R S Aggarwal Solutions for Exercise 3: EXERCISE 5C

Express the following in the form $(a+ib)$ and find its conjugate.

$\frac{1}{(4+3i)}$

Express the following in the form $(a+ib)$ and find its conjugate.

$(2+3i{)}^2$

For all $z\in C$, prove that

$\frac{1}{2i}(z-\overline{z})=Im\left(z\right)$

For all $z\in C$, prove that

$z\overline{z}=|z{|}^2$

For all $z\in C$, prove that $(z+\overline{z})$ is real.

For all $z\in C$, prove that $(z-\overline{z})$ is $0$ or imaginary.

Note:The question given in the book seems to have an error and the modified question should be as given below.

 If $z_1=(1+i)$ and $z_2=(-2+4i)$ prove that $Im\left(\frac{z_1{z}_2}{{\overline{z}}_1}\right)=-2$

If $a$ and $b$ are real numbers such that $a^2+b^2=1$ then show that a real value of $x$ will satisfy the equation, $\frac{1-ix}{1+ix}=(a-ib)$