Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation : $$z\,{z^{ - 3}}\, + \,\overline z \,{z^3}\, = 350$$ is

Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta $$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }$$ is

Let A, B, C be three sets of complex numbers as defined below
$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, \ge \,1} \right\}$$
$$B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$$
$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$

Let z be any point in $$A \cap B \cap C$$

Then, $${\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}$$ lies between

Let A, B, C be three sets of complex numbers as defined below
$$A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,\,z\,\, \ge \,1} \right\}$$
$$B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$$
$$C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$$

Let z be any point $$A \cap B \cap C$$ and let w be any point satisfying $$\left| {w - 2 - i} \right| < 3\,$$. Then, $$\left| z \right| - \left| w \right| + 3$$ lies between