Sketching polar curvesAQA A-Level Further Maths: Subtopic test

1. 1
   A curve $C$ has polar equation $r=4(1+\cos\theta)$ for $0\le\theta<2\pi$.

   (a)

   Find the greatest value of $r$ on $C$.

   [1 mark]

   • A$4$
   • B$0$
   • C$8$
   • D$12$

   (b)

   Find the value of $\theta$, in the given range, at which $C$ passes through the pole.

   [1 mark]

   • A$0$
   • B$\pi$
   • C$\dfrac{\pi}{2}$
   • D$\dfrac{3\pi}{2}$

   (c)

   Explain why $C$ is symmetrical about the initial line, and find the value of $r$ when $\theta=\frac{\pi}{2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has polar equation $r=3\cos2\theta$ for $0\le\theta<2\pi$.

   (a)

   How many petals does the sketch of $C$ have?

   [1 mark]

   • A$2$
   • B$3$
   • C$8$
   • D$4$

   (b)

   Find the smallest positive value of $\theta$ at which $C$ passes through the pole.

   [1 mark]

   • A$\dfrac{\pi}{4}$
   • B$\dfrac{\pi}{2}$
   • C$\dfrac{\pi}{8}$
   • D$\pi$

   (c)

   When $\theta=\frac{\pi}{2}$, $r$ is negative. Find the Cartesian coordinates of the point on $C$ with $\theta=\frac{\pi}{2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has polar equation $r=2+3\cos\theta$ for $0\le\theta<2\pi$.

   (a)

   Find the values of $\theta$ at which $C$ passes through the pole.

   [3 marks]

   (b)

   Explain why the sketch of $C$ contains an inner loop. State the greatest value of $r$, the value of $r$ when $\theta=\pi$, and a line of symmetry.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C_1$ has polar equation $r^2=16\cos2\theta$ and a circle $C_2$ has polar equation $r=2\sqrt2$. In this question take $r\ge0$ and $-\pi<\theta\le\pi$.

   (a)

   Describe the main features of the sketch of $C_1$, justifying the values of $\theta$ for which the curve exists.

   [6 marks]

   (b)

   Find the polar coordinates of the points where $C_1$ and $C_2$ intersect.

   [6 marks]

   Total for question 4: 12 marks

End of questions