The locus of the point z = x + iy satisfying the equation

$\left|\frac{\mathrm{z} - 1}{\mathrm{z} + 1}\right| = 1 \mathrm{is} \mathrm{given} \mathrm{by}$ :

• x = 0

• y = 0

• x = y

• x + y = 0

The equation of the locus of z such that $\left|\frac{\mathrm{z} + \mathrm{i}}{\mathrm{z} - \mathrm{i}}\right| = 2$, where z= x + iy is a complex number, is

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 + 10\mathrm{y} - 3 = 0$

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 + 10\mathrm{y} + 3 = 0$

• $3{\mathrm{x}}^2 - 3{\mathrm{y}}^2 - 10\mathrm{y} - 3 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 5\mathrm{y} + 3 = 0$

The product of the distinct (2n)th roots of 1 + i$\sqrt{3}$ is equal to :

• 0

• $- 1 - \mathrm{i}\sqrt{3}$

• $1 + \mathrm{i}\sqrt{3}$

• $- 1 + \mathrm{i}\sqrt{3}$

If a = $\frac{1 - \mathrm{i}\sqrt{3}}{2}$, then the correct matching of List-I from List-II is

         List-I                       List-II

(i)      $\mathrm{a}\overline{\mathrm{a}}$                            $- \frac{\mathrm{\pi }}{3}$

(ii)    $\arg \left(\frac{1}{\overline{\mathrm{a}}}\right)$                       $- \mathrm{i}\sqrt{3}$

(iii)    $\mathrm{a} - \overline{\mathrm{a}}$                        $\raisebox{1ex}{$2\mathrm{i}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.$ 

(iv)     $\mathrm{Im}\left(\frac{4}{3\mathrm{a}}\right)$                    1

                                        $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

                                        $\frac{2}{\sqrt{3}}$

correct match is 

205.

The points in the set $\left\{\mathrm{z} \in \mathrm{C} : \arg \left(\frac{\mathrm{z} - 2}{\mathrm{z} - 6\mathrm{i}}\right) = \frac{\mathrm{\pi }}{2}\right\}$ (where C denotes the set of all complex numbers) lie on the curve which is a

• circle

• pair of lines

• parabola

• hyperbola

If w is a complex cube root of unity, then $\sin \left\{\left({\mathrm{w}}^{10} + {\mathrm{w}}^{23}\right)\mathrm{\pi } - \frac{\mathrm{\pi }}{4}\right\}$ is equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• 1

• $\frac{\sqrt{3}}{2}$

If m₁, m₂, m₃ and m₄ respectively denote the moduli of the complex numbers 1 + 4i, 3 + i, 1 - i and 2 - 3i, then the correct one, among the following is

• m₁ < m₂ < m₃ < m₄

• m₄ < m₃ < m₂ < m₁

• m₃ < m₂ < m₄ < m₁

• m₃ < m₁ < m₂ < m₄

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } + \mathrm{\beta } = - 2 \mathrm{and} {\mathrm{\alpha }}^3 + {\mathrm{\beta }}^3 = - 56, \mathrm{then} \\ \mathrm{the} \mathrm{quadratic} \mathrm{equation} \mathrm{whose} \mathrm{roots} \mathrm{are} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \end{gathered}$$

• x² + 2x - 16 = 0

• x² + 2x + 15 = 0

• x² + 2x - 12 = 0

• x² + 2x - 8 = 0

let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the quadratic equation ax² + bx + c = 0. Observe the lists given below

The correct match of List-I from List-II is

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of x³ + 4x + 1 = 0, then the equation whose roots are $\frac{{\mathrm{\alpha }}^3}{\mathrm{\beta } + \mathrm{\gamma }}, \frac{{\mathrm{\beta }}^2}{\mathrm{\gamma } + \mathrm{\alpha }}, \frac{{\mathrm{\gamma }}^2}{\mathrm{\alpha } + \mathrm{\beta }}$ is

• x³ - 4x - 1 = 0

• x³ - 4x + 1 = 0

• x³ + 4x - 1 = 0

• x³ + 4x + 1 = 0