51.

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \log \left({\mathrm{e}}^{\mathrm{x}}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} + 2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right), \mathrm{then} \mathrm{f}\text{'}\left(0\right) = ?$

• $\frac{1}{4}$

• 4

• $\frac{- 3}{4}$

• 1

$$\begin{gathered} \mathrm{If} \mathrm{xy} \ne 0, \mathrm{x} + \mathrm{y} \ne 0 \mathrm{and} {\mathrm{x}}^{\mathrm{m}}{\mathrm{y}}^{\mathrm{n}} = {\left(\mathrm{x} + \mathrm{y}\right)}^{\mathrm{m} + \mathrm{n}}, \\ \mathrm{where} \mathrm{m}, \mathrm{n} \mathrm{not} \mathrm{belongs} \mathrm{to} \mathrm{N}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{x} + \mathrm{y}}{\mathrm{xy}}$

• xy

• $\frac{\mathrm{x}}{\mathrm{y}}$

$$\begin{gathered} \mathrm{If} {\mathrm{x}}^2 + {\mathrm{y}}^2 = \mathrm{t} + \frac{1}{\mathrm{t}}, {\mathrm{x}}^4 + {\mathrm{y}}^4 = {\mathrm{t}}^2 + \frac{1}{{\mathrm{t}}^2}, \mathrm{then} \\ {\mathrm{x}}^3\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• - 1

• 1

• 0

• t

If the volume of a sphere increases at the rate of 2$\mathrm{\pi }$ cm³/ s, then the rate of increase of its radius(in cm/s), when the volume is 288$\mathrm{\pi }$ cm³, is

• $\frac{1}{36}$

• $\frac{1}{72}$

• $\frac{1}{18}$

• $\frac{1}{9}$

If f : R$\to$R² and g : R $\to$R are such that g{f(x)} = $\left|\sin \left(\mathrm{x}\right)\right|$and f{g(x)} = (sin $\left(\sqrt{\mathrm{x}}\right)$)², then a possible choice for f and g is

• f(x) = x², g(x) = $\sin \left(\sqrt{\mathrm{x}}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right), \mathrm{g}\left(\mathrm{x}\right) = \left|\mathrm{x}\right|$

• $\mathrm{f}\left(\mathrm{x}\right) = {\sin }^2\left(\mathrm{x}\right), \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2, \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{Z} \to \mathrm{Z} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\mathrm{x}}{2}, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{even} \\ 0, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{odd}\right\}, \mathrm{then} \mathrm{f} \mathrm{is} \end{gathered}$$

• Onto but not one-to-one

• One-to-one but not onto

• One-to-one and onto

• Neither one-to-one nor onto

$\mathrm{A} = \left[\begin{array}{cc}\mathrm{i}& - \mathrm{i}\\ - \mathrm{i}& \mathrm{i}\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& - 1\\ - 1& 1\end{array}\right] \Rightarrow {\mathrm{A}}^8$

• 4B

• 8B

• 64B

• 128B

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) \\ \left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x} + 1\\ 2\mathrm{x}& \mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}\left(\mathrm{x} + 1\right)\\ 3\mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)& \left(\mathrm{x} - 1\right)\mathrm{x}\left(\mathrm{x} + 1\right)\end{array}\right| \\ \mathrm{then} \mathrm{f}\left(2012\right) = ? \end{gathered}$$

• 0

• 1

• - 500

• 500

$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left[\begin{array}{ccc}- 1& - 2& - 3\\ 3& 4& 5\\ 4& 5& 6\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& - 2\\ - 1& 2\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right], \mathrm{if} \mathrm{a}, \mathrm{b} \mathrm{and} \mathrm{c} \mathrm{respectively}, \\ \mathrm{denote} \mathrm{the} \mathrm{rank} \mathrm{of} \mathrm{A}, \mathrm{B}, \mathrm{and} \mathrm{C}, \mathrm{then} \mathrm{the} \mathrm{correct} \mathrm{order} \mathrm{of} \mathrm{these} \mathrm{number} \mathrm{is} \end{gathered}$$

• $\mathrm{a} < \mathrm{b} < \mathrm{c}$

• $\mathrm{c} < \mathrm{b} < \mathrm{a}$

• $\mathrm{b} < \mathrm{a} < \mathrm{c}$

• $\mathrm{a} < \mathrm{c} < \mathrm{b}$

$$\begin{gathered} \mathrm{If} \frac{1}{2} \le \mathrm{x} \le 1, \mathrm{then} \\ {\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\frac{\mathrm{x}}{2} + \frac{1}{2}\sqrt{3 - 3{\mathrm{x}}^2}\right) = ? \end{gathered}$$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• 0