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- Question 9, Exercise 1.2
- =3-2i$. We will use the following formulas: \[\text{Re}(z^{-2}) = \frac{(\text{Re}(z))^2 - (\text{Im}(z))^2}{|z|^4},\] \[\text{Im}(z^{-2}) = \frac{-2 \text{Re}(z) \text{Im}(z)}{|z|^4}. \] First, note $
- Question 2, Exercise 1.2
- ign} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \quad \text {Let }z_5=z_3 z_4 \\ =&z_1 (z_2 z_5) \quad \text{Multiplicative assocative law}\\ =&z_1\left(z_2 (z_3 z... 3 z_4 \\ =&z_1 \left((z_2 z_3) z_4 \right) \quad \text{Multiplicative assocative law}\\ =&z_1 \left((z_3 z_2) z_4 \right) \quad \text{Multiplicative comutative law}\\ =&z_1 \left(z_3
- Question 3, Exercise 1.2
- f $z=\bar{z}$. **Solution.** Let $$z=a+ib\quad \text{where}\quad a,b\in \mathbb{R}\, ... (1)$$ First s... $z$ is real or pure imaginary, then $$z=x \quad \text{ or } \quad z=iy.$$ This gives $$\bar{z}=x \quad \text{ or } \quad \bar{z}=-iy,$$ implies $$(\bar{z})^2=x^2 \quad \text{ or } \quad (\bar{z})^2=-y^2. ... (i)$$ Also, we
- Question 3, Exercise 1.4
- \begin{align*} &|z_r|=\sqrt{x_r^2+y_r^2} \quad \text{and}\quad |z|=\sqrt{a^2+b^2}. \\ &\theta_k = \arg... ) = \tan^{-1}\left(\dfrac{y_r}{x_r}\right) \quad \text{and}\quad \theta=\tan^{-1}\left(\dfrac{b}{a}\righ... as: \begin{align*} z_r=|z_r| e^{i\theta_k} \quad \text{and}\quad z=|z|e^{i\theta} \,\,-- (1) \end{align*... \theta_3+\ldots+\theta_n = \theta + 2k\pi, \quad \text{where } k\in \mathbb{Z}.\,\, -- (3) $$ Taking squ
- Question 5, Exercise 1.1
- real and imaginary parts. \begin{align}x=4 \quad \text{and}\quad 7y &=-7 \,\text{ i.e. }\,y=-1.\end{align} Thus we have $z=x+iy=4-i$. GOOD ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-1-p4|< Question 4]]</btn></text> <text align="right"><btn type="success">[[math-1
- Question 5, Exercise 1.2
- *Solution.** Suppose \begin{align}z_1&=x_1+iy_1 \text{ and } z_2&=x_2+iy_2\end{align} Now \begin{align}... (z_2)+4Im(z_1)Im(z_2).\end{align} ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-2-p4|< Question 4]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-2-p6|Question 6 >]]</btn></text>
- Question 6, Exercise 1.2
- **Solution.** Given: \begin{align} &z_{1}=3+i\text{ and } z_{2}=1+i.\end{align} Now \begin{align} \... {-3\pm i\sqrt{3}}{2} \end{align} ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-2-p5|< Question 5]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-2-p7|Question 7 >]]</btn></text>
- Question 3, Exercise 1.3
- 4ac}}}}{2a},$$ where $$a = 1,\quad b = 6,\quad \text{and}\quad c = -48.$$ Then \begin{align} z& = \d... ac{9 \pm \sqrt{37}}{2}\right\}$ ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-3-p2|< Question 2]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-3-p4|Question 4 >]]</btn></text>
- Question 7, Exercise 1.4
- 1-z) \\ = & \arg(1-i(x+iy))-\arg(1-(x+iy)) \quad \text{ as } z=x+iy \\ = & \arg(1-ix-i^2y)-\arg(1-x-iy) ... xy=0. \end{align*} As required. ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-4-p7|< Question 6(x-xvii)]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-4-p9|Question 8 >]]</btn></text>
- Question 2, Exercise 1.1
- frac{3}{13}i.\\ \end{align} GOOD ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-1-p1|< Question 1]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-1-p3|Question 3 >]]</btn></text> ======
- Question 3, Exercise 1.1
- }{5}\\ =&5+i. \end{align} GOOD ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-1-p2|< Question 2]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-1-p4|Question 4 >]]</btn></text>
- Question 4, Exercise 1.1
- +10i\end{align} Now do yourself ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-1-p3|< Question 3]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-1-p5|Question 5 >]]</btn></text>
- Question 6, Exercise 1.1
- ac{5 }{2}i-\dfrac{7}{8}$. GOOD ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-1-p5|< Question 5]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-1-p7|Question 7 >]]</btn></text>
- Question 4, Exercise 1.2
- c{16}{\sqrt{13}}\end{align} GOOD ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-2-p3|< Question 3]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-2-p5|Question 5 >]]</btn></text>
- Question 7, Exercise 1.2
- t{2} |z| \geq |Re(z)|+|Im(z)|.$$ ====Go to ==== <text align="left"><btn type="primary">[[math-11-nbf:sol:unit01:ex1-2-p6|< Question 6]]</btn></text> <text align="right"><btn type="success">[[math-11-nbf:sol:unit01:ex1-2-p8|Question 8 >]]</btn></text>