Complex Numbers: math test solutions, Grade 11 – download the PDF

1 Radicals and powers of i / 4 pts

1. \(\sqrt{-98} = i\sqrt{49\cdot 2} = 7i\sqrt{2}\). (1 pt)
2. \(58 = 4\cdot 14 + 2\), so \(i^{58} = -1\). (1 pt)
3. \(83 = 4\cdot 20 + 3\), so \(i^{83} = -i\). (1 pt)
4. \(i^7\cdot i^{12} = i^{19}\), and \(19 = 4\cdot 4 + 3\), so the result is \(-i\). (1 pt)

2 Operations / 4 pts

1. \(5 + 5i\). (1 pt)
2. \((6 - 4) + (1 + 5)i = 2 + 6i\). (1 pt)
3. \(6 + 15i - 8i - 20i^2 = 6 + 7i + 20 = 26 + 7i\). (1 pt)
4. \(16 - 24i + 9i^2 = 16 - 24i - 9 = 7 - 24i\). (1 pt)

3 Quotients / 3 pts

1. Multiply by \(3 + i\): \(\dfrac{10(3 + i)}{9 + 1} = \dfrac{30 + 10i}{10} = 3 + i\). (1 pt)
2. Multiply by \(1 - 2i\): numerator \(11 - 22i + 2i - 4i^2 = 15 - 20i\) (1 pt); denominator \(1 + 4 = 5\); result \(3 - 4i\). (1 pt)

4 Quadratic equations / 4 pts

1. \(x^2 = -75\), so \(x = \pm i\sqrt{25\cdot 3} = \pm 5i\sqrt{3}\). (1 pt)
2. Discriminant: \(64 - 100 = -36\) (1 pt). \(x = \dfrac{8 \pm 6i}{2} = 4 \pm 3i\). (1 pt)
3. Sum: \((4 + 3i) + (4 - 3i) = 8 = -\dfrac{b}{a}\); product: \(16 + 9 = 25 = \dfrac{c}{a}\). (1 pt)

5 Complex plane / 3 pts

1. \(|z_1| = \sqrt{9 + 16} = 5\). (1 pt)
2. \(\overline{z_2} = 2 + i\); it is the mirror image of \(z_2\) across the real axis. (1 pt)
3. \(z_1 - z_2 = -5 + 5i\), so the distance is \(\sqrt{25 + 25} = 5\sqrt{2} \approx 7.07\). (1 pt)

6 Find the number / 2 pts

Divide: \(a + bi = \dfrac{7 - 4i}{1 - 2i} = \dfrac{(7 - 4i)(1 + 2i)}{1 + 4}\). (1 pt)

Numerator: \(7 + 14i - 4i - 8i^2 = 15 + 10i\), so \(a + bi = 3 + 2i\). (1 pt)