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- Question 5, Exercise 1.4
- Question 5 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... tan. =====Question 5===== If $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$, show that: (i) $\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$. (
- Question 4, Exercise 1.3
- Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... 3 ; 2 z-(2+5 i) \omega=2+3 i$. ** Solution. ** \begin{align} &(1-i) z+(1+i) \omega=3 \quad \cdots(1)... cdots(2) \end{align} Multiplying Eq. (1) by $2$: \begin{align} &(2-2i)z+(2+2i) \omega=6 \quad \cdots (3) \end{align} Multiplying Eq. (2) by $(1-i)$: \begin{align} &2(1-i)z-(1-i) (2+5 i)\omega=(1-i) (2+3
- Question 10, Exercise 1.2
- uestion 10 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... left|-\overline{z_{!}}\right|.$$ **Solution.** \begin{align} |z_1| &= \sqrt{(-3)^2 + (2)^2} \\ &= \... + 4} = \sqrt{13} \,\, -- (1) \end{align} Now \begin{align} -z_1 &= -(-3 + 2i) = 3 - 2i\\ \implies ... + 4} = \sqrt{13} \,\, -- (2) \end{align} Also \begin{align} \overline{z_1} &= -3 - 2i \\ \implies
- Question 1, Exercise 1.4
- Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... tan. =====Question 1(i)===== Write a complex number $2+i 2 \sqrt{3}$ in polar form. ** Solution. ** Let $z=x+iy=2 + i 2 \sqrt{3}$. We have \begin{align} r & = \sqrt{x^2 + y^2} = \sqrt{2^2 + (... = \sqrt{4 + 12} = \sqrt{16} = 4. \end{align} and \begin{align} \alpha & = \tan^{-1}\left|\frac{y}{x}\r
- Question 1, Exercise 1.3
- Question 1 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... o linear functions: $z^{2}+169$. **Solution.** \begin{align} & z^{2} + 169 \\ = & z^{2} - (13i)^2 \... linear functions: $2 z^{2}+18$. **Solution.** \begin{align} & 2z^2 + 18 \\ = &2(z^2 - (3i)^2)\\ = ... near functions: $3 z^{2}+363$. **Solution.** \begin{align} & 3z^2 + 363 \\ = & 3(z^2 - (11i)^2)\\
- Question 6(i-ix), Exercise 1.4
- on 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... =====Question 6(i)===== Write a given complex number in the algebraic form: $\sqrt{2}\left(\cos 315^{... i \sin 315^{\circ}\right)$ ** Solution. ** \begin{align} &\sqrt{2}\left(\cos 315^{\circ}+i \sin ... =====Question 6(ii)===== Write a given complex number in the algebraic form: $5\left(\cos 210^{\circ}+
- Question 2, Exercise 1.1
- Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... =Question 2(i)==== Write the following complex number in the form $x+iy$: $(3+2i)+(2+4i)$ ** Solution. ** \begin{align}&(3+i2)+(2+i4)\\ =&(3+2)+(i2+i4)\\ =&5+i... Question 2(ii)==== Write the following complex number in the form $x+iy$: $(4+3i)-(2+5i)$ **Solution.
- Question 9, Exercise 1.2
- Question 9 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... +4 i)^{-1}$. **Solution.** Suppose $z=2+4i$. \begin{align} Re(2+4i)^{-1} & = Re(z^{-1}) = \dfrac{R... \dfrac{2}{20}\\ &= \dfrac{1}{10}. \end{align} \begin{align} Im(2+4i)^{-1} & = Im(z^{-1}) = -\dfrac{... |z|^4}. \] First, note $Re(z)=3$, $Im(z)=-2$ and \begin{align} |z| &= \sqrt{3^2 + (-2)^2} \\&= \sqrt{1
- Question 4, Exercise 1.1
- Question 4 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... ====Question 4(i)==== Find the values of real number $x$ and $y$ in each of the following: $(2+3i)x+(1+3i)y+2=0$ **Solution.** \begin{align}&(2+3i)x+(1+3i)y+2=0\\ \implies &(2x+y+2... 0.\end{align} Comparing real and imaginary parts \begin{align} 2x+y+2&=0 \quad \cdots(1)\\ 3x+3y&=0\qu
- Question 7, Exercise 1.4
- Question 7 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... {\pi}{4}$ ** Solution. ** Suppose $z=x+iy$, as \begin{align*} &\arg (z-1)=-\dfrac{\pi}{4} \\ \implie... n. ** Suppose $z=x+iy$, then $\bar{z}=x-iy$. As \begin{align*} &z \bar{z}=4\left|e^{i \theta}\right| ... arg (z-4) \leq \dfrac{\pi}{3}$ ** Solution. ** \begin{align*} &-\frac{\pi}{3} \leq \arg (z-4) \leq \
- Question 6(x-xvii), Exercise 1.4
- 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... =====Question 6(x)===== Write a given complex number in the algebraic form: $7 \sqrt{2}\left(\cos \df... =====Question 6(xi)===== Write a given complex number in the algebraic form: $10 \sqrt{2}\left(\cos \d... ====Question 6(xii)===== Write a given complex number in the algebraic form: $2\left(\cos\dfrac{5\pi}{
- Question 8, Exercise 1.2
- Question 8 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... ** Given: $$|2z-i|=4.$$ Put $z=x+i y$, we have \begin{align} & |2(x+iy)-i|=4 \\ \implies & |2x+i(2y-... 2+(2y-1)^2}=4 \end{align} Squaring on both sides \begin{align} & (2x)^2+(2y-1)^2 = 16\\ \implies & 4x^... en: $$|z-1|=|\bar{z}+i|.$$ Put $z=x+iy$, we have \begin{align} & |(x+iy)-1| = |(x-iy)+i| \\ \implies &
- Question 2, Exercise 1.2
- Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... 2==== Use the algebraic properties of complex numbers to prove that $$ \left(z_{1} z_{2}\right)\left(... eft(z_{1} z_{2}\right) z_{4} $$ **Solution.** \begin{align} &(z_1 z_2)(z_3 z_4) \\ =&(z_1 z_2)z_5 \... ve law}\\ =&z_1\left(z_2 (z_3 z_4) \right) \quad \because\,\, z_5=z_3 z_4 \\ =&z_1 \left((z_2 z_3) z_4
- Question 2, Exercise 1.3
- Question 2 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... pleting square: $z^{2}-6 z+2=0$. **Solution.** \begin{align} & z^2 - 6z + 2 = 0 \\ \implies & z^2 - ... \end{align} Take the square root of both sides: \begin{align} &z - 3 = \pm \sqrt{7} \\ \implies &z =... : $-\dfrac{1}{2} z^{2}-5 z+2=0$. **Solution.** \begin{align} -\dfrac{1}{2} z^{2} - 5z + 2& = 0 \end{
- Question 3, Exercise 1.4
- Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathematics ... z_r=x_r+iy_r$, $r=1,2,...,n$ and $z=a+ib$. Then \begin{align*} &|z_r|=\sqrt{x_r^2+y_r^2} \quad \text{... ight). \end{align*} We can write these complex numbers in polar form as: \begin{align*} z_r=|z_r| e^{i\theta_k} \quad \text{and}\quad z=|z|e^{i\theta} \,\