C3 January 2013 Q6
6.
| Scheme | Marks |
|---|---|
| \((\sin 22.5 + \cos 22.5)^2 = \sin^2 22.5 + \cos^2 22.5 + \ldots\) \(= \sin^2 22.5 + \cos^2 22.5 + 2\sin 22.5\cos 22.5\) | M1 |
| States or uses \(\sin^2 22.5 + \cos^2 22.5 = 1\) | B1 |
| Uses \(2\sin x\cos x = \sin 2x \Rightarrow 2\sin 22.5\cos 22.5 = \sin 45\) | M1 |
| \((\sin 22.5 + \cos 22.5)^2 = 1 + \sin 45\) | A1 |
| \(= 1 + \dfrac{\sqrt{2}}{2}\) or \(1 + \dfrac{1}{\sqrt{2}}\) cso | A1 |
| (5) |
Notes
M1 Attempts to expand \((\sin 22.5 + \cos 22.5)^2\). Award if you see \(\sin^2 22.5 + \cos^2 22.5 + \ldots\)
There must be > two terms. Condone missing brackets ie \(\sin 22.5^2 + \cos 22.5^2 + \ldots\)
B1 Stating or using \(\sin^2 22.5 + \cos^2 22.5 = 1\). Accept \(\sin 22.5^2 + \cos 22.5^2 = 1\) as the intention is clear.
Note that this may also come from using the double angle formula
\(\sin^2 22.5 + \cos^2 22.5 = \left(\dfrac{1 - \cos 45}{2}\right) + \left(\dfrac{1 + \cos 45}{2}\right) = 1\)
M1 Uses \(2\sin x\cos x = \sin 2x\) to write \(2\sin 22.5\cos 22.5\) as sin 45 or sin(2\(\times\)22.5)
A1 Reaching the intermediate answer \(1 + \sin 45\)
A1 Cso \(1 + \dfrac{\sqrt{2}}{2}\) or \(1 + \dfrac{1}{\sqrt{2}}\). Be aware that both 1.707 and \(\dfrac{2 + \sqrt{2}}{2}\) can be found by using a calculator for 1+sin45. Neither can be accepted on their own without firstly seeing one of the two answers given above. Each stage should be shown as required by the mark scheme.
Note that if the candidates use \((\sin\theta + \cos\theta)^2\) they can pick up the first M and B marks, but no others until they use \(\theta = 22.5\). All other marks then become available.
6.alt 1
| Scheme | Marks |
|---|---|
| (i) \((\sin 22.5 + \cos 22.5)^2 = \sin^2 22.5 + \cos^2 22.5 + \ldots\) \(= \sin^2 22.5 + \cos^2 22.5 + 2\sin 22.5\cos 22.5\) | M1 |
| States or uses \(\sin^2 22.5 + \cos^2 22.5 = 1\) | B1 |
| Uses \(2\sin x\cos x = 2\sqrt{\dfrac{1 - \cos 2x}{2}}\sqrt{\dfrac{\cos 2x + 1}{2}} \Rightarrow \sqrt{1 - \cos 45}\sqrt{1 + \cos 45}\) | M1 |
| \(= \sqrt{1 - \cos^2 45}\) | A1 |
| Hence \((\sin 22.5 + \cos 22.5)^2 = 1 + \dfrac{\sqrt{2}}{2}\) or \(1 + \dfrac{1}{\sqrt{2}}\) | A1 |
| (5) |
6.alt 2
| Scheme | Marks |
|---|---|
| (i) Uses Factor Formula \((\sin 22.5 + \sin 67.5)^2 = \left(2\sin 45\cos 22.5\right)^2\) | M1,A1 |
| Reaching the stage \(= 2\cos^2 22.5\) | B1 |
| Uses the double angle formula \(= 2\cos^2 22.5 = 1 + \cos 45\) | M1 |
| \(= 1 + \dfrac{\sqrt{2}}{2}\) or \(1 + \dfrac{1}{\sqrt{2}}\) | A1 |
| (5) |
6.alt 3
| Scheme | Marks |
|---|---|
| (i) Uses Factor Formula \((\cos 67.5 + \cos 22.5)^2 = \left(2\cos 45\cos 22.5\right)^2\) | M1,A1 |
| Reaching the stage \(= 2\cos^2 22.5\) | B1 |
| Uses the double angle formula \(= 2\cos^2 22.5 = 1 + \cos 45\) | M1 |
| \(= 1 + \dfrac{\sqrt{2}}{2}\) or \(1 + \dfrac{1}{\sqrt{2}}\) | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| (a) \(\cos 2\theta + \sin\theta = 1 \Rightarrow 1 - 2\sin^2\theta + \sin\theta = 1\) | M1 |
| \(\sin\theta - 2\sin^2\theta = 0\) \(2\sin^2\theta - \sin\theta = 0\) or \(k = 2\) | A1* |
| (2) | |
| (b) \(\sin\theta(2\sin\theta - 1) = 0\) | M1 |
| \(\sin\theta = 0, \quad \sin\theta = \dfrac{1}{2}\) | A1 |
| Any two of 0,30,150,180 | B1 |
| All four answers 0,30,150,180 | A1 |
| (4) | |
| (11 marks) |
Notes
(iia) M1 Substitutes \(\cos 2\theta = 1 - 2\sin^2\theta\) in \(\cos 2\theta + \sin\theta = 1\) to produce an equation in \(\sin\theta\) only.
It is acceptable to use \(\cos 2\theta = 2\cos^2\theta - 1\) or \(\cos^2\theta - \sin^2\theta\) as long as the \(\cos^2\theta\) is subsequently replaced by \(1 - \sin^2\theta\)
A1* Obtains the correct simplified equation in \(\sin\theta\).
\(\sin\theta - 2\sin^2\theta = 0\) or \(\sin\theta = 2\sin^2\theta\) must be written in the form \(2\sin^2\theta - \sin\theta = 0\) as required by the question. Also accept \(k = 2\) as long as no incorrect working is seen.
(iib) M1 Factorises or divides by \(\sin\theta\). For this mark \(1 = 'k'\sin\theta\) is acceptable. If they have a 3 TQ in \(\sin\theta\) this can be scored for correct factorisation
A1 Both \(\sin\theta = 0\), and \(\sin\theta = \dfrac{1}{2}\)
B1 Any two answers from 0, 30, 150, 180.
A1 All four answers 0, 30, 150, 180 with no extra solutions inside the range. Ignore solutions outside the range.