Complex Numbers: practice solutions, Grade 11 – download the PDF

1 Square roots of negatives ★★★

1. \(\sqrt{-36} = i\sqrt{36} = 6i\).
2. \(\sqrt{-50} = i\sqrt{25\cdot 2} = 5i\sqrt{2}\).
3. \(-\sqrt{-81} = -9i\).
4. \(\sqrt{-12} = i\sqrt{4\cdot 3} = 2i\sqrt{3}\).

2 Powers of i ★★★

Divide each exponent by \(4\) and keep the remainder.

1. \(15 = 4\cdot 3 + 3\), so \(i^{15} = i^3 = -i\).
2. \(22 = 4\cdot 5 + 2\), so \(i^{22} = i^2 = -1\).
3. \(40 = 4\cdot 10 + 0\), so \(i^{40} = 1\).
4. \(101 = 4\cdot 25 + 1\), so \(i^{101} = i\).

3 Add and subtract ★★★

1. Real parts: \(5 + 3 = 8\). Imaginary parts: \(2 - 7 = -5\). Answer: \(8 - 5i\).
2. \(4 - 9 = -5\) and \(-1 - 3 = -4\). Answer: \(-5 - 4i\).
3. \(-2 - (-2) = 0\) and \(6 - (-1) = 7\). Answer: \(7i\).

4 Multiply and distribute ★★★

1. \(3i(4 - 5i) = 12i - 15i^2 = 12i + 15 = 15 + 12i\).
2. \((1 + 3i)(2 - i) = 2 - i + 6i - 3i^2 = 2 + 5i + 3 = 5 + 5i\).

5 Conjugates ★★★

1. The conjugate of \(7 - 3i\) is \(7 + 3i\).
2. \(-4i = 0 - 4i\), so its conjugate is \(4i\).
3. \(9 = 9 + 0i\), so its conjugate is \(9\) itself.

Product: \((7 - 3i)(7 + 3i) = 7^2 + 3^2 = 49 + 9 = 58\).

6 Reading the complex plane ★★★

A point \((a, b)\) represents \(a + bi\).

• \(A(2, 3)\): \(2 + 3i\), real part \(2\), imaginary part \(3\).
• \(B(-3, 1)\): \(-3 + i\), real part \(-3\), imaginary part \(1\).
• \(C(-2, -2)\): \(-2 - 2i\), real part \(-2\), imaginary part \(-2\).
• \(D(4, -1)\): \(4 - i\), real part \(4\), imaginary part \(-1\).

7 Absolute values ★★★

1. \(\sqrt{9 + 16} = \sqrt{25} = 5\).
2. \(\sqrt{25 + 144} = \sqrt{169} = 13\).
3. \(|-8i| = \sqrt{0 + 64} = 8\).
4. \(\sqrt{36 + 64} = \sqrt{100} = 10\).

8 Matching parts ★★★

Two complex numbers are equal when their real parts match and their imaginary parts match.

Real parts: \(x + 2 = 7\), so \(x = 5\).

Imaginary parts: \(3y - 1 = 5\), so \(3y = 6\) and \(y = 2\).

9 Products and squares ★★★

1. \(12 + 15i - 8i - 10i^2 = 12 + 7i + 10 = 22 + 7i\).
2. \((1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i\).
3. \((2 - 3i)^2 = 4 - 12i + 9i^2 = 4 - 12i - 9 = -5 - 12i\).

10 Dividing complex numbers ★★★

1. Multiply by the conjugate \(2 + i\): \(\dfrac{(8 + i)(2 + i)}{(2 - i)(2 + i)} = \dfrac{16 + 8i + 2i + i^2}{4 + 1} = \dfrac{15 + 10i}{5} = 3 + 2i\).
2. Multiply by \(1 - i\): \(\dfrac{6(1 - i)}{(1 + i)(1 - i)} = \dfrac{6 - 6i}{2} = 3 - 3i\).

