AP Precalculus Unit 3 Part C Passwater's MCQ Exam Review

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates (-3, 3). Which of the following is one way to express the complex number using its polar coordinates (r, θ)?

• (A) (3√2 cos(π/4) + i(3√2 sin(π/4)))

• (B) (3cos(π/4) + i(3sin(π/4)))

• (C) (3√2 cos(3π/4) + i(3√2 sin(3π/4))

• (D) (3cos(3π/4) + i(3sin(3π/4))

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates (1/2, -√3/2). Which of the following is one way to express the complex number using its polar coordinates (r, θ)?

• (A) (cos(-π/6) + i(sin(-π/6))

• (B) (cos(π/6) + i(sin(π/6))

• (C) (cos(5π/3) + i(sin(5π/3))

• (D) (2cos(5π/3) + i(2sin(5π/3))

A complex number is represented by a point in the complex plane. In polar coordinates, the complex number can be expressed as (3 cos(5π/4) + i(3 sin(5π/4)). Express the complex number using its rectangular coordinates (x, y).

• (A) (-3√2/2, -3√2/2)

• (B) (-3√2/2, 3√2/2)

• (C) (-3√2, -3√2)

• (D) (3√2, 3√2)

A complex number is represented by a point in the complex plane. In polar coordinates, the complex number can be expressed as (4 cos(-2π/3)) + i(4 sin(-2π/3)). Express the complex number using its rectangular coordinates (x, y).

• (-2√3, -2)

• (-2, -2√3)

• (-2, 2√3)

• (2, -2√3)

The figure shows the graph of the polar function r = f(θ), for 0 ≤ θ ≤ 2π, in the polar coordinate system. Which of the following could be an expression for f(θ)?

• 1.5 + 1.5 sin θ

• 1.5 - 1.5 sin θ

• 1.5 + 1.5 cos θ

• 1.5 - 1.5 cos θ

The figure shows the graph of the polar function r = g(θ), for 0 ≤ θ ≤ 2π, in the polar coordinate system. Which of the following could be an expression for g(θ)?

• 1 - 2 sin θ

• 1 + 2 sin θ

• 1 - 2 cos θ

• 1 + 2 cos θ

The figure shows the graph of the polar function r = f(θ), where f(θ) = 1 + 2 cos(2θ), in the polar coordinate system for 0 ≤ θ ≤ 2π. There are four points labeled A, B, C, and D. If the domain of f is restricted to 0 ≤ θ ≤ π/2, the portion of the given graph that remains consists of two pieces. One of those pieces is the portion of the graph in Quadrant I from A to B. Which of the following describes the other remaining piece?

• The portion of the graph in Quadrant I from B to D

• The portion of the graph in Quadrant II from D to B

• The portion of the graph in Quadrant III from B to E

• The portion of the graph in Quadrant IV from B to A

The figure shows the graph of the polar function r = g(θ), where g(θ) = 3sin(3θ), in the polar coordinate system for 0 ≤ θ ≤ 2π. There are four points labeled A, B, C, and D. If the domain of g is restricted to π/2 ≤ θ ≤ 2π/3, which of the following describes the portion of the given graph that remains?

• (A) The top portion of the graph in Quadrant II from D to B

• (B) The bottom portion of the graph in Quadrant II from B to D

• (C) The portion of the graph in Quadrant III from D to C

• (D) The portion of the graph in Quadrant IV from C to D

A portion of the graph of the polar function r = f(θ), where f(θ) = 1.5cos(3θ), is shown in the polar coordinate system for a ≤ θ ≤ b. If 0 ≤ a < b ≤ 2π, which of the following could be the values for a and b?

• (A) a = 7π/6 and b = 4π/3

• (B) a = 5π/6 and b = π

• (C) a = 11π/6 and b = 2π

• (D) a = 3π/2 and b = 2π

Which of the following is the graph of the polar function r = f(θ), where f(θ) = 1.5 + 1.5 cos(θ), in the polar coordinate system for 0 ≤ θ ≤ 2π?

• A

• B

• C

• D