1. **Problem:** Define a complex number. A complex number is a number of the form $z = x + iy$, where $x$ and $y$ are real numbers and $i$ is the imaginary unit with the property $i^2 = -1$. Example: $z = 2 + 3i$, $z = 2 + 7i$. 2. **Problem:** Express a complex number in polar form. The polar form of a complex number is given by: $$z = r(\cos \theta + i \sin \theta)$$ where $r = \sqrt{x^2 + y^2}$ is the modulus (distance from origin) and $\theta = \tan^{-1}(\frac{y}{x})$ is the argument (angle with the positive real axis). 3. **Problem:** State the rectangular form of a complex number. The rectangular form is the standard form: $$z = x + iy$$ where $x$ is the real part and $y$ is the imaginary part. 4. **Problem:** Express $z = 2 - 2\sqrt{3}i$ in polar form. Step 1: Calculate modulus: $$r = \sqrt{2^2 + (-2\sqrt{3})^2} = \sqrt{4 + 4 \times 3} = \sqrt{4 + 12} = \sqrt{16} = 4$$ Step 2: Calculate argument: $$\theta = \tan^{-1}\left(\frac{-2\sqrt{3}}{2}\right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}$$ Since $x > 0$ and $y < 0$, angle is in the fourth quadrant, so $\theta = -\frac{\pi}{3}$ or equivalently $\frac{5\pi}{3}$. Step 3: Write polar form: $$z = 4 \left(\cos \left(-\frac{\pi}{3}\right) + i \sin \left(-\frac{\pi}{3}\right)\right)$$ 5. **Problem:** Express $z = 1 - i$ in polar form. Step 1: Calculate modulus: $$r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$$ Step 2: Calculate argument: $$\theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}$$ Since $x > 0$ and $y < 0$, angle is in the fourth quadrant, so $\theta = -\frac{\pi}{4}$ or equivalently $\frac{7\pi}{4}$. Step 3: Write polar form: $$z = \sqrt{2} \left(\cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right)\right)$$ 6. **Problem:** State De Moivre's theorem. De Moivre's theorem states: $$\left(\cos \theta + i \sin \theta\right)^n = \cos (n\theta) + i \sin (n\theta)$$ for any integer $n$. 7. **Problem:** Solve $(1 + i\sqrt{3})^{20}$. Step 1: Express base in polar form. Calculate modulus: $$r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$$ Calculate argument: $$\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$$ Step 2: Apply De Moivre's theorem: $$\left(1 + i\sqrt{3}\right)^{20} = \left(2 \left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\right)\right)^{20} = 2^{20} \left(\cos \left(20 \times \frac{\pi}{3}\right) + i \sin \left(20 \times \frac{\pi}{3}\right)\right)$$ Step 3: Simplify angle: $$20 \times \frac{\pi}{3} = \frac{20\pi}{3} = 6\pi + \frac{2\pi}{3}$$ Since $\cos$ and $\sin$ are $2\pi$ periodic: $$\cos \left(6\pi + \frac{2\pi}{3}\right) = \cos \frac{2\pi}{3} = -\frac{1}{2}$$ $$\sin \left(6\pi + \frac{2\pi}{3}\right) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$$ Step 4: Final answer: $$= 2^{20} \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 2^{20} \times \left(-\frac{1}{2}\right) + i 2^{20} \times \frac{\sqrt{3}}{2} = -2^{19} + i 2^{19} \sqrt{3}$$ 8. **Problem:** Solve $(\cos 7 + i \sin 7)(\cos 3 + i \sin 3) - 7$. Step 1: Use product formula for complex numbers in polar form: $$(\cos a + i \sin a)(\cos b + i \sin b) = \cos(a+b) + i \sin(a+b)$$ Step 2: Calculate: $$= \cos(7+3) + i \sin(7+3) - 7 = \cos 10 + i \sin 10 - 7$$ This is the simplified form. 9. **Problem:** Given $\omega^2 = -1$, find value of $(1 + \omega + \omega^2)^{54} (1 - \omega + \omega^2)^{54}$. Step 1: Substitute $\omega^2 = -1$: Calculate $1 + \omega + \omega^2 = 1 + \omega - 1 = \omega$ Calculate $1 - \omega + \omega^2 = 1 - \omega - 1 = -\omega$ Step 2: Expression becomes: $$(\omega)^{54} \times (-\omega)^{54} = \omega^{54} \times (-1)^{54} \times \omega^{54} = (-1)^{54} \times \omega^{108}$$ Since $(-1)^{54} = 1$ (even power), Step 3: Simplify $\omega^{108}$: Since $\omega^2 = -1$, then $\omega^4 = 1$. Divide 108 by 4: $$108 = 4 \times 27$$ So, $$\omega^{108} = (\omega^4)^{27} = 1^{27} = 1$$ Step 4: Final answer: $$1 \times 1 = 1$$ 10. **Problem:** Find cube roots of unity. Cube roots of unity satisfy: $$z^3 = 1$$ They are: $$z = 1, \quad z = \omega = \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}, \quad z = \omega^2 = \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}$$ Where $\omega$ and $\omega^2$ are complex cube roots of unity. 11. **Problem:** Write total arrangements of 2-digit even numbers from digits 1, 2, 3, 4, 5, 6, 7. Step 1: Even digits for units place: 2, 4, 6 (3 choices). Step 2: Tens place can be any digit except the units digit (since digits are distinct): 6 choices. Step 3: Total arrangements: $$6 \times 3 = 18$$ 12. **Problem:** Write total arrangements of word "ONI MIOBE" where all vowels come together. Step 1: Identify vowels: O, I, I, O, E (5 vowels) Step 2: Treat all vowels as one block plus consonants N, M, B (3 consonants) Step 3: Number of letters to arrange: 1 vowel block + 3 consonants = 4 items Number of ways to arrange these 4 items: $$4! = 24$$ Step 4: Number of ways to arrange vowels inside the block: Vowels: O, I, I, O, E with repetitions of O and I twice each. Number of arrangements: $$\frac{5!}{2! \times 2!} = \frac{120}{4} = 30$$ Step 5: Total arrangements: $$24 \times 30 = 720$$