Bluebook SAT Test Complex Numbers 3

Question 1

Which of the following is a solution to x² + 4 = 0?

Accepted Answer

x = 2i

Answer

x² = −4 gives x = ±√(−4) = ±2i, and 2i is one of the solutions.

Question 2

If (a + bi)(3 − 2i) = 12 + 5i, what is the value of a + b?

Accepted Answer

5

Answer

Expanding gives (3a + 2b) + (−2a + 3b)i = 12 + 5i; solving 3a + 2b = 12 and −2a + 3b = 5 yields a = 2, b = 3, so a + b = 5.

Question 3

What is the modulus (absolute value) of the complex number 3 + 4i?

Accepted Answer

5

Answer

|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Question 4

Which quadratic equation has solutions x = 2 + i and x = 2 − i?

Accepted Answer

x² − 4x + 5 = 0

Answer

(x − (2 + i))(x − (2 − i)) = (x − 2)² + 1 = x² − 4x + 4 + 1 = x² − 4x + 5 = 0.

Question 5

What is i^100?

Accepted Answer

1

Answer

100 is divisible by 4 (100 = 4 × 25), so i^100 = (i⁴)^25 = 1^25 = 1.

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