Further Maths polar coordinates test to download as PDF

This Further Maths polar coordinates test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 16-18, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Polar and Cartesian
1. Convert the point \((-2\sqrt3,\,2)\) to polar coordinates with \(-\pi\lt \theta\le\pi\). [2]
2. Find a Cartesian equation of the curve \(r=6\sin\theta\) and describe the curve fully. [2]
Question 2 – A limaçon
The curve \(C\) has equation \(r=3+2\sin\theta\), \(0\le\theta<2\pi\).
1. State the greatest and least values of \(r\), and the values of \(\theta\) at which they occur. [2]
2. Find the exact area enclosed by \(C\). [2]
Question 3 – One petal
The curve \(r=4\sin2\theta\), \(0\le\theta\le\tfrac\pi2\), forms one petal of a rose.
1. Find the polar coordinates of the tip of the petal. [1]
2. Show that the area of the petal is \(2\pi\). [3]
Question 4 – Tangents parallel to the initial line
The cardioid \(C\) has equation \(r=4(1+\cos\theta)\), \(-\pi\lt \theta\le\pi\).
1. Find the values of \(\theta\), other than \(\pi\), where the tangent to \(C\) is parallel to the initial line. [3]
2. Give the Cartesian coordinates of the point with \(\theta>0\). [1]
Question 5 – Problem: a lawn around a pond
A circular lawn has boundary \(r=4\) (in metres) about a sprinkler at the pole. A pond has the cardioid shape \(r=4(1-\cos\theta)\). The region \(G\) inside the circle and outside the pond is grass.
1. Find the values of \(\theta\) at which the boundaries meet. [1]
2. Show that the area of grass in the region \(-\tfrac\pi2\le\theta\le\tfrac\pi2\) is \(32-4\pi\) m². [3]

How to use this Further Maths polar coordinates test

First, revise with the polar coordinates lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the polar coordinates test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Further Maths polar coordinates test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.