FP2: Polar coordinatesEdexcel International A Level Further Maths: Topic test

1. 1
   The circle $C$ has polar equation $r=8\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   Find the radius of $C$.

   [1 mark]

   • A$8$
   • B$4$
   • C$2$
   • D$\sqrt8$

   (b)

   Which is the cartesian equation of $C$?

   [1 mark]

   • A$x^2+y^2=8x$
   • B$x^2+y^2=8y$
   • C$x^2+y^2=64$
   • D$x^2+y^2=8$

   (c)

   Find the coordinates of the centre of $C$ in cartesian form, and state its radius.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The straight line $l$ has polar equation $r=5\sec\left(\frac{\pi}{4}-\theta\right)$.

   (a)

   Find the perpendicular distance from the pole to $l$.

   [1 mark]

   • A$5$
   • B$5\sqrt2$
   • C$\frac{5}{\sqrt2}$
   • D$\frac{\pi}{4}$

   (b)

   Which is the cartesian equation of $l$?

   [1 mark]

   • A$x+y=5$
   • B$x-y=5\sqrt2$
   • C$x+y=5\sqrt2$
   • D$x+\sqrt3y=10$

   (c)

   Find the polar coordinates of the point on $l$ that is nearest to the pole.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has polar equation $r^2=4\cos2\theta$, for $-\frac{\pi}{4}\le\theta\le\frac{\pi}{4}$, and $r\ge0$.

   (a)

   Find the area of the region enclosed by $C$.

   [3 marks]

   (b)

   Find the equation of the tangent to $C$ that is parallel to the initial line and above it, giving your answer in the form $r\sin\theta=k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The circle $C_1$ has polar equation $r=5$ and the circle $C_2$ has polar equation $r=10\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

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

   [6 marks]

   (b)

   Find the equation of the tangent to $C_2$ that is parallel to the initial line and above it, and the equation of the tangent to $C_2$ that is perpendicular to the initial line and furthest from the pole. Give both answers in the form $r\sin\theta=k$ or $r\cos\theta=k$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has polar equation $r=6\cos2\theta$, for $-\frac{\pi}{4}\le\theta\le\frac{\pi}{4}$.

   (a)

   Find the greatest distance of a point on $C$ from the pole.

   [1 mark]

   • A$3$
   • B$12$
   • C$6$
   • D$\frac{\pi}{4}$

   (b)

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

   [1 mark]

   • A$\theta=0$
   • B$\theta=\pm\frac{\pi}{2}$
   • C$\theta=\pm\frac{\pi}{8}$
   • D$\theta=\pm\frac{\pi}{4}$

   (c)

   Show that $C$ is symmetrical about the initial line.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The spiral $S$ has polar equation $r=\frac12\theta$, for $\theta\ge0$.

   (a)

   Which expression gives the area of the region bounded by $S$ and the half-lines $\theta=0$ and $\theta=\pi$?

   [1 mark]

   • A$\frac12\int_0^{\pi}\frac{\theta}{2}\,d\theta$
   • B$\int_0^{\pi}\frac{\theta^2}{4}\,d\theta$
   • C$\frac12\int_0^{\pi}\theta^2\,d\theta$
   • D$\frac12\int_0^{\pi}\frac{\theta^2}{4}\,d\theta$

   (b)

   Find the area of that region.

   [1 mark]

   • A$\frac{\pi^3}{12}$
   • B$\frac{\pi^3}{24}$
   • C$\frac{\pi^3}{6}$
   • D$\frac{\pi^2}{8}$

   (c)

   Find the exact area of the region bounded by $S$ and the half-lines $\theta=\pi$ and $\theta=2\pi$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has polar equation $r=2(1-\cos\theta)$, for $0\le\theta\le2\pi$.

   (a)

   Find the exact area of the region enclosed by $C$.

   [3 marks]

   (b)

   Find the polar coordinates of the point on $C$ at which the tangent is parallel to the initial line and above it, and the equation of that tangent.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The circle $C$ has polar equation $r=6\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$, and the straight line $l$ has polar equation $r=\frac32\sec\theta$.

   (a)

   The line $l$ cuts $C$ into two regions. Find the exact area of the smaller region, the part of $C$ on the same side of $l$ as the pole.

   [6 marks]

   (b)

   Hence find the exact area of the region inside $C$ that lies on the far side of $l$ from the pole.

   [6 marks]

   Total for question 8: 12 marks

End of questions