61.

• k

• k²

• $\frac{1}{\mathrm{k}}$

$$\begin{gathered} \mathrm{If} \mathrm{x} > 0, \mathrm{y} > 0, \mathrm{z} > 0, \mathrm{xy} + \mathrm{yz} + \mathrm{zx} < 1 \mathrm{and} \mathrm{if} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }, \\ \mathrm{then} \mathrm{x} + \mathrm{y} +\hspace{0.17em}\mathrm{z} =? \end{gathered}$$

• 0

• xyz

• 3xyz

• $\sqrt{\mathrm{xyz}}$

${\mathrm{sech}}^{-1}\left(\frac{1}{2}\right) - {\mathrm{cosec}}^{-1}\left(\frac{3}{4}\right) = ?$

• ${\log }_{\mathrm{e}}\left(3\left(2 + \sqrt{3}\right)\right)$

• ${\log }_{\mathrm{e}}\left(\frac{1 + \sqrt{3}}{3}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2 + \sqrt{3}}{3}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2 - \sqrt{3}}{3}\right)$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{1 + \mathrm{x}} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right), \mathrm{then} \mathrm{g}\text{'}\left(\mathrm{x}\right) =$

• $\frac{1}{{\left(2\mathrm{x} + 3\right)}^2}$

• $\frac{1}{{\left(\mathrm{x} + 1\right)}^2}$

• $\frac{1}{{\mathrm{x}}^2}$

• $\frac{1}{{\left(2\mathrm{x} + 1\right)}^2}$

$\mathrm{If} \int \frac{\mathrm{dx}}{\sqrt{{\sin }^3\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}} = \mathrm{g}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{g}\left(\mathrm{x}\right) = ?$

• $- \frac{2}{\sqrt{\cot \left(\mathrm{x}\right)}}$

• $- \frac{2}{\sqrt{\tan \left(\mathrm{x}\right)}}$

• $\frac{2}{\sqrt{\cot \left(\mathrm{x}\right)}}$

• $\frac{2}{\sqrt{\tan \left(\mathrm{x}\right)}}$

$\mathrm{If} \int \frac{\mathrm{dx}}{\left(1 + \sqrt{\mathrm{x}}\right)\sqrt{\mathrm{x} - {\mathrm{x}}^2}} = \frac{\mathrm{A}\sqrt{\mathrm{x}}}{\sqrt{1 -\mathrm{x}}} + \frac{\mathrm{B}}{\sqrt{1 - \mathrm{x}}} + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{real} \mathrm{constant}, \mathrm{then} \mathrm{A} + \mathrm{B} = ?$

• 3

• 0

• 1

• 2

For any integer n $\ge$ 2, let I_n = $\int$tanⁿ(x)dx. If  ${\mathrm{I}}_{\mathrm{n}} = \frac{1}{\mathrm{a}}{\tan }^{\mathrm{n} - 1}\left(\mathrm{x}\right) - {\mathrm{bI}}_{\mathrm{n} - 2} \mathrm{for} \mathrm{n}\ge 2,$ then the ordered par (a, b) equals to

• $\left(\mathrm{n} - 1, \frac{\mathrm{n} - 1}{\mathrm{n} - 2}\right)$

• $\left(\mathrm{n} - 1, \frac{\mathrm{n} - 2}{\mathrm{n} - 1}\right)$

• (n, 1)

• (n - 1, 1)

$$\begin{gathered} \mathrm{If} \frac{\left({\mathrm{x}}^2 - 1\right)}{{\left(\mathrm{x} + 1\right)}^2\sqrt{\mathrm{x}\left({\mathrm{x}}^2 + \mathrm{x}\hspace{0.17em}+\hspace{0.17em}1\right)}}\mathrm{dx} \\ = {\mathrm{Atan}}^{-1}\left(\sqrt{\frac{{\mathrm{x}}^2 + \mathrm{x} +\hspace{0.17em}1}{\mathrm{x}}}\right) \mathrm{C}, \mathrm{in} \mathrm{which} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant}, \mathrm{then} \mathrm{A} =? \end{gathered}$$

• $\frac{1}{2}$

• 3

• 2

• 1

$$\begin{gathered} \mathrm{By} \mathrm{the} \mathrm{defination} \mathrm{of} \mathrm{the} \mathrm{definite} \mathrm{integral}, \mathrm{the} \mathrm{value} \mathrm{of} \\ \underset{\mathrm{n}\to \infty }{\lim }\left(\frac{1^4}{1^5 +\hspace{0.17em}{\mathrm{n}}^5} + \frac{2^4}{2^5 + {\mathrm{n}}^5} + \frac{3^4}{3^5 + {\mathrm{n}}^6} + ... + \frac{{\mathrm{n}}^4}{{\mathrm{n}}^5 + {\mathrm{n}}^5}\right) \mathrm{is} \end{gathered}$$

• $\log \left(2\right)$

• $\frac{1}{5}\log \left(2\right)$

• $\frac{1}{4}\log \left(2\right)$

• $\frac{1}{3}\log \left(2\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{\cos }^4\left(3\mathrm{\theta }\right) . {\sin }^2\left(6\mathrm{\theta }\right)\mathrm{d\theta } = ?$

• $\frac{\mathrm{\pi }}{96}$

• $\frac{5}{192}$

• $\frac{5\mathrm{\pi }}{256}$

• $\frac{5\mathrm{\pi }}{192}$