(i) For any two complex numbers \(z_1, z_2\) and any real numbers \(a, b\), \(\; |az_1 - bz_2|^2 + |bz_1 + az_2|^2 = ____\).
\((a^2 + b^2)\bigl(|z_1|^2 + |z_2|^2\bigr)\)
(ii) The value of \(\sqrt{-25} \times \sqrt{-9}\) is ____.
-15
(iii) The number \(\dfrac{(1 - i)^3}{1 - i^3}\) is equal to ____.
-2
(iv) The sum of the series \(i + i^2 + i^3 + \dots\) up to 100 terms is ____.
0
(v) The multiplicative inverse of \(1 + i\) is ____.
\(\dfrac{1}{2} - \dfrac{i}{2}\)
(vi) If \(z_1\) and \(z_2\) are complex numbers such that \(z_1 + z_2\) is a real number, then \(z_2 = ____\).
\(\overline{z_1}\)
(vii) \(\arg(z) + \arg(\overline{z})\) \((\overline{z} \ne 0)\) is ____.
0
(viii) If \(|z + 4| \le 3\), then the greatest and least values of \(|z + 1|\) are ____ and ____ respectively.
6 and 0
(ix) If \(\left|\dfrac{z - 2}{z + 2}\right| = \dfrac{\pi}{6}\), then the locus of \(z\) is ____.
a circle
(x) If \(|z| = 4\) and \(\arg(z) = \dfrac{5\pi}{6}\), then \(z = ____\).
\(-2\sqrt{3} + 2i\)