Polar Coordinates Practice Questions
A-Levels · A-Level Further Mathematics · 150 free MCQs with instant results and detailed explanations.
150
Total
49
Easy
77
Medium
24
Hard
Start Practicing Polar Coordinates
Take a timed quiz or customize your practice session
Sample Questions from Polar Coordinates
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the polar coordinate representation of the point (3, 60ยฐ)?
A.
(3, 60ยฐ)
B.
(1.5, 30ยฐ)
C.
(3, -60ยฐ)
D.
(3, 120ยฐ)
Show Answer & Explanation
Correct Answer: A
The polar coordinate (3, 60ยฐ) directly indicates a radius of 3 units and an angle of 60 degrees from the positive x-axis. It is already in the correct form.
Q2
Easy
Convert the polar coordinates (4, 135ยฐ) into Cartesian coordinates. What is the x-coordinate?
A.
-2โ2
B.
2โ2
C.
4
D.
-4
Show Answer & Explanation
Correct Answer: A
To convert from polar to Cartesian, use x = r * cos(ฮธ). Here, x = 4 * cos(135ยฐ) = 4 * (-โ2/2) = -2โ2.
Q3
Easy
If a point in polar coordinates is given as (r, ฮธ) = (5, 210ยฐ), which quadrant does this point lie in?
A.
First Quadrant
B.
Second Quadrant
C.
Third Quadrant
D.
Fourth Quadrant
Show Answer & Explanation
Correct Answer: C
An angle of 210ยฐ is in the third quadrant, where both x and y coordinates are negative, matching the given range of ฮธ.
Q4
Medium
What is the polar coordinate representation of the point (3, 60ยฐ)?
A.
(3, 60ยฐ)
B.
(1.5, 30ยฐ)
C.
(3, ฯ/3)
D.
(3, 120ยฐ)
Show Answer & Explanation
Correct Answer: A
The point (3, 60ยฐ) is already in polar coordinates, representing a distance of 3 from the origin at an angle of 60 degrees.
Q5
Medium
Convert the Cartesian coordinates (2, 2โ3) to polar coordinates.
A.
(4, ฯ/3)
B.
(4, 45ยฐ)
C.
(4, 60ยฐ)
D.
(2โ3, 2)
Show Answer & Explanation
Correct Answer: C
Using the formulas r = โ(xยฒ + yยฒ) and ฮธ = tanโปยน(y/x), we find r = โ(2ยฒ + (2โ3)ยฒ) = 4 and ฮธ = tanโปยน(โ3) = 60ยฐ.
Q6
Medium
Which of the following equations represents the curve defined in polar coordinates as r = 2 + 3sin(ฮธ)?
A.
xยฒ + yยฒ = 2 + 3y
B.
xยฒ + yยฒ = 2 + 3x
C.
xยฒ + yยฒ = 3 + 2y
D.
y = 2 + 3x
Show Answer & Explanation
Correct Answer: A
To convert from polar to Cartesian, use rยฒ = xยฒ + yยฒ and r = โ(xยฒ + yยฒ). Substitute r = 2 + 3sin(ฮธ) to get the equation in Cartesian form.
Q7
Medium
If the polar coordinates of a point are (5, 150ยฐ), what are the Cartesian coordinates?
A.
(โ5โ3/2, โ5/2)
B.
(5โ3/2, 5/2)
C.
(5, โ5โ3/3)
D.
(5/2, 5โ3/2)
Show Answer & Explanation
Correct Answer: A
The Cartesian coordinates (x, y) can be found with x = r cos(ฮธ) and y = r sin(ฮธ). Here, x = 5 cos(150ยฐ) = -5โ3/2 and y = 5 sin(150ยฐ) = -5/2.
Q8
Hard
Given the polar equation r = 4 - 2sin(ฮธ), identify the type of conic section it represents.
A.
Ellipse
B.
Hyperbola
C.
Circle
D.
Parabola
Show Answer & Explanation
Correct Answer: A
The equation r = 4 - 2sin(ฮธ) is in the form of the polar equation for a conic section. Since the coefficient of sin(ฮธ) is negative, it represents an ellipse with eccentricity less than 1.
Q9
Hard
Convert the polar coordinates (5, ฯ/3) to Cartesian coordinates and determine the resulting x-coordinate.
A.
2.5
B.
5/2
C.
2.5โ3
D.
5โ3/2
Show Answer & Explanation
Correct Answer: C
To convert polar coordinates (r, ฮธ) to Cartesian coordinates (x, y), use x = r * cos(ฮธ). Here, x = 5 * cos(ฯ/3) = 5 * 1/2 = 2.5, which simplifies to 2.5โ3 when expressed as a function of โ3.
Q10
Hard
A point P moves in a plane according to the polar coordinates given by r(ฮธ) = ฮธ^2 for 0 โค ฮธ โค 2ฯ. What is the Cartesian coordinate of the point P at ฮธ = ฯ?
A.
(0, -ฯ^2)
B.
(ฯ^2, 0)
C.
(-ฯ^2, 0)
D.
(0, ฯ^2)
Show Answer & Explanation
Correct Answer: B
To find the Cartesian coordinates, we convert from polar to Cartesian using x = r(ฮธ)cos(ฮธ) and y = r(ฮธ)sin(ฮธ). At ฮธ = ฯ, r(ฯ) = ฯ^2, so x = ฯ^2cos(ฯ) = -ฯ^2 and y = ฯ^2sin(ฯ) = 0, leading to the coordinates (ฯ^2, 0).
Showing 10 of 150 questions. Start a quiz to practice all questions with scoring and timer.
Practice All 150 Questions →More A-Level Further Mathematics Chapters
Polar Coordinates โ A-Levels A-Level Further Mathematics Practice Questions Online
This page contains 150 practice MCQs for the chapter Polar Coordinates in A-Levels A-Level Further Mathematics. The questions are organized by difficulty โ 49 easy, 77 medium, 24 hard โ so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.