31.

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation 1 + x + x² = 0, then the matrix product $\left[\begin{array}{cc}1& \mathrm{\beta }\\ \mathrm{\alpha }& \mathrm{\alpha }\end{array}\right]\left[\begin{array}{cc}\mathrm{\alpha }& \mathrm{\beta }\\ 1& \mathrm{\beta }\end{array}\right]$ = ?

• $\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$

• $\left[\begin{array}{cc} - 1& - 1\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc}1& - 1\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc} - 1& - 1\\ - 1& - 2\end{array}\right]$

The sum of all real roots of the equation

${\left|\mathrm{x} - 3\right|}^2 + \left|\mathrm{x} - 3\right| - 2 = 0 \mathrm{is} :$

• 2

• 3

• 4

• 6

It is given that the roots of the equation x² - 4x - log₃(P) = 0 are real. For this, the minimum value of P is

• $\frac{1}{27}$

• $\frac{1}{64}$

• $\frac{1}{81}$

• 1

The smallest positive integer n for which = ${\left(\frac{1 + \mathrm{i}}{1 - \mathrm{i}}\right)}^{\mathrm{n}} = 1$ is :

• 1

• 4

• 8

• 16

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation 3x² + 2x + 1 = 0, then the equation whose roots are $\mathrm{\alpha } + {\mathrm{\beta }}^{ - 1}$ is :

• 3x² + 8x + 16 = 0

• 3x² - 8x - 16 = 0

• 3x² + 8x - 16 = 0

• x² + 8x + 16 = 0

If x is any real number, then $\frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^4}$ belongs to which one of the following intervals ?

• $\left(0, 1\right)$

• $\left[0, \frac{1}{2}\right]$

• $\left(0, \frac{1}{2}\right)$

• [0, 1]

Suppose $\mathrm{\omega }$ is a cube root unity with $\mathrm{\omega } \ne 1$. Suppose P and Q are the points on the complex plane defined by $\mathrm{\omega } \mathrm{and} {\mathrm{\omega }}^2$. If O is the origin,then what is the angle between OP and OQ ?

• $60°$

• $90°$

• $120°$

• $150°$

If ${\mathrm{x}}^2 - \mathrm{px} + 4 > 0 \mathrm{for} \mathrm{all} \mathrm{real} \mathrm{values} \mathrm{of} \mathrm{x}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\left|\mathrm{p}\right| < 4$

• $\left|\mathrm{p}\right| \le 4$

• $\mathrm{p} > 4$

• $\mathrm{p} \ge 4$

If z = x + iy = ${\left(\frac{1}{\sqrt{2}} - \frac{\mathrm{i}}{\sqrt{2}}\right)}^{ - 25}, \mathrm{where} \mathrm{i} = \sqrt{ - 1}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{fundamental} \mathrm{amplitude} \mathrm{of} \frac{\mathrm{z} - \sqrt{2}}{\mathrm{z} - \mathrm{i}\sqrt{2}} ?$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

Let z₁ , z₂ and z₃ be non-zero complex numbers satisfying

z² = iz, where i = $\sqrt{ - 1}$.

What is z₁ + z₂ + z₃ = ?

• i

•  - i

• 0

• 1