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

193.

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

The values of p for which the difference between the roots of the equation x² + px + 8 = 0 is 2, are

• $\pm 2$

• $\pm 4$

• $\pm 6$

• $\pm 8$

For any two complex numbers z₁ and z₂ and any real numbers a and b, ${\left|\left({\mathrm{az}}_1 - {\mathrm{az}}_2\right)\right|}^2 + {\left|\left({\mathrm{bz}}_1 + {\mathrm{az}}_2\right)\right|}^2$ is equal to

• $\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)\left(\left|{\mathrm{z}}_1\right| + \left|{\mathrm{z}}_2\right|\right)$

• $\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)\left({\left|{\mathrm{z}}_1\right|}^2 + {\left|{\mathrm{z}}_2\right|}^2\right)$

• $\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)\left({\left|{\mathrm{z}}_1\right|}^2 - {\left|{\mathrm{z}}_2\right|}^2\right)$

• None of these

If w ($\left(\ne 1\right)$ )is a cube root of unity and (1 + w²)ⁿ = (1 + w⁴)ⁿ, then the least positive value of n is

• 2

• 3

• 5

• 6

The complex numbers sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other for

• x = $\mathrm{n\pi }$

• $\mathrm{x} = \left(\mathrm{n} + \frac{1}{2}\right)\mathrm{\pi }$

• x = 0

• No value of x

The locus of the points z which satisfy the condition $\arg \left(\frac{\mathrm{z} - 1}{\mathrm{z} + 1}\right) = \frac{\mathrm{\pi }}{3}$

• a straight line

• a circle

• a parabola

• None of the above