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

937.

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

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

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

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

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

If f(x) = $$\begin{gathered} \left\{{\mathrm{kx}}^2 \mathrm{if} \mathrm{x} \le 2 \\ 3 \mathrm{if} \mathrm{x} > 2\right\} \end{gathered}$$is continuous at x = 2, then the value of k is

• 3/4

• 4

• 4/3

• 3

If y = log(log(x)), then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

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

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

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

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