What is the minimum value of a²x + b²y where xy = c² ?

• abc

• 2abc

• 3abc

• 4abc

$$\begin{gathered} \mathrm{Three} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{trapezium} \mathrm{are} \mathrm{each} \mathrm{equal} \mathrm{to} 6\mathrm{cm}. \mathrm{Let} \mathrm{\alpha } \in \left(0, \frac{\mathrm{\pi }}{2}\right) \\ \mathrm{be} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{pair} \mathrm{of} \mathrm{adjacent} \mathrm{sides}. \end{gathered}$$

If the area of the trapezium is the maximum possible, then what is $\mathrm{\alpha }$ equal to ?

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

$$\begin{gathered} \mathrm{Three} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{trapezium} \mathrm{are} \mathrm{each} \mathrm{equal} \mathrm{to} 6\mathrm{cm}. \mathrm{Let} \mathrm{\alpha } \in \left(0, \frac{\mathrm{\pi }}{2}\right) \\ \mathrm{be} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{pair} \mathrm{of} \mathrm{adjacent} \mathrm{sides}. \end{gathered}$$

If the area of the trapezium is the maximum, then what is the length of fourth side ?

• 8cm

• 9cm

• 10cm

• 12cm

4.

$$\begin{gathered} \mathrm{Three} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{trapezium} \mathrm{are} \mathrm{each} \mathrm{equal} \mathrm{to} 6\mathrm{cm}. \mathrm{Let} \mathrm{\alpha } \in \left(0, \frac{\mathrm{\pi }}{2}\right) \\ \mathrm{be} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{pair} \mathrm{of} \mathrm{adjacent} \mathrm{sides}. \end{gathered}$$

What is the maximum area of the trapezium ?

• $36\sqrt{3}{\mathrm{cm}}^2$

• $30\sqrt{3}{\mathrm{cm}}^2$

• $27\sqrt{3}{\mathrm{cm}}^2$

• $24\sqrt{3}{\mathrm{cm}}^2$

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} 16\sin \left(\mathrm{\theta }\right) - 12{\sin }^2\left(\mathrm{\theta }\right) ?$

• $\frac{3}{4}$

• $\frac{4}{3}$

• $\frac{16}{3}$

• 4

Which one of the following is correct in respect of the function $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{xsin}\left(\mathrm{x}\right) \hspace{0.17em}+ \cos \left(\mathrm{x}\right) + \frac{1}{2}{\cos }^2\left(\mathrm{x}\right) ?$

• Tt is increasing in the interval (0, $\frac{\mathrm{\pi }}{2}$)

• It remains constant in the interval (0, $\frac{\mathrm{\pi }}{2}$)

• It is decreasing in the interval $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• It is decreasing in the interval $\left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right)$

In which one of the following intervals is the function

f(x) = x² - 5x + 6 decreasing ?

• $( - \infty , 2]$

• $[3, \infty )$

• $( - \infty , \infty )$

• (2, 3)

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{minimum} \mathrm{value} \mathrm{of} {\left[\mathrm{x}\left(\mathrm{x} - 1\right) + 1\right]}^{\frac{1}{3}}, \mathrm{where} 0 \le \mathrm{x} \le 1 ?$

• ${\left(\frac{3}{4}\right)}^{\frac{1}{3}}$

• 1

• $\frac{1}{2}$

• ${\left(\frac{3}{8}\right)}^{\frac{1}{3}}$

The maximum value of $\sin \left(\mathrm{x} + \frac{\mathrm{\pi }}{6}\right) + \cos \left(\mathrm{x} + \frac{\mathrm{\pi }}{6}\right)$$\mathrm{in} \mathrm{the} \mathrm{interval} \left(0, \frac{\mathrm{\pi }}{2}\right) \mathrm{is} \mathrm{attained} \mathrm{at}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

What is the length of the longest interval in which the function f(x) = 3sin(x) - 4sin(3x) is increasing ?

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $\mathrm{\pi }$