Moderate -0.8 This is a straightforward algebraic expansion question disguised as trigonometry. Students expand both squares, collect like terms, and apply the Pythagorean identity sin²x + cos²x = 1 to get 10. The algebra is routine and the identity application is standard C2 material, making this easier than average.
8 Prove that, for all values of \(x\), the value of the expression
$$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$
is an integer and state its value.
\(\ldots = 9\sin^2 x + 6\sin x\cos x + \cos^2 x + \sin^2 x - 6\sin x\cos x + 9\cos^2 x\)
M1
Attempt at expanding both sets of brackets. Minimum requirement: either one of the two expansions correct or 4 of the 6 terms seen. Expanding and simplifying the given expression in one step to get the correct two terms scores this M1 and next A1
\(\ldots = 10\cos^2 x + 10\sin^2 x\)
A1
Either correct pair of expansions and simplification to remove \(\sin x \cos x\) terms or full collecting of like terms within the original correct expansion
\(= 10(1 - \sin^2 x) + 10\sin^2 x\)
M1
\(\cos^2 x + \sin^2 x = 1\) clearly used. If identity applied correctly but not directly, it must be stated at the relevant point in the proof
\(= 10\) (which is an integer)
A1 (Total: 4)
CSO [all previous 3 marks must have been scored]; Condone absence of statement after 10 obtained correctly
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\ldots = 9\sin^2 x + 6\sin x\cos x + \cos^2 x + \sin^2 x - 6\sin x\cos x + 9\cos^2 x$ | M1 | Attempt at expanding both sets of brackets. Minimum requirement: either one of the two expansions correct or 4 of the 6 terms seen. Expanding and simplifying the given expression in one step to get the correct two terms scores this M1 and next A1 |
| $\ldots = 10\cos^2 x + 10\sin^2 x$ | A1 | Either correct pair of expansions and simplification to remove $\sin x \cos x$ terms or full collecting of like terms within the original correct expansion |
| $= 10(1 - \sin^2 x) + 10\sin^2 x$ | M1 | $\cos^2 x + \sin^2 x = 1$ clearly used. If identity applied correctly but not directly, it must be stated at the relevant point in the proof |
| $= 10$ (which is an integer) | A1 (Total: 4) | CSO [all previous 3 marks must have been scored]; Condone absence of statement after 10 obtained correctly |
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8 Prove that, for all values of $x$, the value of the expression
$$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$
is an integer and state its value.
\hfill \mbox{\textit{AQA C2 2011 Q8 [4]}}