| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Direct single expansion substitution |
| Difficulty | Moderate -0.3 This is a straightforward binomial expansion question requiring standard application of the formula for fractional powers, followed by routine substitution and a basic validity check. Part (a) is direct recall, part (b) is simple substitution (recognizing 1.18 = 1 + 0.03×6), and part (c) tests understanding that |x| < 1 is required for convergence. The question is slightly easier than average as it's entirely procedural with no problem-solving or novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks |
|---|---|
| 5(a) | Uses binomial expansion, |
| Answer | Marks | Guidance |
|---|---|---|
| correct, may be un-simplified | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| answer | AO1.1b | A1 |
| (b) | Determines the correct value |
| Answer | Marks | Guidance |
|---|---|---|
| ‘their’ answer to part (a) | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| have been awarded | AO1.1b | A1F |
| (c) | Explains the limitation of the |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | AO2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 5 | |
| Q | Marking Instructions | AO |
Question 5:
--- 5(a) ---
5(a) | Uses binomial expansion,
with at least two terms
correct, may be un-simplified | AO1.1a | M1 | 1 1 1 −2 (6x)2
(1+6x)3 ≈1+ ⋅6x+ ⋅ ⋅
3 3 3 2
1
(1+6x)3 ≈1+2x−4x2
Obtains correct simplified
answer | AO1.1b | A1
(b) | Determines the correct value
for x and substitutes this into
‘their’ answer to part (a) | AO3.1a | M1 | x=0.03
31.18 1 + 2(0.03) – 4(0.03)2
1.0564
≈
≈
Obtains correct
approximation for ‘their’
answer to part (a)
FT allowed only if M1 from
part (a) and M1 from part (b)
have been awarded | AO1.1b | A1F
(c) | Explains the limitation of the
expansion found in part (a)
1
with reference to x =
2 | AO2.4 | E1 | 1
13
Although 1+6× = 3 4
2
1
x = cannot be used since the
2
1
expansion is only valid for x <
6
Total | 5
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item Find the first three terms, in ascending powers of $x$, in the binomial expansion of $(1 + 6x)^{\frac{1}{3}}$ [2 marks]
\item Use the result from part (a) to obtain an approximation to $\sqrt[3]{1.18}$ giving your answer to 4 decimal places. [2 marks]
\item Explain why substituting $x = \frac{1}{2}$ into your answer to part (a) does not lead to a valid approximation for $\sqrt[3]{4}$. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 Q5 [11]}}