})dy=(y+y{x}^{2}-3{x}^{4})dx,x>2$ If $y(3)=3,$ then $y(4)$ is equal to:

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 $y=y(x)$ be the solution of the differential equation $((x+2){e}^{(\frac{y+1}{x+2})}+(y+1))dx=(x+2)dy,y(1)=1.$ If the domain of $y=y(x)$ is an open interval $(\alpha ,\beta ),$ then $|\alpha +\beta |$ is equal to ___________.

Let $y=y(x)$ be the solution of the differential equation $x\mathrm{tan}(\frac{y}{x})dy=(y\mathrm{tan}(\frac{y}{x})-x)dx$, $-1\leq x\leq 1,y(\frac{1}{2})=\frac{\pi }{6}.$ Then the area of the region bounded by the curves $x=0,x=\frac{1}{\sqrt{2}}$ and $y=y(x)$ in the upper half plane 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 $y=y(x)$ be the solution of the differential equation $\mathrm{cos}x(3\mathrm{sin}x+\mathrm{cos}x+3)dy=(1+y\mathrm{sin}x(3\mathrm{sin}x+\mathrm{cos}x+3))dx,0\leq x\leq \frac{\pi }{2},y(0)=0.$ Then, $y(\frac{\pi }{3})$ is equal to:

Let $y(x)$ be the solution of the differential equation $2{x}^{2}dy+({e}^{y}-2x)dx=0,x>0.$ If $y(e)=1,$ then $y(1)$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $xdy-ydx=\sqrt{({x}^{2}-{y}^{2})}dx,x\geq 1$, with $y(1)=0$. If the area bounded by the line $x=1,x={e}^{\pi },y=0$ and $y=y(x)$ is $\alpha {e}^{2\pi }+\beta ,$ then the value of $10(\alpha +\beta )$ is equal to ___ .

Let $y=y(x)$ be the solution of the differential equation ${cosec}^{2}xdy+2dx=(1+y\mathrm{cos}2x){cosec}^{2}xdx,$ with $y(\frac{\pi }{4})=0.$ Then, the value of ${(y(0)+1)}^{2}$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $xdy=(y+{x}^{3}\mathrm{cos}x)dx$ with $y(\pi )=0,$ then $y(\frac{\pi }{2})$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\mathrm{sin}x-5)x-2\mathrm{cos}x$ such that $y(0)=7.$ Then $y(\pi )$ is equal to

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 $y=y(x)$ be the solution of the differential equation $dy={e}^{\alpha x+y}dx;\alpha \in N.$ If $y({\mathrm{log}}_{e}2)={\mathrm{log}}_{e}2$ and $y(0)={\mathrm{log}}_{e}(\frac{1}{2}),$ then the value of $\alpha$ is equal to ___.

Let $T$ be the tangent to the ellipse $E:{x}^{2}+4{y}^{2}=5$ at the point $P(1,1)$. If the area of the region bounded by the tangent $T$, ellipse $E$, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\mathrm{cos}}^{-1}(\frac{1}{\sqrt{5}})$, then $|\alpha +\beta +\gamma |$ is equal to______.

Let $a,b\in R,b\neq 0$. Defined a function, $f(x)={\begin{matrix}a\mathrm{sin}\frac{\pi }{2}(x-1),\mathrm{for}x\leq 0 \\ \frac{\mathrm{tan}2x-\mathrm{sin}2x}{b{x}^{3}},\mathrm{for}x>0\end{matrix}$ If $f$ is continuous at$x=0$, then $10-ab$ is equal to

Let $[t]$ denote the greatest integer less than or equal to $t.$ Let $f(x)=x-[x],g(x)=1-x+[x],$ and $h(x)=\mathrm{min}{f(x),g(x)},x\in [-2,2].$ Then $h$ is :

Let $[t]$ denote the greatest integer $\leq t.$ The number of points where the function $f(x)=[x]|{x}^{2}-1|+\mathrm{sin}(\frac{\pi }{[x]+3})-[x+1],x\in (-2,2)$ is not continuous is _____ .

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

Let $f:R\rightarrow R$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in R$ and $f(x)\neq 0$ for any $x\in R$. If the function $f$ is differentiable at $x=0$ and ${f}^{'}(0)=3$, then $\underset{h\rightarrow 0}{\mathrm{lim}}\frac{1}{h}(f(h)-1)$ 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 :

Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis is $\frac{4\sqrt{8}}{3}$, then the value of $y(1)$ is equal to ________.

