Continuity and Differentiability

The function f(x) = $\left|\mathrm{x} - 2\right| +\hspace{0.17em}\mathrm{x}$ is

• differentiable at both x = 2 and x = 0

• differentiable at x = 2 but not at x = 0

• continuous at x = 2 but not at x = 0

• continuous at both x = 2and x = 0

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\sqrt{{\mathrm{x}}^2 - 1}\right), \mathrm{then} \mathrm{the} \mathrm{ratio} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} : \frac{\mathrm{dy}}{\mathrm{dx}}$ is

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

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

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

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

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

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