Comprehensive assessment covering De Moivre's theorem, roots of unity, the nth roots of complex numbers, polynomial equations with complex roots, the conjugate root theorem, and trigonometric identities derived using complex numbers.
De Moivre's theorem states that $\big[r(\cos\theta + i\sin\theta)\big]^n$ equals:
Using De Moivre's theorem, $(\cos\theta + i\sin\theta)^5$ equals:
The value of $\left[2\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)\right]^3$ is:
A non-zero complex number has exactly how many distinct nth roots?
The three cube roots of unity are:
The sum of the three cube roots of unity is:
If $\omega$ is a complex cube root of unity, then $\omega^3$ equals:
On the Argand plane, the nth roots of unity lie at:
By the conjugate root theorem, if a polynomial with real coefficients has $a + bi$ as a root, it must also have the root:
The solutions of $x^2 + 4 = 0$ are:
A monic quadratic with real coefficients has $3 - 2i$ as one root. The quadratic is:
Expanding $(\cos\theta + i\sin\theta)^3$ and equating real parts gives $\cos 3\theta$ equal to:
If a complex number $z$ has modulus $r$, then each of its nth roots has modulus:
One of the square roots of $i$ is:
If $z = \cos\theta + i\sin\theta$, then $z + \dfrac{1}{z}$ equals:
1. Use De Moivre's theorem to evaluate $\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)^{6}$. (2 marks)
2. Express $(1 + i)^8$ in the form $a + bi$. (2 marks)
3. Find the three cube roots of unity in the form $a + bi$. (2 marks)
4. Solve $z^3 = 8$, giving all three roots. (3 marks)
5. Given that $2 - i$ is a root of $z^2 + bz + c = 0$ where $b$ and $c$ are real, find $b$ and $c$. (2 marks)
6. Use De Moivre's theorem to show that $\cos 2\theta = \cos^2\theta - \sin^2\theta$. (2 marks)
7. Find the square roots of $-5 + 12i$ in the form $a + bi$. (3 marks)
8. If $\omega$ is a non-real cube root of unity, simplify $(1 + \omega)(1 + \omega^2)$. (2 marks)
9. State the five fifth-roots of unity in mod-arg form. (2 marks)
10. Given that $z = 1 + i$, find $z^{10}$ in the form $a + bi$. (2 marks)