Core Pure: Polar coordinatesEdexcel A-Level Further Maths: Topic test

1. 1
   The point $A$ has polar coordinates $\left(6,\frac{\pi}{3}\right)$ and the point $B$ has Cartesian coordinates $\left(-2\sqrt3,\,2\right)$.

   (a)

   What are the Cartesian coordinates of $A$?

   [1 mark]

   • A$\left(3,\,3\sqrt3\right)$
   • B$\left(3\sqrt3,\,3\right)$
   • C$\left(3,\,-3\sqrt3\right)$
   • D$\left(6,\,3\sqrt3\right)$

   (b)

   What are the polar coordinates of $B$, with $r>0$ and $0\le\theta<2\pi$?

   [1 mark]

   • A$\left(4,\frac{\pi}{6}\right)$
   • B$\left(4,-\frac{\pi}{6}\right)$
   • C$\left(4,\frac{5\pi}{6}\right)$
   • D$\left(\sqrt{14},\frac{5\pi}{6}\right)$

   (c)

   Find the distance $AB$, giving your answer in the form $k\sqrt{13}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has polar equation $r=5\sec\left(\theta-\frac{\pi}{3}\right)$ for $-\frac{\pi}{6}<\theta<\frac{5\pi}{6}$.

   (a)

   Which statement describes $C$?

   [1 mark]

   • AA circle passing through the pole
   • BA straight line at a perpendicular distance of $5$ from the pole
   • CA cardioid
   • DA spiral

   (b)

   What is the value of $r$ at the point where $C$ meets the initial line?

   [1 mark]

   • A$5$
   • B$\frac52$
   • C$\frac{10}{\sqrt3}$
   • D$10$

   (c)

   Show that a Cartesian equation of $C$ is $x+\sqrt3\,y=10$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has polar equation $r=4\cos^2\theta$ for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   State the greatest value of $r$ and the value of $\theta$ at which it occurs, state the values of $\theta$ at which $C$ passes through the pole, and find $r$ when $\theta=\frac{\pi}{3}$.

   [3 marks]

   (b)

   Find the area enclosed by $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has polar equation $r=3(1-\cos\theta)$ for $0\le\theta<2\pi$, and the circle $S$ has polar equation $r=3$.

   (a)

   Find the polar coordinates of the points on $C$, other than the pole, at which the tangent is parallel to the initial line.

   [6 marks]

   (b)

   Find the exact area of the region that lies inside $C$ and outside $S$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $H$ has Cartesian equation $x^2-y^2=8$.

   (a)

   Which is a polar equation of $H$?

   [1 mark]

   • A$r^2=8\cos2\theta$
   • B$r^2=8\sin2\theta$
   • C$r^2=\frac{8}{\cos2\theta}$
   • D$r=\frac{8}{\cos2\theta}$

   (b)

   What is the smallest value of $r$ on $H$?

   [1 mark]

   • A$2\sqrt2$
   • B$8$
   • C$4$
   • D$\sqrt2$

   (c)

   Find the polar coordinates of the point on $H$ with $\theta=\frac{\pi}{6}$ and $r>0$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The region $R$ is bounded by the curve with polar equation $r=4\sec\theta$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{3}$.

   (a)

   Which integral gives the area of $R$?

   [1 mark]

   • A$\frac12\int_0^{\pi/3}4\sec\theta\,d\theta$
   • B$\int_0^{\pi/3}16\sec^2\theta\,d\theta$
   • C$\frac12\int_0^{\pi/3}16\cos^2\theta\,d\theta$
   • D$\frac12\int_0^{\pi/3}16\sec^2\theta\,d\theta$

   (b)

   What is the area of $R$?

   [1 mark]

   • A$16\sqrt3$
   • B$8\sqrt3$
   • C$\frac{8}{\sqrt3}$
   • D$4\sqrt3$

   (c)

   Find the exact area of the part of $R$ between the half-lines $\theta=\frac{\pi}{6}$ and $\theta=\frac{\pi}{4}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The circles $C_1$ and $C_2$ have polar equations $r=6\cos\theta$ and $r=6\sin\theta$ respectively, for $0\le\theta\le\frac{\pi}{2}$.

   (a)

   Find the polar coordinates of the point, other than the pole, where $C_1$ and $C_2$ intersect.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The circle $C$ has polar equation $r=8\cos\theta$ and the line $L$ has polar equation $r=2\sec\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   Show that $C$ has Cartesian equation $(x-4)^2+y^2=16$, and find the polar coordinates of the points where $L$ meets $C$.

   [6 marks]

   (b)

   Find the exact area of the region that is inside $C$ and on the opposite side of $L$ from the pole.

   [6 marks]

   Total for question 8: 12 marks

End of questions