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- Question 1, Exercise 1.1
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- Question 7, Exercise 1.1
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- Question 6(i-ix), Exercise 1.4
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Fulltext results:
- Question 2, Exercise 1.3
- ====== Question 2, Exercise 1.3 ====== Solutions of Question 2 of Exercise 1.3 of Unit 01: Complex Numb... ard, Islamabad, Pakistan. ====Question 2(i)==== Solve the equation by completing square: $z^{2}-6 z+2=0$. **Solution.** \begin{align} & z^2 - 6z + 2 = 0 \\ \imp... \\ \implies &z = 3 \pm \sqrt{7}\end{align} Hence Solutioin set=$\{3 \pm \sqrt{7}\}$. ====Question 2(
- Question 3, Exercise 1.3
- ====== Question 3, Exercise 1.3 ====== Solutions of Question 3 of Exercise 1.3 of Unit 01: Complex Numb... ard, Islamabad, Pakistan. ====Question 3(i)==== Solve the quadratic equation: $\dfrac{1}{3} z^{2}+2 z-16=0$. **Solution.** Given \begin{align}&\dfrac{1}{3}z^{2}+2 z... 57}}}{2} \\ &= -3 \pm \sqrt{57} \end{align} Hence Solution set $=\{ -3 \pm \sqrt{57} \}$. ====Questio
- Question 1, Exercise 1.3
- ====== Question 1, Exercise 1.3 ====== Solutions of Question 1 of Exercise 1.3 of Unit 01: Complex Numb... polynomial into linear functions: $z^{2}+169$. **Solution.** \begin{align} & z^{2} + 169 \\ = & z^{2... olynomial into linear functions: $2 z^{2}+18$. **Solution.** \begin{align} & 2z^2 + 18 \\ = &2(z^2 - ... nomial into linear functions: $3 z^{2}+363$. **Solution.** \begin{align} & 3z^2 + 363 \\ = & 3(z^2
- Question 6(i-ix), Exercise 1.4
- ====== Question 6(i-ix), Exercise 1.4 ====== Solutions of Question 6(i-ix) of Exercise 1.4 of Unit 01: ... os 315^{\circ}+i \sin 315^{\circ}\right)$ ** Solution. ** \begin{align} &\sqrt{2}\left(\cos 315^{... t(\cos 210^{\circ}+i \sin 210^{\circ}\right)$ ** Solution. ** \begin{align*} &5\left(\cos 210^\circ +... c{3 \pi}{2}+i \sin \dfrac{3 \pi}{2}\right)$ ** Solution. ** \begin{align*} &2\left(\cos \frac{3\pi}
- Question 2, Exercise 1.1
- ====== Question 2, Exercise 1.1 ====== Solutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numb... ex number in the form $x+iy$: $(3+2i)+(2+4i)$ ** Solution. ** \begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+... lex number in the form $x+iy$: $(4+3i)-(2+5i)$ **Solution.** \begin{align}&(4+3i)-(2+5i)\\ =&(4-2)+(... lex number in the form $x+iy$: $(4+7i)+(4-7i)$ **Solution.** \begin{align} &(4+7i)+(4-7i)\\ =&(4+4)+
- Question 6(x-xvii), Exercise 1.4
- ====== Question 6(x-xvii), Exercise 1.4 ====== Solutions of Question 6(x-xvii) of Exercise 1.4 of Unit ... rac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$ ** Solution. ** //Do yourself as previous parts.// ====... rac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$ ** Solution. ** //Do yourself as previous parts.// =... \dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$ ** Solution. ** //Do yourself as previous parts.// ====
- Question 4, Exercise 1.3
- ====== Question 4, Exercise 1.3 ====== Solutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numb... d, Islamabad, Pakistan. =====Question 4(i)===== Solve the simultaneous system of linear equation with... z+(1+i) \omega=3 ; 2 z-(2+5 i) \omega=2+3 i$. ** Solution. ** \begin{align} &(1-i) z+(1+i) \omega=3 \... \dfrac{7}{53}i.$$ GOOD =====Question 4(ii)===== Solve the simultaneous system of linear equation with
- Question 8, Exercise 1.2
