Complex Numbers — Quiz 3

Questions in this quiz (7)

1. Assuming $$i^2 = -1$$, simplify $$\left(\sqrt{3} + i\sqrt{7}\right)^2$$.

   • $$4 + 2i \sqrt{21}$$
   • $$-4 + 2i \sqrt{21}$$
   • $$4 - 2i \sqrt{21}$$
   • $$-4 + 2i \sqrt{7}$$

2. Assuming $$i^2 = -1$$, which of these complex numbers is $$(3 - 4i)^2$$ in simplified form?

   • $$\frac {1}{7 + 24i}$$
   • $$\frac {1}{-7 - 24i}$$
   • $$\frac {-7 + 24i}{625}$$
   • $$\frac {7 + 24i}{625}$$

3. Where $$i^2 = -1$$, which of these numbers equals $$\frac{i}{1 + \sqrt{2}i}$$?

   • $$\frac{i + \sqrt{2}}{3}$$
   • $$\frac{-i + \sqrt{2}}{3}$$
   • $$\frac{2i + \sqrt{2}}{3}$$
   • $$\frac{-i - \sqrt{2}}{3}$$

4. Where $$i^2 = -1$$, which of these complex numbers equals $$\frac{1}{c + di}$$?

   • $$\frac{c + di}{c + d}$$
   • $$\frac{c - di}{c + d}$$
   • $$\frac{c + di}{c^2 + d^2}$$
   • $$\frac{c - di}{c^2 + d^2}$$

5. If $$x \cdot (3 + 2i)^2 = 1$$ and $$i^2 = -1$$, what is $$x$$?

   • $$\frac{5 - 12i}{169}$$
   • $$\frac{5 - 12i}{5 + 12i}$$
   • $$\frac{5 - 12i}{169}$$
   • $$\frac{1}{5 - 12i}$$

6. Assuming that $$y$$ is real and $$i^2 = -1$$, which of these complex numbers equals $$\frac{1}{y^2 - i}$$?

   • $$\frac{y^2 - i}{y^4 - 1}$$
   • $$\frac{y^4 + i}{y^2 - 1}$$
   • $$\frac{y^2 + i}{y^4 + 1}$$
   • $$\frac{y + i}{y^2 + 1}$$

7. What are the complex solutions of $$x^2 + 8x + 13 = 0$$ over complex numbers?

   • $$4 \pm \sqrt{3}$$
   • $$-8 \pm 2\sqrt{3}$$
   • $$-4 \pm \sqrt{3}$$
   • $$8 \pm 2\sqrt{3}$$