Let $$a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}$$ be $$\mathrm{n}$$ positive consecutive terms of an arithmetic progression. If $$\mathrm{d} > 0$$ is its common difference, then

$$\lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right)$$ is

The sum of all the roots of the equation $$\left|x^{2}-8 x+15\right|-2 x+7=0$$ is :

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $$\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$$ is $$\sqrt{6}: 1$$, then the third term from the beginning is :

If $$2 x^{y}+3 y^{x}=20$$, then $$\frac{d y}{d x}$$ at $$(2,2)$$ is equal to :