| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | State validity only |
| Difficulty | Moderate -0.8 Part (a) is a direct application of the binomial expansion formula for negative/fractional powers—a standard textbook exercise. Part (b) requires recognizing that |x| < 1 for validity (basic recall). Part (c) is straightforward substitution and arithmetic. The entire question involves routine procedures with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Uses binomial expansion to obtain \(\left(-\frac{1}{2}\right)x\) or \(\frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)x^2}{2!}\) | M1 | OE |
| \(1 - \frac{1}{2}x + \frac{3}{8}x^2\) | A1 | Must have evaluated coefficients; allow equivalent fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The expansion is valid for \( | x | < 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitutes \(x = -\frac{1}{4}\) into answer to part (a) | M1 | |
| \(1 - \frac{1}{2}\left(-\frac{1}{4}\right) + \frac{3}{8}\left(-\frac{1}{4}\right)^2 = \frac{147}{128}\) | A1 | AWRT 1.148; condone 1.15 if fully correct substituted expansion seen |
| Deduces \(\frac{1}{\sqrt{3}} \approx \frac{147}{256} \approx 0.574\) | A1 | AWRT 0.574 or AWRT 0.580 |
## Question 9:
**Part 9(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Uses binomial expansion to obtain $\left(-\frac{1}{2}\right)x$ or $\frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)x^2}{2!}$ | M1 | OE |
| $1 - \frac{1}{2}x + \frac{3}{8}x^2$ | A1 | Must have evaluated coefficients; allow equivalent fractions |
**Part 9(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| The expansion is valid for $|x| < 1$ | E1 | Must include the word **valid** or **invalid**; accept statement that expansion is not valid for $|x|>1$ |
**Part 9(c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitutes $x = -\frac{1}{4}$ into answer to part (a) | M1 | |
| $1 - \frac{1}{2}\left(-\frac{1}{4}\right) + \frac{3}{8}\left(-\frac{1}{4}\right)^2 = \frac{147}{128}$ | A1 | AWRT 1.148; condone 1.15 if fully correct substituted expansion seen |
| Deduces $\frac{1}{\sqrt{3}} \approx \frac{147}{256} \approx 0.574$ | A1 | AWRT 0.574 or AWRT 0.580 |
---9
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms, in ascending powers of $x$, of the binomial expansion of
$$( 1 + x ) ^ { - \frac { 1 } { 2 } }$$
9
\item A student substitutes $x = 2$ into the expansion of $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ to find an approximation for $\frac { 1 } { \sqrt { 3 } }$
Explain the mistake in the student's approach.\\[0pt]
[1 mark]
9
\item By substituting $x = - \frac { 1 } { 4 }$ in your expansion for $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ find an approximation for $\frac { 1 } { \sqrt { 3 } }$ Give your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 2023 Q9 [6]}}