| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Find constant given one specific term |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the binomial theorem formula to find a single parameter by equating coefficients - a standard textbook exercise. Part (b) requires identifying which terms multiply to give the constant term, adding one modest extra step but still routine for AS-level students who have practiced this type of question. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Term in \(x^5\): \(^8C_5 2^3(ax)^5 = 448a^5x^5\) | M1, A1 | Attempt at selecting correct term; \(^8C_5\) must be numerical |
| Sets \(448a^5 = 3402 \Rightarrow a^5 = \frac{243}{32}\) | M1 | Sets coefficient equal to 3402, proceeds to \(a^k = ...\) where \(k \in \mathbb{N}, k \neq 1\) |
| \(\Rightarrow a = \frac{3}{2}\) | A1 | Correct work leading to \(a = \frac{3}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempts either \(2^8\) or \(^8C_4 2^4 a^4\) | M1 | Allow even with \(a\) still present |
| Sum of both terms: \(2^8 + {}^8C_4 2^4 a^4\) | dM1 | Must attempt sum of both terms |
| \(= 256 + 5670 = 5926\) | A1 | cso |
## Question 8:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Term in $x^5$: $^8C_5 2^3(ax)^5 = 448a^5x^5$ | M1, A1 | Attempt at selecting correct term; $^8C_5$ must be numerical |
| Sets $448a^5 = 3402 \Rightarrow a^5 = \frac{243}{32}$ | M1 | Sets coefficient equal to 3402, proceeds to $a^k = ...$ where $k \in \mathbb{N}, k \neq 1$ |
| $\Rightarrow a = \frac{3}{2}$ | A1 | Correct work leading to $a = \frac{3}{2}$ |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts either $2^8$ or $^8C_4 2^4 a^4$ | M1 | Allow even with $a$ still present |
| Sum of both terms: $2^8 + {}^8C_4 2^4 a^4$ | dM1 | Must attempt sum of both terms |
| $= 256 + 5670 = 5926$ | A1 | cso |
---8.
$$g ( x ) = ( 2 + a x ) ^ { 8 } \quad \text { where } a \text { is a constant }$$
Given that one of the terms in the binomial expansion of $\mathrm { g } ( x )$ is $3402 x ^ { 5 }$
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$.
Using this value of $a$,
\item find the constant term in the expansion of
$$\left( 1 + \frac { 1 } { x ^ { 4 } } \right) ( 2 + a x ) ^ { 8 }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q8 [7]}}