Challenging +1.2 This is a Further Maths FP2 polar coordinates question requiring sketching a limaçon, finding tangent conditions using dy/dx = 0, and computing area with polar integration. While these are standard FP2 techniques, the multi-part nature and requirement to work with polar calculus formulas places it moderately above average A-level difficulty.
8. (a) Sketch the curve \(C\) with polar equation
$$r = 5 + \sqrt { 3 } \cos \theta , \quad 0 \leq \theta \leq 2 \pi$$
(b) Find the polar coordinates of the points where the tangents to \(C\) are parallel to the initial line \(\theta = 0\). Give your answers to 3 significant figures where appropriate.
(c) Using integration, find the area enclosed by the curve \(C\), giving your answer in terms of \(\pi\).
3rd M: Solving a 3 term quadratic to find a value of \(\cos\theta\) (even if called \(\theta\))
Special case: Working with \(r\cos\theta = 5\sin\theta + \sqrt{3}\cos^2\theta\). 1st M1 for \(r\cos\theta = 5\sin\theta + \sqrt{3}\cos^2\theta\). 1st A1 for derivative \(= -5\sin\theta - 2\sqrt{3}\sin\theta\cos\theta\), then no further marks
\(= \frac{1}{2}(50\pi + 3\pi) = \frac{53\pi}{2}\) or equiv. in terms of \(\pi\)
A16
1st M: Attempt to integrate at least one term
2nd M: Requires use of the \(\frac{1}{2}\), correct limits (which could be 0 to \(2\pi\), or \(-\pi\) to \(\pi\), or 'double' 0 to \(\pi\)), and subtraction (which could be implied)
Total: [14]
**(a)** Shape (close curve, approx. symmetrical about the initial line, in all 'quadrants' and 'centred' to the right of the pole/origin) | B1 |
Shape (at least one correct 'intercept' r value... shown on sketch or perhaps seen in a table) | B12 |
(Also allow awrt 3.27 or awrt 6.73) | |
**(b)** $x = r\sin\theta = 5\sin\theta + \sqrt{3}\sin\theta\cos\theta$ | M1 |
$\frac{d\theta}{d\theta} = 5\cos\theta - \sqrt{3}\sin^2\theta + \sqrt{3}\cos^2\theta(= 5\cos\theta + \sqrt{3}\cos 2\theta)$ | A1 |
$5\cos\theta - \sqrt{3}(1 - \cos^2\theta) + \sqrt{3}\cos^2\theta = 0$ | M1 |
$2\sqrt{3}\cos^2\theta + 5\cos\theta - \sqrt{3} = 0$ | |
$(2\sqrt{3}\cos\theta - 1)(\cos\theta + \sqrt{3}) = 0$ | M1 |
$\cos\theta = \ldots (0.288...)$ | | Also allow $\pm \arccos\frac{1}{2\sqrt{3}}$
$\theta = 1.28$ and 5.01 (awrt) (Allow $\pm 1.28$ awrt) | A1 |
$r = 5 + \sqrt{3}\left(\frac{2\sqrt{3}}{2}\right) = \frac{11}{2}$ (Allow awrt 5.50) | A16 |
**2nd M:** Forming a quadratic in $\cos\theta$ | |
**3rd M:** Solving a 3 term quadratic to find a value of $\cos\theta$ (even if called $\theta$) | |
**Special case:** Working with $r\cos\theta = 5\sin\theta + \sqrt{3}\cos^2\theta$. 1st M1 for $r\cos\theta = 5\sin\theta + \sqrt{3}\cos^2\theta$. 1st A1 for derivative $= -5\sin\theta - 2\sqrt{3}\sin\theta\cos\theta$, then no further marks | |
**(c)** $r^2 = 25 + 10\sqrt{3}\cos\theta + 3\cos^2\theta$ | B1 |
$\int(25 + 10\sqrt{3}\cos\theta + 3\cos^2\theta) d\theta = \frac{53\theta}{2} + 10\sqrt{3}\sin\theta + 3\left(\frac{\sin 2\theta}{4}\right)$ | M1 A1ft A1ft |
(It for integration of $(a + b\cos\theta)$ and $c\cos 2\theta$ respectively) | |
$\frac{1}{2}\left[25\theta + 10\sqrt{3}\sin\theta + \frac{3\sin 2\theta}{4} + \frac{3\theta}{2}\right]_0^{\pi/4} = \ldots$ | M1 |
$= \frac{1}{2}(50\pi + 3\pi) = \frac{53\pi}{2}$ or equiv. in terms of $\pi$ | A16 |
**1st M:** Attempt to integrate at least one term | |
**2nd M:** Requires use of the $\frac{1}{2}$, correct limits (which could be 0 to $2\pi$, or $-\pi$ to $\pi$, or 'double' 0 to $\pi$), and subtraction (which could be implied) | |
**Total: [14]**
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8. (a) Sketch the curve $C$ with polar equation
$$r = 5 + \sqrt { 3 } \cos \theta , \quad 0 \leq \theta \leq 2 \pi$$
(b) Find the polar coordinates of the points where the tangents to $C$ are parallel to the initial line $\theta = 0$. Give your answers to 3 significant figures where appropriate.\\
(c) Using integration, find the area enclosed by the curve $C$, giving your answer in terms of $\pi$.\\
\hfill \mbox{\textit{Edexcel FP2 2007 Q8 [14]}}