11 Pure imaginary solutions ★★★

1. \(x^2 = -49\), so \(x = \pm 7i\).
2. \(x^2 = -20\), so \(x = \pm i\sqrt{20} = \pm 2i\sqrt{5}\).
3. \(2x^2 = -18\), so \(x^2 = -9\) and \(x = \pm 3i\).

12 Quadratic formula ★★★

The discriminant is \((-6)^2 - 4(1)(13) = 36 - 52 = -16\).

\(x = \dfrac{6 \pm \sqrt{-16}}{2} = \dfrac{6 \pm 4i}{2} = 3 \pm 2i\).

Check: the sum of the roots is \(6\) and their product is \(3^2 + 2^2 = 13\), as expected.

13 Circuit impedance ★★★

1. \(Z = (3 + 4i) + (2 - i) = 5 + 3i\) ohms.
2. \(|Z| = \sqrt{25 + 9} = \sqrt{34} \approx 5.83\) ohms.

14 True or false? ★★★

1. False. The product rule for radicals does not apply to negative numbers. Correctly, \(\sqrt{-4}\cdot\sqrt{-9} = 2i\cdot 3i = 6i^2 = -6\).
2. True for these numbers. \(z + w = 3 - i\), so \(\overline{z + w} = 3 + i\). Also \(\overline{z} + \overline{w} = (2 - 3i) + (1 + 4i) = 3 + i\). The two results agree.

15 Distance in the plane ★★★

The distance is the absolute value of the difference: \((2 + i) - (-1 + 5i) = 3 - 4i\).

\(|3 - 4i| = \sqrt{9 + 16} = 5\). They are 5 yards apart (about \(4.6\) meters).

16 Complex roots with coefficients ★★★

Discriminant: \(16 - 36 = -20\).

\(x = \dfrac{-4 \pm \sqrt{-20}}{2} = \dfrac{-4 \pm 2i\sqrt{5}}{2} = -2 \pm i\sqrt{5}\).

Check: the sum is \(-4 = -\dfrac{b}{a}\) and the product is \((-2)^2 + (\sqrt{5})^2 = 4 + 5 = 9 = \dfrac{c}{a}\).

17 Build the equation ★★★

The sum of the roots is \(4\) and their product is \((2 + 3i)(2 - 3i) = 4 + 9 = 13\).

An equation with sum \(S\) and product \(P\) is \(x^2 - Sx + P = 0\), so \(x^2 - 4x + 13 = 0\).

Check with the formula: the discriminant is \(16 - 52 = -36\), and \(x = \dfrac{4 \pm 6i}{2} = 2 \pm 3i\).

18 Higher powers ★★★

1. \((2 + i)^2 = 4 + 4i + i^2 = 3 + 4i\). Then \((3 + 4i)(2 + i) = 6 + 3i + 8i + 4i^2 = 2 + 11i\).
2. \((1 - i)^2 = 1 - 2i + i^2 = -2i\). Then \((1 - i)^4 = (-2i)^2 = 4i^2 = -4\).

19 A square root of a complex number ★★★

\((2 + 3i)^2 = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i\). So \(2 + 3i\) is a square root of \(-5 + 12i\).

Since \((-z)^2 = z^2\), the other square root is \(-(2 + 3i) = -2 - 3i\).

20 Reciprocal with the conjugate ★★★

1. \(\overline{z} = 8 + 15i\) and \(z\overline{z} = 8^2 + 15^2 = 64 + 225 = 289\). Since \(z\overline{z} = |z|^2\), \(|z| = \sqrt{289} = 17\).
2. \(\dfrac{1}{z} = \dfrac{\overline{z}}{z\overline{z}} = \dfrac{8 + 15i}{289} = \dfrac{8}{289} + \dfrac{15}{289}i\).

21 Quarter turns ★★★

One move: \(i(4 + i) = 4i + i^2 = -1 + 4i\).

Two moves: \(i(-1 + 4i) = -i + 4i^2 = -4 - i\).

Three moves: \(i(-4 - i) = -4i - i^2 = 1 - 4i\).

Four moves: \(i(1 - 4i) = i - 4i^2 = 4 + i\). The drone is back at its start, because \(i^4 = 1\).