Complex Numbers

Match the statements in Column-I with those in Column-II.

[Note: Here $z$ takes values in the complex plane and Im $z$ and Re $z$ denote, respectively, the imaginary part and the real part of $z$.]

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
$$z\bar z^3+\bar zz^3=350$$
is

Let $z=\cos\theta+i\sin\theta$. Then the value of
$$\sum_{m=1}^{15}\operatorname{Im}\left(z^{2m-1}\right)$$
at $\theta=2°$ is

Match the conics in Column I with the statements/expressions in Column II.

The number of elements in the set $A\cap B\cap C$ is

Let $z$ be any point in $A\cap B\cap C$. Then, $|z+1-i|^2+|z-5-i|^2$ lies between

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

A particle $P$ starts from the point $z_0=1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt2$ units in the direction of the vector $\hat i+\hat j$ and then it moves through an angle $\frac\pi2$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_2$. The point $z_2$ is given by

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point $P$. Then the position of $P$ in the Argand plane is

If $|z|=1$ and $z\ne\pm1$, then all the values of $\dfrac{z}{1-z^2}$ lie on