Domain of y = ${\cot }^{-1}\left(\mathrm{x}\right)$ is

• $\left(- \infty , 0\right)$

• $\left(0, \infty \right)$

• $\left(- \infty , \infty \right)$

• None of these

If $\underset{\mathrm{x}\to 0}{\lim }\frac{\log \left(3 + \mathrm{x}\right) - \log \left(3 - \mathrm{x}\right)}{\mathrm{x}}$ = k, then the value of k will be

• 0

• $- \frac{1}{3}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

For the function 3x⁴ - 8x³ + 12x² - 48x + 25 in the interval [1, 3]. The value of maxima and minima are

• 16, - 39

• - 16, 39

• 6, - 9

• None of these

f(x) = x³ - 27x + 5 is an increasing function when

• x < - 3

• $\left|\mathrm{x}\right| >\hspace{0.17em}3$

• $\mathrm{x} \le - 3$

• $\left|\mathrm{x}\right| <\hspace{0.17em}3$

In the interval $\left(0, \frac{\mathrm{\pi }}{2}\right)$ function log(sin(x)) is

• increasing

• decreasing

• neither increasing nor decreasing

• None of the above

A cone of maximum volume is being cut from sphere, then ratio between height of cone and diameter of sphere is

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{3}{4}$

• $\frac{1}{4}$

The nth derivative of e^{2x + 5} is

• $\frac{2^{2\mathrm{n}}}{\mathrm{n}!} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $2^{\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $2^{2\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

• $\frac{2^{\mathrm{n}}}{\mathrm{n}} {\mathrm{e}}^{2\mathrm{x} + 5}$

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

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

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

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

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

Function f(x) = $$\begin{gathered} \left\{\mathrm{x} - 1, \mathrm{x} < 2 \\ 2\mathrm{x} - 3, \mathrm{x} \ge 2\right\} \end{gathered}$$ is continuous function for

• all real values of x

• only x = 2

• only real values of x, when x $\ne$ 2

• the integer values of x

If g(x) = x² + x - 2 and gof(x) = 2x² - 5x+ 2, then f(x) is equal to

• 2x - 3

• 2x + 3

• 2x² + 3x + 1

• 2x² - 3x - 1