Let the functions $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as : $f(x)={\begin{matrix}x+2, & x<0 \\ {x}^{2}, & x\geq 0\end{matrix}$ and $g(x)={\begin{matrix}{x}^{3}, & x<1 \\ 3x-2, & x\geq 1\end{matrix}$ Then, the number of points in $R$ where $(fog)(x)$ is NOT differentiable 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 $f(x)={x}^{6}+2{x}^{4}+{x}^{3}+2x+3,x\in R$. 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 $f(x)=\mathrm{cos}(2{\mathrm{tan}}^{-1}\mathrm{sin}({\mathrm{cot}}^{-1}\sqrt{\frac{1-x}{x}})),0<x<1$. Then:

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

Let $f:S\rightarrow S$ where $S=(0,\infty )$ be a twice differentiable function such that $f(x+1)=xf(x)$. If $g:S\rightarrow R$ be defined as $g(x)={\mathrm{log}}_{e}f(x)$, then the value of $|{g}^{''}(5)-{g}^{''}(1)|$ is equal to :

Let ${I}_{n}={\int }_{1}^{e}{x}^{19}(\mathrm{log}|x|{)}^{n}dx,$ where $n\in N$. If $(20){I}_{10}=\alpha {I}_{9}+\beta {I}_{8},$ for natural numbers $\alpha$ and $\beta$, then $\alpha -\beta$ 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 :

Let $g(t)={\int }_{-\pi /2}^{\pi /2}(\mathrm{cos}\frac{\pi }{4}t+f(x))dx,$ where $f(x)={\mathrm{log}}_{e}(x+\sqrt{{x}^{2}+1}),x\in R.$ Then which one of the following is correct?

lim(x→0) (sin x)/x is equal to:

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

The area bounded by the lines $y=||x-1|-2|$ and $y=2$ is _____.

The area, enclosed by the curves $y=\mathrm{sin}x+\mathrm{cos}x$ and $y=|\mathrm{cos}x-\mathrm{sin}x|$ and the lines $x=0,x=\frac{\pi }{2},$ is :

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

The area (in sq. units) of the part of the circle ${x}^{2}+{y}^{2}=36$, which is outside the parabola ${y}^{2}=9x$, is equal to

The area (in sq. units) of the region bounded by the curves ${x}^{2}+2y-1=0,{y}^{2}+4x-4=0$ and ${y}^{2}-4x-4=0$ in the upper half plane is _________.

The area (in sq. units) of the region, given by the set ${(x,y)\in R\times R\mid x\geq 0,2{x}^{2}\leq y\leq 4-2x}$ is :

The area of the region bounded by $y-x=2$ and ${x}^{2}=y$ is equal to :-

The area of the region bounded by the parabola $(y-2{)}^{2}=(x-1)$, the tangent to it at the point whose ordinate is $3$ and the $x$-axis, is:

The area of the region: $R={(x,y):5{x}^{2}\leq y\leq 2{x}^{2}+9}$ is

The area of the region $S={(x,y):3{x}^{2}\leq 4y\leq 6x+24}$ is______.

The difference between degree and order of a differential equation that represents the family of curves given by ${y}^{2}=a(x+\frac{\sqrt{a}}{2}),a>0$ is _______.

The function $f(x)=|{x}^{2}-2x-3|\cdot {e}^{9{x}^{2}-12x+4}$ is not differentiable at exactly :

The function $f(x)={x}^{3}-6{x}^{2}+ax+b$ is such that $f(2)=f(4)=0$. Consider two statements: $({S}_{1})$ there exists ${x}_{1},{x}_{2}\in (2,4),{x}_{1}<{x}_{2},$ such that ${f}^{'}({x}_{1})=-1$ and ${f}^{'}({x}_{2})=0.$ $({S}_{2})$ there exists ${x}_{3},{x}_{4}\in (2,4),{x}_{3}<{x}_{4}$, such that $f$ is decreasing in $(2,{x}_{4})$, increasing in $({x}_{4},4)$ and $2{f}^{'}({x}_{3})=\sqrt{3}f({x}_{4})$ then

The function $f(x)=\frac{4{x}^{3}-3{x}^{2}}{6}-2\mathrm{sin}x+(2x-1)\mathrm{cos}x$:

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

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then ${A}^{4}$ is equal to

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 integral $\int \frac{1}{\sqrt[4]{(x-1{)}^{3}(x+2{)}^{5}}}dx$ is equal to : (where $C$ is a constant of integration)

