Let $f:\mathbb{R}-\{0\}\to \mathbb{R}$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}$. If the $\underset{x\to 0}{\lim }\left(\frac{1}{\alpha x}+f(x)\right)=\beta ;\alpha ,\beta \in \mathbb{R}$, then $\alpha +2\beta$ is equal to

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2

Given below are two statements: Statement I: $\underset{x\to 0}{\lim }\left(\frac{{\tan }^{-1}x+{\log }_e\sqrt{\frac{1+x}{1-x}}-2x}{x^5}\right)=\frac{2}{5}$ Statement II: $\underset{x\to 1}{\lim }\left(x^{\frac{2}{1-x}}\right)=\frac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below:

For $\alpha ,\beta ,\gamma \in \mathbf{R}$, if \lim _\limits{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) \mathrm{e}^{x^2}}{\sin 2 x-\beta x}=3, then $\beta +\gamma -\alpha$ is equal to :

\lim _\limits{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _e\left(1+3 x^2\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^2\left(e^{5(x)^{\frac{4}{3}}}-1\right)} is equal to

Let $f(x)=\{\begin{array}{ll}(1+ax{)}^{1/x}& ,x0\end{array}$ be continuous at $x=0$. Then $e^{a}bc$ is equal to:

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1,f^{\prime }(2)=4$. Let $\underset{x\to 0}{\lim }(f(2+x){)}^{3/x}={\mathrm{e}}^{\alpha }$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha -7)x-\alpha$ meets $x$-axis is :

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x\in \mathbb{R}$. If \lim _\limits{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x), then \sum_\limits{r=1}^{10} G\left(r^2\right) is equal to

If \lim _\limits{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^3}=-1, where $\lambda ,\mu \in \mathbb{R}$, then $\lambda +\mu$ is equal to

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left({\left({\sec }^{-1}x\right)}^2+{\left({\mathrm{cosec}}^{-1}x\right)}^2\right)$ is :

Let [x] denote the greatest integer less than or equal to x. Then the domain of $f(x)={\sec }^{-1}(2[x]+1)$ is:

If $\frac{\pi }{2}\le x\le \frac{3\pi }{4}$, then ${\cos }^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$ is equal to

If $\alpha >\beta >\gamma >0$, then the expression ${\cot }^{-1}\left\{\beta +\frac{\left(1+{\beta }^2\right)}{(\alpha -\beta )}\right\}+{\cot }^{-1}\left\{\gamma +\frac{\left(1+{\gamma }^2\right)}{(\beta -\gamma )}\right\}+{\cot }^{-1}\left\{\alpha +\frac{\left(1+{\alpha }^2\right)}{(\gamma -\alpha )}\right\}$ is equal to :

$\cos \left({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{33}{65}\right)$ is equal to:

The value of ${\cot }^{-1}\left(\frac{\sqrt{1+{\tan }^2(2)}-1}{\tan (2)}\right)-{\cot }^{-1}\left(\frac{\sqrt{1+{\tan }^2\left(\frac{1}{2}\right)}+1}{\tan \left(\frac{1}{2}\right)}\right)$ is equal to

The sum of the infinite series ${\cot }^{-1}\left(\frac{7}{4}\right)+{\cot }^{-1}\left(\frac{19}{4}\right)+{\cot }^{-1}\left(\frac{39}{4}\right)+{\cot }^{-1}\left(\frac{67}{4}\right)+\dots$. is :

Considering the principal values of the inverse trigonometric functions, $\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^2}\right),-\frac{1}{2}

The sum of all values of $\theta \in [0,2\pi ]$ satisfying $2{\sin }^2\theta =\cos 2\theta$ and $2{\cos }^2\theta =3\sin \theta$ is

If $\theta \in [-2\pi ,2\pi ]$, then the number of solutions of $2\sqrt{2}{\cos }^2\theta +(2-\sqrt{6})\cos \theta -\sqrt{3}=0$, is equal to: