Complex Numbers: math test, Grade 11

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Math tests Grade 11 : Complex Numbers — Zyro the alien explorer of Planète Maths

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

Sommaire

1 Radicals and powers of i / 4 pts

No calculator. Simplify. (a) \(\sqrt{-98}\) (b) \(i^{58}\) (c) \(i^{83}\) (d) \(i^7\cdot i^{12}\)

2 Operations / 4 pts

Write each result in the form \(a + bi\). (a) \((7 - 3i) + (-2 + 8i)\) (b) \((6 + i) - (4 - 5i)\) (c) \((3 - 4i)(2 + 5i)\) (d) \((4 - 3i)^2\)

3 Quotients / 3 pts

Write in the form \(a + bi\). (a) \(\dfrac{10}{3 - i}\) (b) \(\dfrac{11 + 2i}{1 + 2i}\)

4 Quadratic equations / 4 pts

Solve over the complex numbers. (a) \(x^2 + 75 = 0\) (b) \(x^2 - 8x + 25 = 0\) (c) Check your answers to (b) with the sum and the product of the roots.

5 Complex plane / 3 pts

Let \(z_1 = -3 + 4i\) and \(z_2 = 2 - i\) be two points of the complex plane. (a) Find \(|z_1|\). (b) Give the conjugate of \(z_2\) and describe where it is located compared with \(z_2\). (c) Find the distance between \(z_1\) and \(z_2\).