If the coefficients of the equation whose roots are k times the roots of the equation ${\mathrm{x}}^3 + \frac{1}{4}{\mathrm{x}}^2 - \frac{1}{16}\mathrm{x} + \frac{1}{144} = 0$, are integers, then a possible value of k is

• 3

• 12

• 9

• 4

The sum of all 4-digit numbers that can be formed using the digits 2, 3, 4, 5, 6 without repetition, is

• 533820

• 532280

• 533280

• 532380

If a set A has 5 elements, then the number of ways of selecting two subsets P and Q from A such that P and Q are mutually disjoint, is

• 64

• 128

• 243

• 729

The coefficient of x in the expansion of (1 - x + x² - x³)⁴ is

• 31

• 30

• 25

• - 14

If the middle term in the expansion of (1 + x)²ⁿ is the greatest term, then x lies in the interval

• $\left(\frac{\mathrm{n}}{\mathrm{n}\hspace{0.17em}+ 1}, \frac{\mathrm{n} + 1}{\mathrm{n}}\right)$

• $\left(\frac{\mathrm{n} + 1}{\mathrm{n}}, \frac{\mathrm{n}}{\mathrm{n} + 1}\right)$

• (n - 2, n)

• (n - 1, n)

To find the coefficient of x⁴ in the expansion of $\frac{3\mathrm{x}}{\left(\mathrm{x} - 2\right)\left(\mathrm{x} - 1\right)}$, the interval in which the expansion is valid, is

• $- 2 < \mathrm{x} < \infty$

• $- \frac{1}{2} < \mathrm{x} < \frac{1}{2}$

• $- 1 < \mathrm{x} < 1$

• $- \infty < \mathrm{x} < \infty$

$$\begin{gathered} \mathrm{If} \left(1 + \tan \left(\mathrm{\alpha }\right)\right)\left(1 + \tan \left(4\mathrm{\alpha }\right)\right) = 2, \mathrm{\alpha } \in \left(0, \frac{\mathrm{\pi }}{16}\right), \\ \mathrm{then} \mathrm{\alpha } = ? \end{gathered}$$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{30}$

• $\frac{\mathrm{\pi }}{40}$

• $\frac{\mathrm{\pi }}{50}$

$\mathrm{If} \cos \left(\mathrm{\theta }\right) = \frac{\cos \left(\mathrm{\alpha }\right) - \cos \left(\mathrm{\beta }\right)}{1 - \cos \left(\mathrm{\alpha }\right)\cos \left(\mathrm{\beta }\right)}, \mathrm{then} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{values} \mathrm{of} \tan \left(\frac{\mathrm{\theta }}{2}\right) \mathrm{is}$

• $\cot \left(\frac{\mathrm{\beta }}{2}\right)\tan \left(\frac{\mathrm{\alpha }}{2}\right)$

• $\tan \left(\frac{\mathrm{\alpha }}{2}\right)\tan \left(\frac{\mathrm{\beta }}{2}\right)$

• $\tan \left(\frac{\mathrm{\beta }}{2}\right)\cot \left(\frac{\mathrm{\alpha }}{2}\right)$

• ${\tan }^2\left(\frac{\mathrm{\alpha }}{2}\right){\tan }^2\left(\frac{\mathrm{\beta }}{2}\right)$

19.

The value of the expression $\frac{1 + \sin \left(2\mathrm{\alpha }\right)}{\cos \left(2\mathrm{\alpha } - 2\mathrm{\pi }\right)\tan \left(\mathrm{\alpha } - {\displaystyle \frac{3\mathrm{\pi }}{4}}\right)} - \frac{1}{4}\sin \left(2\mathrm{\alpha }\right)\left[\cot \left(\frac{\mathrm{\alpha }}{2}\right) + \cot \left(\frac{3\mathrm{\pi }}{2} + \frac{\mathrm{\alpha }}{2}\right)\right] \mathrm{is}$

• 0

• 1

• ${\sin }^2\left(\frac{\mathrm{\alpha }}{2}\right)$

• ${\sin }^2\left(\mathrm{\alpha }\right)$

$\mathrm{If} \frac{1}{6}\sin \left(\mathrm{\theta }\right), \cos \left(\mathrm{\theta }\right) \mathrm{and} \tan \left(\mathrm{\theta }\right) \mathrm{are} \mathrm{in} \mathrm{geometric} \mathrm{progression}, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{set} \mathrm{of} \mathrm{\theta } \mathrm{is}$

• $2\mathrm{n\pi } \pm \left(\frac{\mathrm{\pi }}{6}\right)$

• $2\mathrm{n\pi } \pm \left(\frac{\mathrm{\pi }}{3}\right)$

• $\mathrm{n\pi } + {\left( - 1\right)}^{\mathrm{n}}\left(\frac{\mathrm{\pi }}{3}\right)$

• $\mathrm{n\pi } + \left(\frac{\mathrm{\pi }}{3}\right)$