| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2020 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Direct single expansion substitution |
| Difficulty | Moderate -0.8 Part (a) is a straightforward application of the binomial expansion formula for fractional powers with direct substitution—purely procedural with no problem-solving required. Part (b) asks for explanation only (not calculation) of how to use the expansion for approximation, which is a standard textbook application. This is easier than average A-level content. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((1+8x)^{\frac{1}{2}} = 1 + \frac{1}{2}\times 8x + \frac{\frac{1}{2}\times-\frac{1}{2}}{2!}\times(8x)^2 + \frac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}}{3!}\times(8x)^3\) | M1 | Attempts binomial expansion with \(n=\frac{1}{2}\); must obtain correct structure for term 3 or term 4. Do not accept \(^nC_r\) notation. Look for term 3 in form \(\frac{\frac{1}{2}\times-\frac{1}{2}}{2!}\times(*x)^2\) or \(\frac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}}{3!}\times(*x)^3\) |
| \(= 1 + 4x - 8x^2 + 32x^3 + \ldots\) | A1 | Correct unsimplified expression (may be implied by correct simplified). Award if extra terms present (even if incorrect). Award if terms listed as \(1,\ 4x,\ -8x^2,\ 32x^3\) |
| \(= 1 + 4x - 8x^2 + 32x^3 + \ldots\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Substitutes \(x = \frac{1}{32}\) into \((1+8x)^{\frac{1}{2}}\) to give \(\frac{\sqrt{5}}{2}\) | M1 | Score for substituting \(x=\frac{1}{32}\) into \((1+8x)^{\frac{1}{2}}\) to obtain \(\frac{\sqrt{5}}{2}\) or equivalent such as \(\sqrt{\frac{5}{4}}\). Alternatively award for substituting into both sides and making connection via \(=\) or \(\approx\). Also implied by \(\frac{\sqrt{5}}{2}=\frac{1145}{1024}\) for correct expansion. Not sufficient to state "substitute \(x=\frac{1}{32}\) into the expansion" or just the RHS. |
| Explains that \(x=\frac{1}{32}\) is substituted into \(1+4x-8x^2+32x^3\) and multiply result by 2 | A1ft | Requires full (and correct) explanation of how expansion estimates \(\sqrt{5}\). E.g. calculates \(1+4\times\frac{1}{32}-8\times\left(\frac{1}{32}\right)^2+32\times\left(\frac{1}{32}\right)^3\) and multiplies by 2. Can score from incorrect expansion or expansion with more terms. Mathematical explanation acceptable: \(\frac{\sqrt{5}}{2}=\frac{1145}{1024}\rightarrow\sqrt{5}=\frac{1145}{512}\). SC: M1,A0 for "substitute \(x=\frac{1}{32}\) into both sides of (a) and make \(\sqrt{5}\) the subject" |
## Question 1:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1+8x)^{\frac{1}{2}} = 1 + \frac{1}{2}\times 8x + \frac{\frac{1}{2}\times-\frac{1}{2}}{2!}\times(8x)^2 + \frac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}}{3!}\times(8x)^3$ | M1 | Attempts binomial expansion with $n=\frac{1}{2}$; must obtain correct structure for term 3 **or** term 4. Do not accept $^nC_r$ notation. Look for term 3 in form $\frac{\frac{1}{2}\times-\frac{1}{2}}{2!}\times(*x)^2$ or $\frac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}}{3!}\times(*x)^3$ |
| $= 1 + 4x - 8x^2 + 32x^3 + \ldots$ | A1 | Correct unsimplified expression (may be implied by correct simplified). Award if extra terms present (even if incorrect). Award if terms listed as $1,\ 4x,\ -8x^2,\ 32x^3$ |
| $= 1 + 4x - 8x^2 + 32x^3 + \ldots$ | A1 | cao |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Substitutes $x = \frac{1}{32}$ into $(1+8x)^{\frac{1}{2}}$ to give $\frac{\sqrt{5}}{2}$ | M1 | Score for substituting $x=\frac{1}{32}$ into $(1+8x)^{\frac{1}{2}}$ to obtain $\frac{\sqrt{5}}{2}$ or equivalent such as $\sqrt{\frac{5}{4}}$. Alternatively award for substituting into **both sides** and making connection via $=$ or $\approx$. Also implied by $\frac{\sqrt{5}}{2}=\frac{1145}{1024}$ for correct expansion. Not sufficient to state "substitute $x=\frac{1}{32}$ into the expansion" or just the RHS. |
| Explains that $x=\frac{1}{32}$ is substituted into $1+4x-8x^2+32x^3$ and multiply result by 2 | A1ft | Requires full (and correct) **explanation** of how expansion estimates $\sqrt{5}$. E.g. calculates $1+4\times\frac{1}{32}-8\times\left(\frac{1}{32}\right)^2+32\times\left(\frac{1}{32}\right)^3$ and multiplies by 2. Can score from incorrect expansion or expansion with more terms. Mathematical explanation acceptable: $\frac{\sqrt{5}}{2}=\frac{1145}{1024}\rightarrow\sqrt{5}=\frac{1145}{512}$. SC: M1,A0 for "substitute $x=\frac{1}{32}$ into both sides of (a) and make $\sqrt{5}$ the subject" |\begin{enumerate}
\item (a) Find the first four terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$( 1 + 8 x ) ^ { \frac { 1 } { 2 } }$$
giving each term in simplest form.\\
(b) Explain how you could use $x = \frac { 1 } { 32 }$ in the expansion to find an approximation for $\sqrt { 5 }$ There is no need to carry out the calculation.
\hfill \mbox{\textit{Edexcel Paper 1 2020 Q1 [5]}}