| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Ratio of coefficients condition |
| Difficulty | Moderate -0.3 Part (a) is straightforward application of binomial theorem formula requiring recall of nCr and basic simplification. Part (b) involves setting up and solving a simple equation from coefficient comparison, requiring only basic algebraic manipulation. This is a standard textbook-style question with no novel insight needed, making it slightly easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1+kx)^{10} = 1 + \binom{10}{1}(kx)^1 + \binom{10}{2}(kx)^2 + \binom{10}{3}(kx)^3\ldots\) | M1 | Attempt at binomial expansion; may be awarded for second, third or fourth term |
| \(= 1+10kx+45k^2x^2+120k^3x^3\ldots\) | A1 | Correct unsimplified expansion (coefficients must be numerical) |
| (fully simplified as above) | A1 | \(1+10kx+45k^2x^2+120k^3x^3\); allow as a list |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(120k^3 = 3\times10k\) | B1 | Seen or implied; if \(x\) present, allow recovery if \(x\) disappears |
| \(4k^2 = 1 \Rightarrow k = \ldots\) | M1 | Solves cubic \(Ak^3 = Bk\) by factorising/cancelling \(k\) to find at least one value |
| \(k = \pm\frac{1}{2}\) | A1 | Both values; ignore any reference to \(0\) |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1+kx)^{10} = 1 + \binom{10}{1}(kx)^1 + \binom{10}{2}(kx)^2 + \binom{10}{3}(kx)^3\ldots$ | M1 | Attempt at binomial expansion; may be awarded for second, third or fourth term |
| $= 1+10kx+45k^2x^2+120k^3x^3\ldots$ | A1 | Correct unsimplified expansion (coefficients must be numerical) |
| (fully simplified as above) | A1 | $1+10kx+45k^2x^2+120k^3x^3$; allow as a list |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $120k^3 = 3\times10k$ | B1 | Seen or implied; if $x$ present, allow recovery if $x$ disappears |
| $4k^2 = 1 \Rightarrow k = \ldots$ | M1 | Solves cubic $Ak^3 = Bk$ by factorising/cancelling $k$ to find at least one value |
| $k = \pm\frac{1}{2}$ | A1 | Both values; ignore any reference to $0$ |
---\begin{enumerate}
\item (a) Find the first 4 terms, in ascending powers of $x$, in the binomial expansion of
\end{enumerate}
$$( 1 + k x ) ^ { 10 }$$
where $k$ is a non-zero constant. Write each coefficient as simply as possible.
Given that in the expansion of $( 1 + k x ) ^ { 10 }$ the coefficient $x ^ { 3 }$ is 3 times the coefficient of $x$, (b) find the possible values of $k$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2020 Q6 [6]}}