Geometrical Representation of Algebra of Complex Numbers

Q.

Let $z_{1}and{z}_2$be two complex numbers with alpha and beta as their principal arguments such that $\alpha +\beta >\mathrm{\pi }$, then principal $arg\left(z_1{z}_2\right)$ is given by

1. $\alpha +\beta +\pi$

2. $\alpha +\beta -\pi$

3. $\alpha +\beta -2\pi$

4. $\alpha +\beta$

Q. The roots of the equation $t^3+3a{t}^2+3bt+c=0$ are $z_1,z_2,z_3$ which represent the vertices of an equilateral triangle, then

1. $a^2=3b$
2. $b^2=a$
3. $a^2=b$
4. $b^2=3a$

Q. A region $S$ in complex plane is defined by $S=\{x+iy:-1\le x,y\le 1\}$. A complex number $z=x+iy$ is chosen uniformly at random from $S.$ If $P$ be the probability that the complex number $\frac{3}{4}(1+i)z$ is also in $S,$ then the value of $27P$ is

Q. If $a,b$ are the numbers between $0$ and $1$ such that the points $z_1=a+i,z_2=1+bi$ and $z_0=0$ form an equilateral triangle, then

1. $a=2-\sqrt{3}\text{and}b=2-\sqrt{3}$
2. $a=2-\sqrt{3}\text{and}b=2+\sqrt{3}$
3. $a=2+\sqrt{3}\text{and}b=2-\sqrt{3}$
4. $a=2+\sqrt{3}\text{and}b=2+\sqrt{3}$