Polar coordinatesAQA A-Level Further Maths: Topic test

1. 1
   The point $A$ has polar coordinates $\left(6,\dfrac{2\pi}{3}\right)$.

   (a)

   What are the Cartesian coordinates of $A$?

   [1 mark]

   • A$\left(3,\,3\sqrt3\right)$
   • B$\left(-3,\,3\sqrt3\right)$
   • C$\left(-3\sqrt3,\,3\right)$
   • D$\left(3\sqrt3,\,-3\right)$

   (b)

   A point $B$ has Cartesian coordinates $(-4,4)$. Which are the polar coordinates of $B$, with $r>0$ and $-\pi<\theta\le\pi$?

   [1 mark]

   • A$\left(4\sqrt2,\,\frac{\pi}{4}\right)$
   • B$\left(4\sqrt2,\,-\frac{\pi}{4}\right)$
   • C$\left(8,\,\frac{3\pi}{4}\right)$
   • D$\left(4\sqrt2,\,\frac{3\pi}{4}\right)$

   (c)

   Show that $A$ lies on the curve with polar equation $r=4(1-\cos\theta)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has polar equation $r=2\sin3\theta$ for $0\le\theta\le\dfrac{\pi}{3}$.

   (a)

   What is the value of $r$ when $\theta=\frac{\pi}{18}$?

   [1 mark]

   • A$1$
   • B$\frac12$
   • C$\sqrt3$
   • D$2$

   (b)

   At which value of $\theta$ does $r$ take its greatest value?

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{\pi}{9}$
   • C$\frac{\pi}{6}$
   • D$\frac{\pi}{2}$

   (c)

   Find the area enclosed by $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has polar equation $r=\dfrac{6}{2+\cos\theta}$ for $0\le\theta<2\pi$.

   (a)

   Show that a Cartesian equation of $C$ is $3x^2+4y^2+12x-36=0$.

   [3 marks]

   (b)

   Find the greatest and least values of $r$, and state, with a reason, a line of symmetry of $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The circles $C_1$ and $C_2$ have polar equations $r=2\cos\theta$ and $r=2\sin\theta$ respectively, for $0\le\theta\le\dfrac{\pi}{2}$.

   (a)

   Find a Cartesian equation of each circle, stating its centre and radius, and find the polar coordinates of the point, other than the pole, where $C_1$ and $C_2$ meet.

   [6 marks]

   (b)

   Find the area of the region that lies inside both $C_1$ and $C_2$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The point $Q$ has Cartesian coordinates $(3,-3)$.

   (a)

   Which are the polar coordinates of $Q$, with $r>0$ and $-\pi<\theta\le\pi$?

   [1 mark]

   • A$\left(3\sqrt2,\,\frac{\pi}{4}\right)$
   • B$\left(6,\,-\frac{\pi}{4}\right)$
   • C$\left(3\sqrt2,\,-\frac{\pi}{4}\right)$
   • D$\left(3\sqrt2,\,\frac{3\pi}{4}\right)$

   (b)

   The point $Q$ lies on the circle with Cartesian equation $x^2+y^2=-6y$. Which is a polar equation of this circle?

   [1 mark]

   • A$r=-6\sin\theta$
   • B$r=6\sin\theta$
   • C$r=-6\cos\theta$
   • D$r=6\cos\theta$

   (c)

   Write down the polar equation and the Cartesian equation of the circle with centre at the pole that passes through $Q$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve $C$ has polar equation $r=2+\sin\theta$ for $0\le\theta<2\pi$.

   (a)

   What is the value of $r$ when $\theta=\frac{7\pi}{6}$?

   [1 mark]

   • A$2.5$
   • B$2+\frac{\sqrt3}{2}$
   • C$1$
   • D$1.5$

   (b)

   Which line is a line of symmetry of $C$?

   [1 mark]

   • A$\theta=0$
   • B$\theta=\frac{\pi}{2}$
   • C$\theta=\frac{\pi}{4}$
   • D$\theta=\frac{3\pi}{4}$

   (c)

   Find the least value of $r$ and the value of $\theta$ at which it occurs.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A curve $C$ has polar equation $r=3+2\sin\theta$ for $0\le\theta<2\pi$.

   (a)

   Show that the area enclosed by $C$ is $11\pi$.

   [3 marks]

   (b)

   Find the exact area of the region bounded by $C$ and the lines $\theta=0$ and $\theta=\frac{\pi}{2}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The circle $C_1$ has polar equation $r=6\sin\theta$ and the cardioid $C_2$ has polar equation $r=2+2\sin\theta$, for $0\le\theta\le\pi$.

   (a)

   Find the values of $\theta$ at which $C_1$ and $C_2$ meet, and the Cartesian coordinates of the points of intersection.

   [6 marks]

   (b)

   Find the exact area of the region that is inside $C_1$ and outside $C_2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions