Standard +0.8 This question requires sketching the Archimedean spiral r=θ, setting up and evaluating a polar area integral (∫½r²dθ), then solving an equation involving π² to find where areas are equal. It combines visualization, integration technique, and algebraic manipulation beyond routine exercises, though the concepts are standard for Further Maths polar coordinates.
5 Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\).
The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\). Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.
\(\Rightarrow \alpha^3 = \pi^3/16 \Rightarrow \alpha = \pi(16)^{-1/3}\), or equivalent
A1
Sketch with correct shape, location and orientation | B1 |
Shows tangency to the initial line at the pole | B1 |
Ignore extra in diagram |
Draws, in the same diagram, a straight line passing through the origin and with positive gradient (distinct half-line and not a construction line) | B1 |
$(1/2)\int_0^{\pi/2} \theta^2 d\theta = \pi^3/48$ | M1A1 |
$(1/2)\int_0^\alpha \theta^2 d\theta = \alpha^3/6$ | A1 |
$\alpha^3/6 = \pi^3/96 \Rightarrow \alpha = \pi 2^{-4/3}$ (acf) | A1 |
**or for previous 4 marks:**
$(1/2)\int_0^\alpha \theta^2 d\theta = (1/2)\int_{\pi/2}^\pi \theta^2 d\theta$ | M1 |
$\alpha^3/6 = (1/6)[(\pi/2)^3 - \alpha^3]$ | A1A1 |
$\Rightarrow \alpha^3 = \pi^3/16 \Rightarrow \alpha = \pi(16)^{-1/3}$, or equivalent | A1 |
5 Draw a sketch of the curve $C$ whose polar equation is $r = \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
On the same diagram draw the line $\theta = \alpha$, where $0 < \alpha < \frac { 1 } { 2 } \pi$.
The region bounded by $C$ and the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $R$. Find the exact value of $\alpha$ for which the line $\theta = \alpha$ divides $R$ into two regions of equal area.
\hfill \mbox{\textit{CAIE FP1 2009 Q5 [7]}}