The integral $\int \frac{{e}^{3{\mathrm{log}}_{e}2x}+5{e}^{2{\mathrm{log}}_{e}2x}}{{e}^{4{\mathrm{log}}_{e}x}+5{e}^{3{\mathrm{log}}_{e}x}-7{e}^{2{\mathrm{log}}_{e}x}}dx,x>0$, is equal to (where $c$ is a constant of integration)

The local maximum value of the function, $f(x)={(\frac{2}{x})}^{{x}^{2}},x>0,$ is

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

The minimum value of $\alpha$ for which the equation $\frac{4}{\mathrm{sin}x}+\frac{1}{1-\mathrm{sin}x}=\alpha$ has at least one solution in $(0,\frac{\pi }{2})$ is______.

The number of distinct real roots of the equation $3{x}^{4}+4{x}^{3}-12{x}^{2}+4=0$ is _________.

The number of points, at which the function $f(x)=|2x+1|-3|x+2|+|{x}^{2}+x-2|,x\in R$ is not differentiable, is

The number of real roots of the equation ${e}^{6x}-{e}^{4x}-2{e}^{3x}-12{e}^{2x}+{e}^{x}+1=0$ 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:

The range of $a\in R$ for which the function $f(x)=(4a-3)(x+{\mathrm{log}}_{e}5)+2(a-7)\mathrm{cot}(\frac{x}{2}){\mathrm{sin}}^{2}(\frac{x}{2}),x\neq 2n\pi ,n\in N$, has critical points, is :

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t=0.$ The number of bacteria is increased by $20%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{{\mathrm{log}}_{e}(\frac{6}{5})}$ hours, then ${(\frac{k}{{\mathrm{log}}_{e}2})}^{2}$ is equal to:

The sum of all the local minimum values of the twice differentiable function $f:R\rightarrow R$ defined by $f(x)={x}^{3}-3{x}^{2}-\frac{3{f}^{"}(2)}{2}x+{f}^{"}(1)$ is:

The triangle of maximum area that can be inscribed in a given circle of radius '$r$' is :

The value of $\underset{x\rightarrow 0}{\mathrm{lim}}(\frac{x}{\sqrt[8]{1-\mathrm{sin}x}-\sqrt[8]{1+\mathrm{sin}x}})$ is equal to :

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\mathrm{cos}}^{2}x}{1+{3}^{x}}dx$ is:

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{1+{\mathrm{sin}}^{2}x}{1+{\pi }^{\mathrm{sin}x}})dx$ is :

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

The value of ${\int }_{-2}^{2}|3{x}^{2}-3x-6|dx$ is

The value of $\underset{h\rightarrow 0}{\mathrm{lim}}{\frac{\sqrt{3}\mathrm{sin}(\frac{\pi }{6}+h)-\mathrm{cos}(\frac{\pi }{6}+h)}{\sqrt{3}h(\sqrt{3}\mathrm{cos}h-\mathrm{sin}h)}}$ is :

The value of the definite integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{dx}{(1+{e}^{x\mathrm{cos}x})({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x)}$ is equal to :

The value of the definite integral ${\int }_{\pi /24}^{5\pi /24}\frac{dx}{1+\sqrt[3]{\mathrm{tan}2x}}$ is

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

The value of the integral ${\int }_{0}^{\pi }|\mathrm{sin}2x|dx$ is ________.

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

The value of the integral ${\int }_{0}^{1}\frac{\sqrt{x}dx}{(1+x)(1+3x)(3+x)}$ is:

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)

The value of the integral, ${\int }_{1}^{3}[{x}^{2}-2x-2]dx,$ where $[x]$ denotes the greatest integer less than or equal to $x,$ 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 :

The value of $\underset{x\rightarrow {0}^{+}}{\mathrm{lim}}\frac{{\mathrm{cos}}^{-1}(x-[x{]}^{2})\cdot {\mathrm{sin}}^{-1}(x-[x{]}^{2})}{x-{x}^{3}},$ where $[x]$ denotes the greatest integer $\leq x$ is:

The value of ${\int }_{-1}^{1}{x}^{2}{e}^{[{x}^{3}]}dx,$ where $[t]$ denotes the greatest integer $\leq t,$ 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 :

$\int \frac{2{e}^{x}+3{e}^{-x}}{4{e}^{x}+7{e}^{-x}}dx=\frac{1}{14}(ux+v{\mathrm{log}}_{e}(4{e}^{x}+7{e}^{-x}))+C$, where $C$ is a constant of integration, then $u+v$ is equal to

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y;y(0)=0$

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