Let $$C$$ be the circle of minimum area touching the parabola $$y=6-x^2$$ and the lines $$y=\sqrt{3}|x|$$. Then, which one of the following points lies on the circle $$C$$ ?

If $\underset{0}{\overset{\frac{\pi }{3}}{\int }}{\cos }^{4}x\mathrm{d}x=\mathrm{a}\pi +\mathrm{b}\sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers, then $9\mathrm{a}+8\mathrm{b}$ is equal to :

The value of $\underset{0}{\overset{1}{\int }}{\left(2x^3-3x^2-x+1\right)}^{\frac{1}{3}}\mathrm{d}x$ is equal to :

The value of the integral $\underset{0}{\overset{\pi /4}{\int }}\frac{x\mathrm{d}x}{{\sin }^4(2x)+{\cos }^4(2x)}$ equals :

If $\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}\mathrm{d}x=\mathrm{a}+\mathrm{b}\sqrt{2}+\mathrm{c}\sqrt{3}$, where $\mathrm{a},\mathrm{b},\mathrm{c}$ are rational numbers, then $2\mathrm{a}+3\mathrm{b}-4\mathrm{c}$ is equal to :

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and ${\mathrm{I}}_1=\underset{\mathrm{a}}{\overset{\mathrm{b}}{\int }}x\sin \left(4x-x^2\right)\mathrm{d}x,{\mathrm{I}}_2=\underset{\mathrm{a}}{\overset{\mathrm{b}}{\int }}\sin \left(4x-x^2\right)\mathrm{d}x$, then $36\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}$ is equal to :

For $$0

Let $$f,g:(0,\infty )\to \mathbb{R}$$ be two functions defined by $$f(x)=\underset{-x}{\overset{x}{\int }}\left(|t|-t^2\right)e^{-t^2}dt$$ and $$g(x)=\underset{0}{\overset{x^2}{\int }}{t}^{1/2}{e}^{-t}dt$$. Then, the value of $$9\left(f\left(\sqrt{{\log }_{e}9}\right)+g\left(\sqrt{{\log }_{e}9}\right)\right)$$ is equal to :

\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right) is equal to

If the value of the integral \int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2, then the value of $$a$$ is

Let $$f:\mathbb{R}\to \mathbb{R}$$ be a function defined by $$f(x)=\frac{x}{{\left(1+x^4\right)}^{1/4}}$$, and $$g(x)=f(f(f(f(x))))$$. Then, $$18{\int }_0^{\sqrt{2\sqrt{5}}}{x}^{2}g(x)dx$$ is equal to

Let $$y=f(x)$$ be a thrice differentiable function in $$(-5,5)$$. Let the tangents to the curve $$y=f(x)$$ at $$(1,f(1))$$ and $$(3,f(3))$$ make angles $$\pi /6$$ and $$\pi /4$$, respectively with positive $$x$$-axis. If 27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} where $$\alpha ,\beta$$ are integers, then the value of $$\alpha +\beta$$ equals

Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x)=\{\begin{array}{ll}{x}^2+3x+a& ,x\le 1\\ bx+2& ,x>1\end{array}$$ be differentiable on $$\mathbb{R}$$. Then, the value of \int_\limits{-2}^2 f(x) d x equals

Let $$\mathrm{f}:\mathbb{R}\to \mathbb{R}$$ be defined as $$f(x)=a{e}^{2x}+b{e}^x+cx$$. If $$f(0)=-1,f^{\prime }\left({\log }_{e}2\right)=21$$ and $${\int }_0^{{\log }_{e}4}(f(x)-cx)dx=\frac{39}{2}$$, then the value of $$|a+b+c|$$ equals

The value of \lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)} is :

Let $$f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathbf{R}$$ be a differentiable function such that $$f(0)=\frac{1}{2}$$. If the \lim _\limits{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t}) \mathrm{dt}}{\mathrm{e}^{x^2}-1}=\alpha, then $$8{\alpha }^2$$ is equal to :

The integral \int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x is equal to

\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right) is equal to

The value of the integral \int_\limits{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x is

$$\text { Let } f(x)=\left\{\begin{array}{lr} -2, & -2 $\le$ x $\le$ 0 \\ x-2, & 0