If the harmonic mean between the roots of (5 + $\sqrt{2}$) x² - bx + (8 + 2$\sqrt{5}$) = 0 is 4, then the value of b is

• 2

• 3

• 4 - $\sqrt{5}$

• $4 + \sqrt{5}$

The set of solutions satisfying both x² + 5x + 6 $\ge$ 0 and x² + 3x - 4 < 0 is

• (- 4, 1)

• $\left(- 4, - 3\right) \cup (- 2, 1)$

• $\left(- 4, - 3\right) \cup \left(- 2, 1\right)$

• $\left(- 4, - 3\right) \cup \left( - 2, 1\right)$

If the roots of ${\mathrm{x}}^3 - 42{\mathrm{x}}^2 + 336\mathrm{x} - 512 = 0$, are in increasing geometric progression, then its common ratio is

• 2 : 1

• 3 : 1

• 4 : 1 

• 6 : 1

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation ${\mathrm{x}}^2 - 2\mathrm{x} + 4 = 0$, then ${\mathrm{\alpha }}^9 + {\mathrm{\beta }}^9$ is equal to

• - 2⁸

• 2⁹

• - 2¹⁰

• 2¹⁰

If a complex number z satisfied $\left|{\mathrm{z}}^2 - 1\right| = {\left|\mathrm{z}\right|}^2 + 1$, then z lies on

• the real axis

• the imaginary axis

• y = x

• a circle

The period of f(x) = $\cos \left(\frac{\mathrm{x}}{3}\right) + \sin \left(\frac{\mathrm{x}}{2}\right)$ is

• $2\mathrm{\pi }$

• $4\mathrm{\pi }$

• $8\mathrm{\pi }$

• $12\mathrm{\pi }$

If $\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right) = \mathrm{p} \mathrm{and} {\sin }^3\left(\mathrm{\theta }\right) + {\cos }^3\left(\mathrm{\theta }\right) = \mathrm{q}$, then p(p² - 3) is equal to

• q

• 2q

• - q

• - 2q

18.

$\mathrm{If} \tan \left(\left(\mathrm{\pi cos}\left(\mathrm{\theta }\right)\right)\right) = \cot \left(\mathrm{\pi sin}\left(\mathrm{\theta }\right)\right), \mathrm{then} \mathrm{a} \mathrm{value} \mathrm{of} \cos \left(\mathrm{\theta } - \frac{\mathrm{\pi }}{4}\right) \mathrm{among} \mathrm{the} \mathrm{following} \mathrm{is}$

• $\frac{1}{2\sqrt{2}}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{4}$

$$\begin{gathered} \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ \mathrm{x} + \mathrm{y} = \frac{2\mathrm{\pi }}{3} \\ \mathrm{and} \cos \left(\mathrm{x}\right) + \cos \left(\mathrm{y}\right) = \frac{3}{2}, \\ \mathrm{where} \mathrm{x}, \mathrm{y} \mathrm{are} \mathrm{real}, \mathrm{is} \end{gathered}$$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\sin \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\mathrm{x} - \mathrm{y}\right) = \frac{1}{2}\right\}$

• Empty set

The origin is translated to (1, 2). The point(7, 5) in the old system undergoes the following transformations successively.

I. Moves to the new point under the given translation of origin.

II. Translated through 2 units along the negative direction of the new X-axis.

III. Rotated through an angle - about the 4 origin of new system in the clockwise direction. The final position of the point (7, 5) is

• $\left(\frac{9}{\sqrt{2}}, \frac{- 1}{\sqrt{2}}\right)$

• $\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$

• $\left(\frac{7}{\sqrt{2}}, \frac{- 1}{\sqrt{2}}\right)$

• $\left(\frac{5}{\sqrt{2}}, \frac{- 1}{\sqrt{2}}\right)$