| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a straightforward completing-the-square question with standard follow-up parts. Part (a) requires routine algebraic manipulation, part (b) is a basic sketch using the completed square form, and part (c) is direct recall. All techniques are standard P1 content with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(f(x)=11-4x-2x^2 \Rightarrow \ldots\underline{-2}(2x+x^2)\ldots\) or \(\Rightarrow \ldots\underline{-2}(2x+x^2\ldots)\) | B1 | \(b=-2\) |
| \(\ldots(2x+x^2)\Rightarrow\ldots\left((x+1)^2\pm\ldots\right)\) | M1 | Attempts to complete the square on \(x^2\pm 2x\); score for \((x\pm1)^2\pm\ldots\) or attempts to compare coefficients to find \(c\). Condone \(11-4x-2x^2 \Rightarrow \ldots-2(2x-x^2)\ldots\Rightarrow -2(x\pm1)^2\pm\ldots\) |
| \((f(x)=)\ 13-2(x+1)^2\) | A1 | If \(a\), \(b\) and \(c\) stated with contradiction, mark final expression. Condone \(13+-2(x+1)^2\) or just values of \(a\), \(b\) and \(c\) stated. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(\cap\) shaped curve on a set of axes | M1 | \(\cap\) shape anywhere on axes. Cannot be part of other functions (e.g. cubic). |
| Correct shape cutting \(x\)-axis once either side of origin, maximum in quadrant 2, \(y\)-intercept \((0,11)\) or 11 marked on \(y\)-axis | A1 | Condone \((11,0)\) if intercept in correct place. Line of symmetry must appear to left of origin; curve should appear broadly symmetrical about this line. Ignore \(x\)-intercepts or maximum stated. Do not accept graphs symmetrical about \(y\)-axis. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(x=-1\) | B1ft | Follow through their numeric \(c\), so allow \(x=-\)"\(c\)" |
# Question 2(a):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $f(x)=11-4x-2x^2 \Rightarrow \ldots\underline{-2}(2x+x^2)\ldots$ or $\Rightarrow \ldots\underline{-2}(2x+x^2\ldots)$ | B1 | $b=-2$ |
| $\ldots(2x+x^2)\Rightarrow\ldots\left((x+1)^2\pm\ldots\right)$ | M1 | Attempts to complete the square on $x^2\pm 2x$; score for $(x\pm1)^2\pm\ldots$ or attempts to compare coefficients to find $c$. Condone $11-4x-2x^2 \Rightarrow \ldots-2(2x-x^2)\ldots\Rightarrow -2(x\pm1)^2\pm\ldots$ |
| $(f(x)=)\ 13-2(x+1)^2$ | A1 | If $a$, $b$ and $c$ stated with contradiction, mark final expression. Condone $13+-2(x+1)^2$ or just values of $a$, $b$ and $c$ stated. |
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# Question 2(b):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\cap$ shaped curve on a set of axes | M1 | $\cap$ shape anywhere on axes. Cannot be part of other functions (e.g. cubic). |
| Correct shape cutting $x$-axis once either side of origin, maximum in quadrant 2, $y$-intercept $(0,11)$ or 11 marked on $y$-axis | A1 | Condone $(11,0)$ if intercept in correct place. Line of symmetry must appear to left of origin; curve should appear broadly symmetrical about this line. Ignore $x$-intercepts or maximum stated. Do not accept graphs symmetrical about $y$-axis. |
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# Question 2(c):
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $x=-1$ | B1ft | Follow through their numeric $c$, so allow $x=-$"$c$" |
---2.
$$f ( x ) = 11 - 4 x - 2 x ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form
$$a + b ( x + c ) ^ { 2 }$$
where $a , b$ and $c$ are integers to be found.
\item Sketch the graph of the curve $C$ with equation $y = \mathrm { f } ( x )$, showing clearly the coordinates of the point where the curve crosses the $y$-axis.
\item Write down the equation of the line of symmetry of $C$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2022 Q2 [6]}}