Bluebook SAT Test Complex Numbers 2

Question 1

What is (3 + 2i) − (1 − 4i)?

Accepted Answer

2 + 6i

Answer

Subtract real parts (3 − 1 = 2) and imaginary parts (2 − (−4) = 6) to get 2 + 6i.

Question 2

Which of the following equals i^17?

Accepted Answer

i

Answer

Since i cycles with period 4, 17 = 4 × 4 + 1, so i^17 = i^1 = i.

Question 3

What is (1 + i) ÷ (1 − i)?

Accepted Answer

i

Answer

Multiplying numerator and denominator by (1 + i): (1 + i)² / ((1)² + (1)²) = (1 + 2i − 1)/2 = 2i/2 = i.

Question 4

If z = 3 − 4i, what is the product z × z̄ (where z̄ is the complex conjugate of z)?

Accepted Answer

25

Answer

(3 − 4i)(3 + 4i) = 9 − (4i)² = 9 − (−16) = 25, which is always a real number.

Question 5

What is the real part of the product (5 + 3i)(2 − i)?

Accepted Answer

13

Answer

(5 + 3i)(2 − i) = 10 − 5i + 6i − 3i² = 10 + i + 3 = 13 + i, so the real part is 13.

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