| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Standard +0.3 Part (a) is a straightforward binomial expansion requiring routine application of the formula. Parts (b) and (c) involve multiplying the expansion by (a+bx) and equating coefficients to find unknowns—a standard technique but requiring careful algebraic manipulation across multiple steps. This is slightly easier than average as it's methodical rather than requiring insight. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04b Binomial probabilities: link to binomial expansion |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(2-\frac{x}{16}\right)^9 = 2^9 + \binom{9}{1}2^8\left(-\frac{x}{16}\right) + \binom{9}{2}2^7\left(-\frac{x}{16}\right)^2 + ...\) | M1 | Attempts binomial expansion; correct binomial coefficient with correct power of 2 and correct power of \(\left(\pm\frac{x}{16}\right)\) |
| \(= 512 + ...\) | B1 | For 512 |
| \(= ... - 144x + ...\) | A1 | For \(-144x\) |
| \(= ... + ... + 18x^2 (+...)\) | A1 | For \(+18x^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Sets \(512a = 128 \Rightarrow a = ...\) | M1 | Setting their \(512a = 128\) and finding \(a\); or substitute \(x=0\) into both sides |
| \(a = \frac{1}{4}\) oe | A1 ft | Follow through on \(\frac{128}{\text{their } 512}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Sets \(512b + (-144)a = 36 \Rightarrow b = ...\) | M1 | Condone \(512b \pm 144 \times a = 36\) following through on their 512, their \(-144\) and using their value of \(a\) |
| \(b = \frac{9}{64}\) oe | A1 |
## Question 11(a):
$\left(2-\frac{x}{16}\right)^9 = 2^9 + \binom{9}{1}2^8\left(-\frac{x}{16}\right) + \binom{9}{2}2^7\left(-\frac{x}{16}\right)^2 + ...$ | M1 | Attempts binomial expansion; correct binomial coefficient with correct power of 2 and correct power of $\left(\pm\frac{x}{16}\right)$
$= 512 + ...$ | B1 | For 512
$= ... - 144x + ...$ | A1 | For $-144x$
$= ... + ... + 18x^2 (+...)$ | A1 | For $+18x^2$
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## Question 11(b):
Sets $512a = 128 \Rightarrow a = ...$ | M1 | Setting their $512a = 128$ and finding $a$; or substitute $x=0$ into both sides
$a = \frac{1}{4}$ oe | A1 ft | Follow through on $\frac{128}{\text{their } 512}$
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## Question 11(c):
Sets $512b + (-144)a = 36 \Rightarrow b = ...$ | M1 | Condone $512b \pm 144 \times a = 36$ following through on their 512, their $-144$ and using their value of $a$
$b = \frac{9}{64}$ oe | A1 |
---\begin{enumerate}
\item (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
\end{enumerate}
$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$
giving each term in its simplest form.
$$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$
Given that the first two terms, in ascending powers of $x$, in the series expansion of $\mathrm { f } ( x )$ are 128 and $36 x$,\\
(b) find the value of $a$,\\
(c) find the value of $b$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2018 Q11 [8]}}