| Exam Board | AQA |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify expression to polynomial form |
| Difficulty | Standard +0.8 This question requires knowledge of small angle approximations and series expansions for composite angles (4θ, 3θ, 2θ), then substitution of a specific value. While the technique is standard A-level content, working with multiple composite angles and combining three different trigonometric functions requires careful algebraic manipulation and is more demanding than typical small angle questions that use simple θ. The application in part (b) is straightforward once part (a) is complete. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks |
|---|---|
| 9(a) | Substitutes at least one small |
| Answer | Marks | Guidance |
|---|---|---|
| cos4+2sin3−tan2 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| ACF | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1+4−82 | 2.1 | R1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 9(b) | Substitutes . = 0 0 7 into their | |
| 1+4−82 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| CSO | 1.1b | A1 |
| Subtotal | 2 | |
| Question 9 Total | 5 | |
| Q | Marking instructions | AO |
Question 9:
--- 9(a) ---
9(a) | Substitutes at least one small
angle identity correctly into
cos4+2sin3−tan2 | 1.1a | M1 | ( 4 2)
c o s 4 2 s in 3 ta n 2 1 2 ( 3 ) ( 2 ) + − − + −
2
1 4 8 2 = + −
Obtains a correct expression in
terms of
ACF | 1.1b | A1
Completes argument to obtain
1+4−82 | 2.1 | R1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 9(b) ---
9(b) | Substitutes . = 0 0 7 into their
1+4−82 | 3.1a | M1 | 1+40.07−80.07 2 =1.2408
1.241
Obtains AWRT 1.241
CSO | 1.1b | A1
Subtotal | 2
Question 9 Total | 5
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item Show that, for small values of $\theta$ measured in radians
$$\cos 4\theta + 2 \sin 3\theta - \tan 2\theta \approx A + B\theta + C\theta^2$$
where $A$, $B$ and $C$ are constants to be found.
[3 marks]
\item Use your answer to part (a) to find an approximation for
$$\cos 0.28 + 2 \sin 0.21 - \tan 0.14$$
Give your answer to three decimal places.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 2024 Q9 [5]}}