Polar coordinates and curve sketchingEdexcel International A Level Further Maths: Subtopic test

1. 1
   The straight line $l$ has polar equation $r=4\sec\left(\theta-\frac{\pi}{3}\right)$, where $r\ge0$ and the pole is $O$.

   (a)

   Which is a Cartesian equation of $l$?

   [1 mark]

   • A$x+y=4$
   • B$x+\sqrt3\,y=8$
   • C$\sqrt3\,x+y=8$
   • D$x+\sqrt3\,y=4$

   (b)

   What is the shortest distance from $O$ to $l$?

   [1 mark]

   • A$8$
   • B$2$
   • C$\frac{8}{\sqrt3}$
   • D$4$

   (c)

   Find the polar coordinates of the point where $l$ meets the initial line.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $S$ has polar equation $r=3\theta$, for $\theta\ge0$.

   (a)

   What is the value of $r$ when $\theta=\frac{\pi}{2}$?

   [1 mark]

   • A$\frac{\pi}{6}$
   • B$\frac{3\pi}{4}$
   • C$\frac{3\pi}{2}$
   • D$3\pi$

   (b)

   $S$ crosses the positive $x$-axis ($\theta=0,2\pi,4\pi,\dots$) at points that get further from $O$. What is the distance between two consecutive crossing points?

   [1 mark]

   • A$6\pi$
   • B$3\pi$
   • C$2\pi$
   • D$12\pi$

   (c)

   Find the Cartesian coordinates of the point on $S$ where $\theta=\frac{3\pi}{2}$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Find the greatest and least values of $r$, stating the value of $\theta$ at each, and explain why $C$ does not pass through the pole.

   [3 marks]

   (b)

   Find the values of $\theta$, to 3 significant figures, at which the tangent to $C$ is parallel to the initial line.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has polar equation $r=a(1+\cos\theta)$, where $a>0$ and $0\le\theta\le\pi$.

   (a)

   Find the polar coordinates of the point on $C$, with $0<\theta<\pi$, at which the tangent is parallel to the initial line.

   [6 marks]

   (b)

   Find the equation of the tangent to $C$, with $0<\theta<\pi$, that is perpendicular to the initial line. Give your answer in Cartesian form.

   [6 marks]

   Total for question 4: 12 marks

End of questions