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- Unit 01: Complex Numbers (Solutions)
- ===== Unit 01: Complex Numbers (Solutions) ===== {{ :math-11-nbf:sol:math-11-nbf-sol-unit01.jpg?nolink&400x335|Unit 01: Complex Numbers (Solutions)}} This is a first unit of the... his unit the students will be able to * Recall complex number $z$ and recognize its real and imaginary part. * Know the condition for equality of two complex numbers. * Revising the basic operations on com
- Question 1, Exercise 1.4 @math-11-nbf:sol:unit01
- lutions of Question 1 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... abad, Pakistan. =====Question 1(i)===== Write a complex number $2+i 2 \sqrt{3}$ in polar form. ** Soluti... \sqrt{3}) = \frac{\pi}{3}. \end{align} Since the complex number \( 2 + i 2 \sqrt{3} \) lies in the first q... OOD =====Question 1(ii)===== Write the following complex number $3-i \sqrt{3}$ in polar form. ** Solution
- Question 6(i-ix), Exercise 1.4 @math-11-nbf:sol:unit01
- s of Question 6(i-ix) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... akistan. =====Question 6(i)===== Write a given complex number in the algebraic form: $\sqrt{2}\left(\cos... end{align} =====Question 6(ii)===== Write a given complex number in the algebraic form: $5\left(\cos 210^{\... d{align*} =====Question 6(iii)===== Write a given complex number in the algebraic form: $2\left(\cos \dfrac
- Question 2, Exercise 1.1 @math-11-nbf:sol:unit01
- lutions of Question 2 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... stan. ====Question 2(i)==== Write the following complex number in the form $x+iy$: $(3+2i)+(2+4i)$ ** So... GOOD ====Question 2(ii)==== Write the following complex number in the form $x+iy$: $(4+3i)-(2+5i)$ **Sol... GOOD ====Question 2(iii)==== Write the following complex number in the form $x+iy$: $(4+7i)+(4-7i)$ **Sol
- Question 6(x-xvii), Exercise 1.4 @math-11-nbf:sol:unit01
- of Question 6(x-xvii) of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... Pakistan. =====Question 6(x)===== Write a given complex number in the algebraic form: $7 \sqrt{2}\left(\c... s parts.// =====Question 6(xi)===== Write a given complex number in the algebraic form: $10 \sqrt{2}\left(\... parts.// =====Question 6(xii)===== Write a given complex number in the algebraic form: $2\left(\cos\dfrac{
- Question 1, Review Exercise @math-11-nbf:sol:unit01
- ions of Question 1 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... * (a) natural * (b) integer * %%(c)%% complex * (d) rational \\ <btn type="link" collapse="... </btn><collapse id="a1" collapsed="true">%%(c)%%: complex</collapse> ii. Every complex number has $\operatorname{part}(\mathrm{s})$. * (a) one * (b) t
- Question 6, Exercise 1.1 @math-11-nbf:sol:unit01
- lutions of Question 6 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... ====Question 6(i)==== Find the conjugate of the complex number $4-3 i$. **Solution.** Given: $z=4-3 i$,... ====Question 6(ii)==== Find the conjugate of the complex number $3 i+8$. **Solution.** Do Yourself ====Question 6(iii)==== Find the conjugate of the complex number $2+\sqrt{\dfrac{-1}{5}}$. **Solution.**
- Question 2, Exercise 1.2 @math-11-nbf:sol:unit01
- lutions of Question 2 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... ===Question 2==== Use the algebraic properties of complex numbers to prove that $$ \left(z_{1} z_{2}\right)... e the required result. **Remark:** For any three complex numbers $z_1$, $z_2$ and $z_3$, we have $$z_1 (z_... ells us that the order in which we multiply three complex numbers doesn't matter; we will always end up wit
- Question 4, Exercise 1.3 @math-11-nbf:sol:unit01
- lutions of Question 4 of Exercise 1.3 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... e the simultaneous system of linear equation with complex coefficients: $(1-i) z+(1+i) \omega=3 ; 2 z-(2+5 ... e the simultaneous system of linear equation with complex coefficients: $2 i z+(3-2 i) \omega=1+i ;(1-2 i) ... e the simultaneous system of linear equation with complex coefficients: $\dfrac{3}{i} z-(6+2 i) \omega=5 ;
- Question 1, Exercise 1.2 @math-11-nbf:sol:unit01
- lutions of Question 1 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... akistan. ====Question 1(i)==== Show that for any complex number, $\operatorname{Re}(i z)=-\operatorname{Im... {align} ====Question 1(ii)==== Show that for any complex number, $\operatorname{Im}(i z)=\operatorname{Re}
- Question 2, Exercise 1.4 @math-11-nbf:sol:unit01
- lutions of Question 2 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... ad, Pakistan. =====Question 2(i)===== Write the complex number $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\... angular form. =====Question 2(ii)===== Write the complex number $\dfrac{\cos \dfrac{\pi}{6} - i \sin \dfra
- Question 4, Review Exercise @math-11-nbf:sol:unit01
- ions of Question 4 of Review Exercise of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... ad, Pakistan. ===== Question 4 ===== Locate the complex number $z=x+i y$ on the complex plane if $\left|\dfrac{z+2 i}{z-2 i}\right|=1$ ** Solution. ** Given $z
- Question 5, Exercise 1.1 @math-11-nbf:sol:unit01
- lutions of Question 5 of Exercise 1.1 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... Islamabad, Pakistan. ====Question 5==== Find the complex number $z$ if $4z-3\bar{z}=\dfrac{1-18i}{2-i}$ *
- Question 5, Exercise 1.2 @math-11-nbf:sol:unit01
- lutions of Question 5 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... ====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=
- Question 3, Exercise 1.4 @math-11-nbf:sol:unit01
- lutions of Question 3 of Exercise 1.4 of Unit 01: Complex Numbers. This is unit of Model Textbook of Mathem... rac{b}{a}\right). \end{align*} We can write these complex numbers in polar form as: \begin{align*} z_r=|z_r