Calculus PYQ — Page 50

Let $f(x)={\mathrm{sin}}^{4}⁡x+{\mathrm{cos}}^{4}⁡x.$ Then, $f$ is an increasing function in the interval:

If $2\int _{0}^{1}{\mathrm{tan}}^{-1}xdx=\int _{0}^{1}{\mathrm{cot}}^{-1}(1-x+{x}^{2})dx$, then $\int _{0}^{1}{\mathrm{tan}}^{-1}(1-x+{x}^{2})dx$ is equal to

If $\int \frac{\mathrm{log}(t+\sqrt{1+{t}^{2}})}{\sqrt{1+{t}^{2}}}dt=\frac{1}{2}{(g(t))}^{2}+c$, where $c$ is a constant, then $g(2)$, is equal to

Let $y(x)$ be the solution of the differential equation $(x\mathrm{log}x)\frac{dy}{dx}+y=2x\mathrm{log}x, (x\geq 1)$. Then $y(e)$ is equal to

If the function $g(x)={\begin{matrix}k\sqrt{x+1}\text{ , } & 0\leq x\leq 3 \\ mx+2 \text{, } & 3<x\leq 5\end{matrix}$ is differentiable, then the value of $k+m$ is

For $x>0,$ let $f(x)={\int }_{1}^{x}\frac{\mathrm{log}⁡t}{1+t} dt.$ Then $f(x)+f(\frac{1}{x})$ is equal to

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{(1-cos2x)(3+cosx)}{xtan4x}=$

The area (in sq. units) of the region described by $[(x,y):{y}^{2}\leq 2x\mathrm{and}y\geq 4x-1]$ is

The solution of the differential equation $ydx-(x+2{y}^{2})dy=0$ is $x=f(y)$. If $f(-1)=1,$ then $f(1)$ is equal to

If $y (x)$ is the solution of the differential equation $(x+2)\frac{dy}{dx}={x}^{2}+4x-9, x \neq -2$ and $y(0)=0,$ then $y(-4)$ is equal to

$\underset{x\rightarrow 0}{\mathrm{lim}}⁡\frac{{e}^{{x}^{2}}-\mathrm{cos}⁡x}{{\mathrm{sin}}^{2}⁡ x}$is equal to

Let $f(x)$ be a polynomial of degree four and having its extreme values at $x=1$ and $x=2$. If $\underset{x\rightarrow 0}{\mathrm{lim}}[1+\frac{f(x)}{{x}^{2}}]=3$, then $f(2)$ is equal to

The integral $\int _{2}^{4}\frac{log{x}^{2}}{log{x}^{2}+log{(6-x)}^{2}}dx$ is equal to

The integral $\int \frac{dx}{{(x+1)}^{\frac{3}{4}}{(x-2)}^{\frac{5}{4}}}$, is equal to

Let $k$ be a non - zero real number. If $f(x)={\begin{matrix}\frac{{({e}^{x}-1)}^{2}}{\mathrm{sin} (\frac{x}{k})\mathrm{log} (1+\frac{x}{4})}\begin{matrix}, & x \neq 0\end{matrix} \\ \begin{matrix}12 , & x = 0\end{matrix}\end{matrix}$ is a continuous function at $x=0$, then the value of $k$ is

The area (in square units) of the region bounded by the curves $y+2{x}^{2}=0$ and $y+3{x}^{2}=1$, is equal to

Let $f:R\rightarrow R$ be a function such that $f(2-x)=f(2+x)$ and $f(4-x)=f(4+x)$, for all $x\in R$ and $\int _{0}^{2}f(x)dx=5$. Then the value of $\int _{10}^{50}f(x)dx$ is

Let $f:(-1, 1)\rightarrow R$ be a continuous function. If ${\int }_{0}^{\mathrm{sin}⁡x}f(t) dt=\frac{\sqrt{3}}{2}x,$ then $f(\frac{\sqrt{3}}{2})$ is equal to:

The integral $\int \frac{dx}{{x}^{2}{({x}^{4}+1)}^{\frac{3}{4}}}$ equals to

Let $k$ and $K$ be the minimum and the maximum values of the function $f(x)=\frac{{(1+x)}^{0.6}}{1+{x}^{0.6}} \text{in} [0, 1]$, respectively, then the ordered pair $(k,K)$ is equal to:

If $f(x)$ is continuous and $f(\frac{9}{2})=\frac{2}{9},$ then $\underset{x\rightarrow 0}{\mathrm{lim}}f(\frac{1-\mathrm{cos}3x}{{x}^{2}})$ equals to

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $=\sqrt{3}$ is:

If [ ] denotes the greatest integer function, then the integral $\int_0^\pi[\cos x d x$ is equal to:

If $\lim _{x \rightarrow 2} \frac{\tan \left(x-2\left\{x^2+k+2 x-2 k\right\}\right.}{x^2-4 x+4}=5$, then $\mathrm{k}$ is equal to: