Complex Numbers Test Answer Key with Scoring – Algebra 2

Question 1 Place the numbers in the regions (2 points)

a) 36/4 = 9: A. b) −√100 = −10: B. c) 0.125 = 1/8: C. d) √40 = 2√10: D. e) √(−16) = 4i: E.

Scoring: 1 point for 3 or 4 correct placements, 2 points for all five.

Question 2 Simplify negative square roots (2 points)

a) √150 = 5√6, so √(−150) = 5i√6.

b) √48 = 4√3, so the sum is −3 + 4i√3.

Scoring: 1 point each; half credit for a correct radical with a missing i.

Question 3 Add and subtract (2 points)

a) 5 + 7i.

b) −6 + 9i − 4 + i = −10 + 10i.

Scoring: 1 point each.

Question 4 Multiply complex numbers (3 points)

a) 20 + 8i − 15i − 6i² = 26 − 7i.

b) 36 − 12i + i² = 35 − 12i.

c) 49 − 4i² = 53.

Scoring: 1 point each.

Question 5 Powers of i (2 points)

a) 47 = 4 · 11 + 3, so i⁴⁷ = −i.

b) 82 = 4 · 20 + 2, so i⁸² = −1.

Scoring: 1 point each.

Question 6 Two points on the complex plane (2 points)

a) P = 5 − 3i, Q = −2 + 4i.

b) (5 − 2) + (−3 + 4)i = 3 + i.

Scoring: 1 point each part.

Question 7 Solve for a and b (3 points)

Match parts: a + 2b = 11 and 3a − b = 5.

From the second, b = 3a − 5. Substitute: a + 6a − 10 = 11, so a = 3 and b = 4.

Check: 3 + 8 = 11 and 9 − 4 = 5.

a = 3, b = 4

Scoring: 1 point for the two equations, 1 for solving, 1 for correct values.

Question 8 Verify a solution (2 points)

(2 − 3i)² = 4 − 12i − 9 = −5 − 12i.

−4(2 − 3i) = −8 + 12i.

Total: −5 − 12i − 8 + 12i + 13 = 0.

Scoring: 1 point for the square, 1 point for the total of 0.

Question 9 Circuit calculations (2 points)

a) (9 + 5) + (4 − 7)i = 14 − 3i ohms.

b) (1 + 2i)(3 − 4i) = 3 − 4i + 6i − 8i² = 11 + 2i volts.

Scoring: 1 point each.