Continuity and Differentiability

If f(x) = $$\begin{gathered} \left\{\frac{3\sin \left(\mathrm{\pi x}\right)}{5\mathrm{x}}, \mathrm{x} \ne 0 \\ 2\mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value k is

• $\frac{\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{5}$

If f(x) = $$\begin{gathered} \left\{3\mathrm{x} - 8, \mathrm{if} \mathrm{x} \le 5 \\ 2\mathrm{k}, \mathrm{if} \mathrm{x} > 5\right\} \end{gathered}$$ is continuous, then find k.

• 4/7

• 2/7

• 7/2

• 3/7

The function f(x) = [x], where [x] denotes greatest integer function, is continuous at

• 1

• 4

• 1.5

• - 2

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

• $\frac{1}{1 - {\mathrm{x}}^4}$

• $\frac{- 4\mathrm{x}}{1 - {\mathrm{x}}^4}$

• $\frac{- 4{\mathrm{x}}^3}{1 - {\mathrm{x}}^4}$

• $\frac{4{\mathrm{x}}^3}{1 - {\mathrm{x}}^4}$

The two curves x³ - 3xy² + 2 = 0 and 3x²y - y³ = 2

• cut at angle $\mathrm{\pi }$/3

• touch each other

• cut at angle $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• cut at right angle

If y = ${\mathrm{e}}^{{\sin }^{-1}\left({\mathrm{t}}^2 - 1\right)} \mathrm{and} \mathrm{x} = {\mathrm{e}}^{{\sec }^{-1}\left(\frac{1}{{\mathrm{t}}^2 - 1}\right)}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{- \mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{- \mathrm{x}}{\mathrm{y}}$

If x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

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

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^{\mathrm{x} - \mathrm{y}}}$

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

• $\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{x} - \mathrm{y}}$

The function f(x) = [x], where [x] is the greatest integer function, is continuous at

• 1.5

• 4

• 1

• - 2

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

• $\frac{- \mathrm{x}}{\mathrm{y}}$

• xy

• $\frac{\mathrm{x}}{\mathrm{y}}$

• - xy

If y = $\left|\begin{array}{ccc}\mathrm{f}\left(\mathrm{x}\right)& \mathrm{g}\left(\mathrm{x}\right)& \mathrm{h}\left(\mathrm{x}\right)\\ \mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\left|\begin{array}{ccc}\mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\left(\mathrm{x}\right)\\ \mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{f}\left(\mathrm{x}\right)& \mathrm{g}\left(\mathrm{x}\right)& \mathrm{h}\left(\mathrm{x}\right)\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{l}& \mathrm{a}\\ \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{m}& \mathrm{b}\\ \mathrm{h}\text{'}\left(\mathrm{x}\right)& \mathrm{n}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\left(\mathrm{x}\right)\end{array}\right|$