Standard +0.3 This is a straightforward application of the cosine rule to find r in terms of θ, then using arc length formula s = rθ. It requires two standard formulas and simple algebraic manipulation, making it slightly easier than average but not trivial due to the multi-step nature.
\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]
State \(4^2 = r^2 + r^2 - 2r^2 \cos \theta\). Attempt to make \(r\), or \(r^2\), the subject. Obtain a correct expression for \(r\), or \(r^2\). Attempt to eliminate \(r\) from \(s = r\theta\). Obtain correct arc length, aef.
For expressions that involve \(\sqrt{(\frac{1}{2}\theta)}\):
- B1 – correct expression involving \(r\) and \(\frac{1}{2}\theta\) (e.g. right-angled trig, Sine Rule etc.)
- M1 – attempt to eliminate \(r\) from \(s = r\theta\)
- M1 – attempt to use a correct identity to link \(\sqrt{(\frac{1}{2}\theta)}\) and \(\cos\theta\)
- A1 – obtain correct identity
- A1 – obtain correct arc length, aef
$4^2 = r^2 + r^2 - 2r^2 \cos \theta$; $r^2(1 - \cos \theta) = 8$; Arc $PQ = r\theta = \theta\sqrt{\frac{8}{1-\cos \theta}}$ | B1, M1, A1, M1, A1 [5] | State $4^2 = r^2 + r^2 - 2r^2 \cos \theta$. Attempt to make $r$, or $r^2$, the subject. Obtain a correct expression for $r$, or $r^2$. Attempt to eliminate $r$ from $s = r\theta$. Obtain correct arc length, aef.
For expressions that involve $\sqrt{(\frac{1}{2}\theta)}$:
- B1 – correct expression involving $r$ and $\frac{1}{2}\theta$ (e.g. right-angled trig, Sine Rule etc.)
- M1 – attempt to eliminate $r$ from $s = r\theta$
- M1 – attempt to use a correct identity to link $\sqrt{(\frac{1}{2}\theta)}$ and $\cos\theta$
- A1 – obtain correct identity
- A1 – obtain correct arc length, aef
$P$ and $Q$ are points on the circumference of a circle with centre $O$ and radius $r$. The angle $POQ$ is $\theta$ radians. Given that the chord $PQ$ has length 4, find an expression for the length of the arc $PQ$ in terms of $\theta$ of only. [5]
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q8 [5]}}