Algebra

A value of $\theta \,$ for which ${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$ is purely imaginary, is :

Let $A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $\mathrm{A}$ is :

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

Let O be the origin and A be the point ${z_1} = 1 + 2i$. If B is the point ${z_2}$, ${\mathop{\rm Re}\nolimits} ({z_2}) < 0$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

Let a complex number be $w=1-\sqrt{3}i$. Let another complex number $z$ be such that $\left|zw\right|=1$ and $\arg \left(z\right)-\arg \left(w\right)=\frac{\pi }{2}$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to

Let z $\in$ C be such that |z| < 1.

If $\omega = {{5 + 3z} \over {5(1 - z)}}$z, then :

If the four complex numbers $z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$ and $z-2Re(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :

If ${{3 + i\sin \theta } \over {4 - i\cos \theta }}$, $\theta$ $\in$ [0, 2$\theta$], is a real number, then an argument of
sin$\theta$ + icos$\theta$ is :

For all $z \in C$ on the curve $C_{1}:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $\mathrm{C}_{2}$. Then :