Challenging +1.2 This requires proving a non-standard identity involving double/multiple angles (8θ and 4θ). Students must recognize to use double angle formulas strategically, likely expressing sin 8θ and cos 8θ in terms of 4θ, then simplify using tan 4θ = sin 4θ/cos 4θ. While systematic, it requires more algebraic manipulation and insight than routine identity proofs, placing it moderately above average difficulty.
Using \(\sin 8\theta = 2\sin 4\theta\cos 4\theta\) OR \(\cos 8\theta = 1 - 2\sin^2 4\theta\) or equivalent OR \(\tan 4\theta = \sin 4\theta / \cos 4\theta\)
M1
Not awarded for just writing formula down
Using another identity
M1
Using third identity and completing to show equals 1
A1
SC1 for incorrect factorisation of \(\sin 8\theta = 4\sin 2\theta = 4 \times 2\sin\theta\cos\theta\) (SC1 awarded once only)
*Note: Small angle approximations lead to answer of 1 but score M0M0A0.*
Question 8a:
Answer
Marks
Guidance
\(R\cos\alpha = \sqrt{3}\) and \(R\sin\alpha = 1\)
M1
Must be in terms of \(\alpha\)
\(R = 2\)
B1
\(\alpha = \frac{\pi}{6}\)
M1A1
A1 dependent on scoring M2; award M0 if R is missing
Question 8b:
Answer
Marks
Guidance
Using *their* result from (a) to get equation equal to \(\sqrt{3}\)
M1
Incorrect \(\alpha\) from Q8a can still access next two M1s
*Their* trig expression \(=\) a value (probably \(\frac{\sqrt{3}}{2}\))
M1
Solving *their* equation for at least one solution
M1
Two correct values in radians, no others
A1
*Note: Squaring both sides may produce extraneous solutions which must be discarded. Maximum M3A0 if not discarded, M2A0 if not discarded.*
Question 9a:
Answer
Marks
Finding either \(f(1)\) or \(f(2)\)
B1
Finding other value AND stating sign change
B1
Question 9b:
Answer
Marks
Guidance
Any value between 1.125 and 1.25
B1
Halving the range and using 1.0625 scores B0
Question 9c:
Answer
Marks
Guidance
Valid iteration values e.g. \(f(1.15) = -0.09\), \(f(1.16) = -0.025\), \(f(1.163) = -0.05\), \(f(1.164) = 0.0015\); final answer 1.2 (1dp)
B1
Must give a value for \(f\); 1.2 with no other working scores B0
Question 9di:
Answer
Marks
Guidance
Change in sign
B1
Change in sign is all that is required
Question 9dii:
Answer
Marks
Guidance
Various valid comments
B1
ISW after correct answer seen
## Question 7:
| Using $\sin 8\theta = 2\sin 4\theta\cos 4\theta$ OR $\cos 8\theta = 1 - 2\sin^2 4\theta$ or equivalent OR $\tan 4\theta = \sin 4\theta / \cos 4\theta$ | M1 | Not awarded for just writing formula down |
| Using another identity | M1 | |
| Using third identity and completing to show equals 1 | A1 | SC1 for incorrect factorisation of $\sin 8\theta = 4\sin 2\theta = 4 \times 2\sin\theta\cos\theta$ (SC1 awarded once only) |
*Note: Small angle approximations lead to answer of 1 but score M0M0A0.*
## Question 8a:
| $R\cos\alpha = \sqrt{3}$ and $R\sin\alpha = 1$ | M1 | Must be in terms of $\alpha$ |
| $R = 2$ | B1 | |
| $\alpha = \frac{\pi}{6}$ | M1A1 | A1 dependent on scoring M2; award M0 if R is missing |
## Question 8b:
| Using *their* result from (a) to get equation equal to $\sqrt{3}$ | M1 | Incorrect $\alpha$ from Q8a can still access next two M1s |
| *Their* trig expression $=$ a value (probably $\frac{\sqrt{3}}{2}$) | M1 | |
| Solving *their* equation for at least one solution | M1 | |
| Two correct values in radians, no others | A1 | |
*Note: Squaring both sides may produce extraneous solutions which must be discarded. Maximum M3A0 if not discarded, M2A0 if not discarded.*
## Question 9a:
| Finding either $f(1)$ or $f(2)$ | B1 | |
| Finding other value AND stating sign change | B1 | |
## Question 9b:
| Any value between 1.125 and 1.25 | B1 | Halving the range and using 1.0625 scores B0 |
## Question 9c:
| Valid iteration values e.g. $f(1.15) = -0.09$, $f(1.16) = -0.025$, $f(1.163) = -0.05$, $f(1.164) = 0.0015$; final answer 1.2 (1dp) | B1 | Must give a value for $f$; 1.2 with no other working scores B0 |
## Question 9di:
| Change in sign | B1 | Change in sign is all that is required |
## Question 9dii:
| Various valid comments | B1 | ISW after correct answer seen |