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

• 0

• - 1

• 1

• e

912.

If $\sqrt{\mathrm{r}} = {\mathrm{ae}}^{\mathrm{\theta cot}\left(\mathrm{\alpha }\right)}$ where a and $\mathrm{\alpha }$, are real numbers, then $\frac{{\mathrm{d}}^2\mathrm{r}}{{\mathrm{d\theta }}^2} - 4{\mathrm{rcot}}^2\left(\mathrm{\alpha }\right)$ is

• r

• $\frac{1}{\mathrm{r}}$

• r

• 0

The derivative of ${\tan }^{-1}\left[\frac{\sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right]$ with respect to ${\tan }^{-1}\left[\frac{\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}\right]$ is

• 2

• - 1

• 0

• - 2

$\frac{\mathrm{d}}{\mathrm{dx}}\left[\cos \left({\cot }^{-1}\left(\sqrt{\frac{2 +\hspace{0.17em}\mathrm{x}}{2 - \mathrm{x}}}\right)\right)\right]$ is

• $\frac{1}{4}$

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $- \frac{3}{4}$

If y = ${\log }_{\mathrm{e}}\left[1 + \mathrm{x} + {\mathrm{x}}^2 + .. \infty \right], \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{{\left(1 +\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{1}{{\left(1 -\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{- 1}{{\left(1 +\hspace{0.17em}\mathrm{x}\right)}^2}$

• $\frac{- 1}{{\left(1 -\hspace{0.17em}\mathrm{x}\right)}^2}$

Length of the subtangent at (x₁, y₁) on xⁿy^m = a^{m + n}, m, n > 0, is

• $\frac{\mathrm{n}}{\mathrm{m}}{\mathrm{x}}_1$

• $\frac{\mathrm{m}}{\mathrm{n}}\left|{\mathrm{x}}_1\right|$

• $\frac{\mathrm{n}}{\mathrm{m}}\left|{\mathrm{y}}_1\right|$

• $\frac{\mathrm{n}}{\mathrm{m}}\left|{\mathrm{x}}_1\right|$

If y = ${\tan }^{-1}\left(\frac{1}{1 +\hspace{0.17em}\mathrm{x}\hspace{0.17em}+ {\mathrm{x}}^2}\right) + {\tan }^{-1}\left(\frac{1}{{\mathrm{x}}^2 +\hspace{0.17em}2\mathrm{x}\hspace{0.17em}+ 3}\right) + {\tan }^{-1}\left(\frac{1}{{\mathrm{x}}^2 +\hspace{0.17em}5\mathrm{x}\hspace{0.17em}+ 7}\right) + ... \mathrm{n}$ terms, then y'(0) is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2\mathrm{n}}{1 + {\mathrm{n}}^2}$

• $\frac{{\mathrm{n}}^2}{1 + {\mathrm{n}}^2}$

• $- \frac{{\mathrm{n}}^2}{1 + {\mathrm{n}}^2}$

If f(x) = $$\begin{gathered} \left\{\frac{{\mathrm{x}}^2 - \left(\mathrm{a} + 2\right)\mathrm{x} + \mathrm{a}}{\mathrm{x} - 2}, \mathrm{x} \ne 2 \\ 2, \mathrm{x} = 2\right\} \end{gathered}$$ is continuous at x = 2, then at x = 2, then the value of a is

• - 6

• 0

• 1

• - 1

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\log }_{\mathrm{e}}\left(1 +\hspace{0.17em}\mathrm{x}\right)}{3^{\mathrm{x}} - 1}$ is equal to

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

• 0

• ${\log }_3\left(\mathrm{e}\right)$

• 1

If f(x) = $$\begin{gathered} \left\{\mathrm{x}, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{irrational} \\ 0, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{rational}\right\} \end{gathered}$$, then f is

• continuous everywhere

• discontinuous everywhere

• continuous only at x = 0

• continuous at all rational numbers