- ====== Question 8, Exercise 1.2 ====== Solutions of Question 8 of Exercise 1.2 of Unit 01: Complex Numb... $ in terms of $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|2z-i|=4.$$ Put $z=x+i y$, we ... $ in terms of $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|z-1|=|\bar{z}+i|.$$ Put $z=x+i... $ in terms of $x$ and $y$ by taking $z=x+i y$. **Solution.** Given: $$|z-4i| + |z+4i| = 10.$$ Put $z
- Question 9, Exercise 1.2
- ====== Question 9, Exercise 1.2 ====== Solutions of Question 9 of Exercise 1.2 of Unit 01: Complex Numb... nd real and imaginary parts of $(2+4 i)^{-1}$. **Solution.** Suppose $z=2+4i$. \begin{align} Re(2+4... and imaginary parts of $(3-\sqrt{-4})^{-2}$. **Solution.** Suppose $z=3 - \sqrt{-4}=3-2i$. We wi... ts of $\left(\dfrac{7+2 i}{3-i}\right)^{-1}$. **Solution.** We use the following formulas: \[Re\l
- Question 7, Exercise 1.4
- ====== Question 7, Exercise 1.4 ====== Solutions of Question 7 of Exercise 1.4 of Unit 01: Complex Numb... Cartesian form: $\arg (z-1)=-\dfrac{\pi}{4}$ ** Solution. ** Suppose $z=x+iy$, as \begin{align*} &\a... form: $z \bar{z}=4\left|e^{i \theta}\right|$ ** Solution. ** Suppose $z=x+iy$, then $\bar{z}=x-iy$. ... {\pi}{3} \leq \arg (z-4) \leq \dfrac{\pi}{3}$ ** Solution. ** \begin{align*} &-\frac{\pi}{3} \leq \ar
- Question 3, Exercise 1.1
- ====== Question 3, Exercise 1.1 ====== Solutions of Question 3 of Exercise 1.1 of Unit 01: Complex Numb... plify the following $\dfrac{(2+i)(3-2i)}{1+i}$ **Solution.** \begin{align}&\dfrac{(2+i)(3-2i)}{1+i}\\ ... Simplify the following $\dfrac{1+i}{(2+i)^2}$ **Solution.** \begin{align}&\dfrac{1+i}{(2+i)^2}\\ =&\... the following $\dfrac{1}{3+i}-\dfrac{1}{3-i}$ **Solution.** \begin{align}&\dfrac{1}{3+i}-\dfrac{1}{3
- Question 10, Exercise 1.2
- ====== Question 10, Exercise 1.2 ====== Solutions of Question 10 of Exercise 1.2 of Unit 01: Complex Nu... _{!}}\right|=\left|-\overline{z_{!}}\right|.$$ **Solution.** \begin{align} |z_1| &= \sqrt{(-3)^2 + (2... )}=\frac{\overline{z_{1}}}{\overline{z_{2}}}$. **Solution.** Given \[z_1 = -3 + 2i, \quad z_2 = 1 - ... z_{2}}=\overline{z_{1}}\,\, \overline{z_{2}}$. **Solution.** Given \[ z_1 = -3 + 2i, \quad z_2 = 1
- Question 1, Exercise 1.1
- ====== Question 1, Exercise 1.1 ====== Solutions of Question 1 of Exercise 1.1 of Unit 01: Complex Numb... ====Question 1(i)==== Evaluate ${{i}^{31}}$. **Solution.** \begin{align}{{i}^{31}}&=i\cdot{{i}^{30}... (ii)==== Evaulate ${{\left( -i \right)}^{6}}$. **Solution.** \begin{align} {{\left( -i \right)}^{23}}... luate ${{\left( -1 \right)}^{\frac{-13}{2}}}$. **Solution.** \begin{align}{{\left( -1 \right)}^{\frac
- Question 4, Exercise 1.1
- ====== Question 4, Exercise 1.1 ====== Solutions of Question 4 of Exercise 1.1 of Unit 01: Complex Numb... n each of the following: $(2+3i)x+(1+3i)y+2=0$ **Solution.** \begin{align}&(2+3i)x+(1+3i)y+2=0\\ \im... ollowing: $\dfrac{x}{(1+i)}+\dfrac{y}{1-2i}=1$ **Solution.** \begin{align}&\dfrac{x}{(1+i)}+\dfrac{y... x}{(2+i)}=\dfrac{1-5i}{(3-2i)}+\dfrac{y}{2-i}$ **Solution.** \begin{align}&\dfrac{x}{(2+i)}=\dfrac{(
- Question 6, Exercise 1.1
- ====== Question 6, Exercise 1.1 ====== Solutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numb... d the conjugate of the complex number $4-3 i$. **Solution.** Given: $z=4-3 i$, then $\bar{z}=4+3i$. ... d the conjugate of the complex number $3 i+8$. **Solution.** Do Yourself ====Question 6(iii)==== Fi... f the complex number $2+\sqrt{\dfrac{-1}{5}}$. **Solution.** Given: \begin{align}z=&2+\sqrt{\dfrac{-