51.

Suppose f(x) = x(x + 3)(x - 2), x  [- 1, 4]. Then, a value of c in (- 1, 4) satisfying f'(c) = 10 is

• 2

• 3

• $\frac{7}{2}$

The area included between the parabola $\mathrm{y} = \frac{{\mathrm{x}}^2}{4\mathrm{a}} \mathrm{and} \mathrm{the} \mathrm{curve} \mathrm{y} = \frac{8{\mathrm{a}}^3}{\left({\mathrm{x}}^2 + 4{\mathrm{a}}^2\right)} \mathrm{is}$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } + \frac{2}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{8}{3}\right)$

• ${\mathrm{a}}^2\left(\mathrm{\pi } + \frac{4}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{4}{3}\right)$

If a, b, c are distinct positive real numbers, then the value of the determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right| \mathrm{is}$

• $< 0$

• $> 0$

• 0

• $\ge 0$

$$\begin{gathered} \mathrm{The} \mathrm{equations} \mathrm{x} - \mathrm{y} + 2\mathrm{z} = 4 \\ 3\mathrm{x} + \mathrm{y} + 4\mathrm{z} = 6 \\ \mathrm{x} + \mathrm{y} + \mathrm{z} = 1 \mathrm{have} \end{gathered}$$

• unique solution

• infinitely many solutions

• no solution

• two solutions

$\mathrm{If} \mathrm{x} =\sin \left(2{\tan }^{-1}\left(2\right)\right) \mathrm{and} \mathrm{y} = \sin \left(\frac{1}{2}{\tan }^{-1}\left(\frac{4}{3}\right)\right), \mathrm{then}$

• $\mathrm{x} > \mathrm{y}$

• x = y

• x = 0 = y

• $\mathrm{x}< \mathrm{y}$

$\mathrm{If} \cosh \left(\mathrm{x}\right) = \frac{5}{4}, \mathrm{then} \cosh \left(3\mathrm{x}\right) = ?$

• $\frac{61}{16}$

• $\frac{63}{16}$

• $\frac{65}{16}$

• $\frac{61}{63}$

$\mathrm{In} \mathrm{a}∆\mathrm{ABC}, \mathrm{if} <\mathrm{A} = 90°, \mathrm{then} {\cos }^{-1}\left(\frac{\mathrm{R}}{{\mathrm{r}}_2 +\hspace{0.17em}{\mathrm{r}}_3}\right) = ?$

• $90°$

• $30°$

• $60°$

• $45°$

The value (s) of x for which the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) \left\{= 1 - \mathrm{x}, \mathrm{x} < 1 \\ =\left(1 - \mathrm{x}\right)\left(2 - \mathrm{x}\right), 1 \le \mathrm{x} \le 2 \\ 3 - \mathrm{x}, \mathrm{x} > 2\right\} \\ \mathrm{fails} \mathrm{to} \mathrm{be} \mathrm{continuous} \mathrm{is} \left(\mathrm{are}\right) \end{gathered}$$

• 1

• 2

• 3

• all real numbers

$\mathrm{If} \mathrm{y} = {\log }_2\left({\log }_2\left(\mathrm{x}\right)\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{{\log }_{\mathrm{e}}\left(2\right)}{{\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right)}$

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

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

• $\frac{1}{\mathrm{x}{\left({\log }_2\left(\mathrm{x}\right)\right)}^2}$

The angle of intersection between the curves y² + x² = a²$\sqrt{2}$ and x² - y² = a² is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{12}$