Polar and Cartesian coordinatesAQA A-Level Further Maths: Subtopic test

1. 1
   The point $P$ has polar coordinates $\left(4,\dfrac{5\pi}{6}\right)$.

   (a)

   Find the Cartesian $x$-coordinate of $P$.

   [1 mark]

   • A$2\sqrt3$
   • B$-2\sqrt3$
   • C$2$
   • D$-2$

   (b)

   Find the Cartesian $y$-coordinate of $P$.

   [1 mark]

   • A$2\sqrt3$
   • B$-2$
   • C$2$
   • D$4$

   (c)

   The point $P'$ is the reflection of $P$ in the initial line. Give the polar coordinates of $P'$ with $r>0$ and $-\pi<\theta\le\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The point $R$ has Cartesian coordinates $\left(-\sqrt3,-1\right)$.

   (a)

   Find the value of $r$ for $R$.

   [1 mark]

   • A$2$
   • B$4$
   • C$1+\sqrt3$
   • D$-2$

   (b)

   Find $\theta$ for $R$, with $-\pi<\theta\le\pi$.

   [1 mark]

   • A$\dfrac{\pi}{6}$
   • B$\dfrac{5\pi}{6}$
   • C$-\dfrac{\pi}{6}$
   • D$-\dfrac{5\pi}{6}$

   (c)

   The point $R$ is rotated through $\frac{\pi}{2}$ anticlockwise about the origin. Find the Cartesian coordinates of its new position.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two curves are given. Curve $C$ has Cartesian equation $x^2+y^2=4x$ and curve $D$ has polar equation $r^2=18\sin2\theta$.

   (a)

   Show that $C$ has polar equation $r=4\cos\theta$.

   [3 marks]

   (b)

   Find a Cartesian equation for $D$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A circle $C$ has polar equation $r=6\cos\theta$ for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$, and a line $L$ has polar equation $r(\cos\theta+\sin\theta)=6$.

   (a)

   (i) Find a Cartesian equation for $C$, and state the centre and radius of the circle.
   (ii) Find a Cartesian equation for $L$.

   [6 marks]

   (b)

   Find the polar coordinates of the points where $C$ and $L$ intersect, with $r\ge0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions