11.

$5{\cos }^{-1}\left(\frac{1 - {\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\right) + 7{\sin }^{-1}\left(\frac{2\mathrm{x}}{1 \hspace{0.17em}+ {\mathrm{x}}^2}\right) - 4{\tan }^{-1}\left(\frac{2\mathrm{x}}{1 \hspace{0.17em}+ {\mathrm{x}}^2}\right) - {\tan }^{-1}\left(\mathrm{x}\right) = 5\mathrm{\pi }$, then x is equal to

• 3

• - $\sqrt{3}$

• $\sqrt{2}$

• $\sqrt{3}$

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 - \cos \left(\mathrm{x}\right)}{\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$  is continuous at x = 0, then the value of k is

• 0

• 1/2

• $\frac{1}{4}$

• - $\frac{1}{2}$

Let f(x)= sin(x), g(x) = x and h(x) = log_e(x). If F(x) = (hogof)(x), then F"(x) is equal to

• a ${\csc }^3\left(\mathrm{x}\right)$

• $2\cot \left({\mathrm{x}}^2\right) - 4{\mathrm{x}}^2{\csc }^2\left({\mathrm{x}}^2\right)$

• $2\mathrm{xcot}\left({\mathrm{x}}^2\right)$

• $- 2{\csc }^2\left(\mathrm{x}\right)$

If y = ${\tan }^{-1}\left(\frac{4\mathrm{x}}{1 + 5{\mathrm{x}}^2}\right) + {\tan }^{-1}\left(\frac{2 + 3\mathrm{x}}{3 - 2\mathrm{x}}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{5}{1 + 25{\mathrm{x}}^2}$

• $\frac{1}{1 + 25{\mathrm{x}}^2}$

• 0

• $\frac{5}{1 - 25{\mathrm{x}}^2}$

If $\mathrm{y} = {\log }_{\mathrm{a}}\left(\mathrm{x}\right) + {\log }_{\mathrm{x}}\left(\mathrm{a}\right) + {\log }_{\mathrm{x}}\left(\mathrm{x}\right) + {\log }_{\mathrm{a}}\left(\mathrm{a}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{\mathrm{x}} + \mathrm{xlog}\left(\mathrm{a}\right)$

• $\frac{\log \left(\mathrm{a}\right)}{\mathrm{x}} + \frac{\mathrm{x}}{\log \left(\mathrm{a}\right)}$

• $\frac{1}{\mathrm{xlog}\left(\mathrm{a}\right)} + \mathrm{xlog}\left(\mathrm{a}\right)$

• None of the above

If $\mathrm{y} = {\sin }^{-1}\left(3\mathrm{x} - 4{\mathrm{x}}^3\right) + {\cos }^{-1}\left(4{\mathrm{x}}^3 - 3\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{E}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 5

• 0

• $\frac{2}{\sqrt{1 + {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x} - 1}{4} + \frac{{\left(\mathrm{x} - 1\right)}^3}{12} + \frac{{\left(\mathrm{x} - 1\right)}^5}{20} + \frac{{\left(\mathrm{x} - 1\right)}^7}{28} + ...$, where 0 < x < 2, then f'(x) is equal to

• $\frac{1}{4\mathrm{x}\left(2 - \mathrm{x}\right)}$

• $\frac{1}{4{\left(\mathrm{x} - 2\right)}^2}$

• $\frac{1}{2 - \mathrm{x}}$

• $\frac{1}{2 + \mathrm{x}}$

If f(x) = sin(x), the derivative of f(log(x)) w.r.t. x is

• cos(x)

• f'(log(x))

• cos(log(x))

• $\frac{\cos \left(\log \left(\mathrm{x}\right)\right)}{\mathrm{x}}$

If f(x + y) = 2 f(x) f(y), f'(5) = 1024log(2) and f(2) = 8,  then the value of f'(3) is

• 64(log(2))

• 128(log(2))

• 256

• 256log(2)

Gas is being pumped into a spherical balloon at the rate of 30 ft³/min. Then, the rate at which the radius increases when it reaches the value 15 ft is

• $\frac{1}{15\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{30\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{20} \mathrm{ft}/\min$

• $\frac{1}{25} \mathrm{ft}/\min$