Moderate -0.8 Part (a) is a routine shading exercise requiring identification of regions from inequalities. Part (b) involves solving simultaneous equations with one quadratic—a standard algebraic procedure taught early in A-level. The question requires no problem-solving insight, just careful execution of familiar techniques. Below average difficulty due to its straightforward nature and limited conceptual demand.
The graphs with equations
$$y = 2 + 3x - 2x^2 \text{ and } x + y = 1$$
are shown in the diagram below.
\includegraphics{figure_7}
The graphs intersect at the points \(A\) and \(B\)
\begin{enumerate}[label=(\alph*)]
\item On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities
$$0 \leq y \leq 2 + 3x - 2x^2$$
and
$$x + y \geq 1$$
[2 marks]
\item Find the exact coordinates of \(A\)
[3 marks]
Question 7:
--- 7(a) ---
7(a) | Shades two of regions 1, 2 or 3
only
or
shades one of region 1 or 2 or 3
only
or
shades regions 1, 2 and 3 only | 1.1a | M1
Shades the correct regions 1
and 2 only
Condone missing label R | 2.2a | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 7(b) ---
7(b) | Eliminates y or x correctly
to obtain a quadratic in x or y | 1.1a | M1 | 1 − x = 2 + 3 x − 2 x 2
2 x 2 − 4 x − 1 = 0
2− 6
x =
2
6
y =
2
2− 6 6
So A ,
2 2
2 − 6 2 6
Obtains x = or
2 2
Accept AWFW [–0.225, –0.22]
or
6 6
obtains y = or
2 2
Accept AWFW [1.22, 1.225]
May be unsimplified | 1.1b | A1
2 − 6 6
Obtains ,
2 2
2− 6 6
Accept x = and y =
2 2
ISW
Must be simplified | 1.1b | A1
Subtotal | 3
Question 7 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
The graphs with equations
$$y = 2 + 3x - 2x^2 \text{ and } x + y = 1$$
are shown in the diagram below.
\includegraphics{figure_7}
The graphs intersect at the points $A$ and $B$
\begin{enumerate}[label=(\alph*)]
\item On the diagram above, shade and label the region, $R$, that is satisfied by the inequalities
$$0 \leq y \leq 2 + 3x - 2x^2$$
and
$$x + y \geq 1$$
[2 marks]
\item Find the exact coordinates of $A$
[3 marks]
</end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2024 Q7 [5]}}