Exercise 1.2 (Solutions)

The solutions of the Exercise 1.2 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to real and imaginary part of complex numbers, modulus and conjugate of the complex numbers.

Question 1. Show that for any complex number :
(i) $\operatorname{Re}(i z)=-\operatorname{Im}(z)$ (ii) $\operatorname{Im}(i z)=\operatorname{Re}(z)$
Solution: Question 1

Question 2. Use the algebraic properties of complex numbers to prove that $$\left(z_{1} z_{2}\right)\left(z_{3} z_{4}\right)=\left(z_{1} z_{3}\right)\left(z_{2} z_{4}\right)=z_{3}\left(z_{1} z_{2}\right) z_{4} $$
Solution: Question 2

Question 3. Prove that for $z \in \mathbb{C}$ :
(i) $z$ is real iff $z=\bar{z}$ (ii) $\dfrac{z-\bar{z}}{z+\bar{z}}=i\left(\dfrac{I m z}{R e z}\right)$ (iii) $z$ is either real or pure imaginary iff $(\overline{z})^{2}=z^{2}$.
Solution: Question 3

Question 4. Prove that for $z \in \mathbb{C}$ :
If $z_{1}=2-3 i$ and $\left|z_{1} z_{2}\right|=16$ find $\left|z_{2}\right|$.
Solution: Question 4

Question 5. If $z_1$ and $z_2$ are two any complex numbers then prove that $|z_1+z_2|^2-|z_1-z_2|^2=4Re(z_1)Re(z_2)$
Solution: Question 5

Question 6. Find the value of $\lambda$; if $\left|\dfrac{z_{1}}{z_{2}}+\lambda\right|=\sqrt{\lambda+2}$; where $z_{1}=3+i$ and $z_{2}=1+i$.
Solution: Question 6

Question 7.Verify that $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$ Hint: (Start with $\left(|x|-|y|)^{2} \geq 0\right)$
Verify that $\sqrt{2}|z| \geq|\operatorname{Re}(z)|+|\operatorname{Im}(z)| \quad$ Hint: (Start with $\left(|x|-|y|)^{2} \geq 0\right)$
Solution: Question 7

Question 8. Solve the following
(i) Write $|2 z-i|=4$ in terms of $x$ and $y$ by taking $z=x+i y$
(ii) Write $|z-1|=|\bar{z}+i|$ in terms of $x$ and $y$ by taking $z=x+i y$.
(iii) Write $|z-4 i|+|z+4 i|=10$ in terms of $x$ and $y$ by taking $z=x+i y$.
(iv) Write $\frac{1}{2} \operatorname{Re}(i \bar{z})=4$ in terms of $x$ and $y$ by taking $z=x+i y$.
(v) Write $lm\left(\dfrac{z-1}{2 i}\right)=-5$ in terms of $x$ and $y$ by taking $z=x+i y$.
(vi) Write $-2 \leq Im(z+i) \leq 3$ in terms of $x$ and $y$ by taking $z=x+i y$.
Solution: Question 8

Question 9. Find real and imaginary parts of the following.
(i) $(2+4 i)^{-1}$
(ii) $(3-\sqrt{-4})^{-2}$
(iii) $\left(\dfrac{7+2 i}{3-i}\right)^{-1}$
(iv) $\left(\dfrac{4+2 i}{2+5 i}\right)^{-2}$.
(v) $\left(\dfrac{5-4 i}{5+4 i}\right)^{2}$
(vi) $\dfrac{3-7 i}{2+5 i}$
Solution: Question 9

Question 10. Solve the following.
(i) For $z_{1}=-3+2 i$, verify:
$\left|z_{1}\right|=\left|-z_{1}\right|=\left|\overline{z_{1}}\right|=\left|-\overline{z_{1}}\right|.$
(ii) For $z_{1}=-3+2 i$ and $z_{2}=1-3 i$ verify:
$\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\overline{z_{1}}}{\overline{z_{2}}}$
(iii) For $z_{1}=-3+2 i$ and $z_{2}=1-3 i$ verify:
$\overline{z_{1} z_{2}}=\overline{z_{1}}\,\, \overline{z_{2}}$
(iv) For $z_{1}=-3+2 i$ and $z_{2}=1-3 i$ verify:
$\overline{z_{1}+z_{2}}=\overline{z_{1}}+\overline{z_{2}}$.
(v) For $z_{1}=-3+2 i$ and $z_{2}=1-3 i$ verify:
$\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$
(vi) For $z_{1}=-3+2 i$ and $z_{2}=1-3 i$ verify:
$\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|$
Solution: Question 10