Moderate -0.8 This is a straightforward binomial expansion question requiring recall of the binomial theorem formula and basic algebraic manipulation. Part (a) is routine application of (a+b)^n expansion, and part (b) involves setting up and solving a simple linear equation. The question requires no problem-solving insight and is more mechanical than the average A-level question.
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms, in ascending powers of \(x\), in the expansion of
$$(2 + kx)^5$$
where \(k\) is a positive constant.
[3 marks]
\item Hence, given that the coefficient of \(x\) is four times the coefficient of \(x^2\), find the value of \(k\)
[2 marks]
Question 8:
--- 8(a) ---
8(a) | Obtains the correct constant term
32 | 1.1b | B1 | ( 2+kx)5 =32+80kx+80k2x2+...
Obtains 516kx or 108(kx)2
OE
5 k 5 4 k x 2
PI by x or
2 2 ! 2 | 1.1a | M1
Obtains 3 2 + 8 0 k x + 8 0 k 2 x 2 ( + . . . )
Accept list of correct terms.
No ISW
If more terms are given it must be
obvious which are their first three
terms. | 1.1b | A1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 8(b) ---
8(b) | Forms the equation
their A k = 4 t h e ir B k 2 OE
May recover if x is initially included. | 3.1a | M1 | 8 0 k = 4 8 0 k 2
1
k = o 0 r
4
1
k =
4
Since k 0
1
Deduces k = only
4
A
Or their k = their
4 B
Justification of rejection k = 0 not
required. | 2.2a | A1F
Subtotal | 2
Question 8 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
\begin{enumerate}[label=(\alph*)]
\item Find the first three terms, in ascending powers of $x$, in the expansion of
$$(2 + kx)^5$$
where $k$ is a positive constant.
[3 marks]
\item Hence, given that the coefficient of $x$ is four times the coefficient of $x^2$, find the value of $k$
[2 marks]
</end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 2024 Q8 [5]}}