Complex Numbers and Quadratic Equations

Find the values of 'a' for which the expression x² - (3a - 1)x + 2a² + 2a - 11 is always positive

The value of (1 - w + w²)⁵ + (1 + w - w²)⁵, where w and w² are the complex cube roots of unity, is

• 0

• 32w

• - 32

• 32

Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - 2x$\cos \left(\mathrm{\varphi }\right)$ + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{\mathrm{n}}, {\mathrm{\beta }}^{\mathrm{n}}$ is

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) - 1$ = 0

• ${\mathrm{x}}^2 - 2\mathrm{xcos}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 - 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) + 1 = 0$

• ${\mathrm{x}}^2 + 2\mathrm{xsin}\left(\mathrm{n\varphi }\right) - 1 = 0$

If one root of the equation x² + (1 - 3i)x - 2(1 + i) = 0 is - 1 + i, then the other root is

• - 1 - i

• $\frac{\left(- 1 - \mathrm{i}\right)}{2}$

• i

• 2i

The equation ${\mathrm{x}}^2 - 3\left|\mathrm{x}\right| + 2 = 0$ has

• no real root

• one real root

• two real root

• four real root

For two complex numbers z₁, z₂ the relation $\left|{\mathrm{z}}_1 + {\mathrm{z}}_2\right| = \left|{\mathrm{z}}_1\right| + \left|{\mathrm{z}}_2\right|$ holds, if

• arg(z₁) = arg(z₂)

• arg(z₁) + arg(z₂) = $\frac{\mathrm{\pi }}{2}$

• z₁z₂ = 1

• $\left|{\mathrm{z}}_1\right| = \left|{\mathrm{z}}_2\right|$

If $\left|\mathrm{z} + 4\right| \le 3$, then the greatest and the least value of $\left|\mathrm{z} + 1\right|$ are

• 6, - 6

• 6, 0

• 7, 2

• 0, - 1

The region of the complex plane for which $\left|\frac{\mathrm{z} - \mathrm{a}}{\mathrm{z} + \overline{\mathrm{a}}}\right|$ = 1, [Re (a) $\ne$ 0] is

• x - axis

• y - axis

• the straight line x = a

• None of the above

The number of non-zero integral solutions of the equation ${\left|1 - \mathrm{i}\right|}^{\mathrm{x}}$ = 2^x

• infinite

• 1

• 2

• None of these

Let $\mathrm{\alpha }, {\mathrm{\alpha }}^2$ be the roots of x² + x + 1 = 0, then the equation whose roots are ${\mathrm{\alpha }}^{31}, {\mathrm{\alpha }}^{62}$ is

• x² - x + 1 = 0

• x² + x - 1 = 0

• x² + x + 1 = 0

• x⁶⁰ + x³⁰ + 1 = 0