| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Completing square from standard form |
| Difficulty | Easy -1.2 This is a routine P1 question testing standard completing the square technique, reading coordinates from completed square form, finding a line through two points, and writing simple inequalities. All parts are textbook exercises requiring only direct application of well-practiced methods with no problem-solving or insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02i Represent inequalities: graphically on coordinate plane1.02n Sketch curves: simple equations including polynomials1.03a Straight lines: equation forms y=mx+c, ax+by+c=0 |
11.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-30_595_869_255_568}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}
Figure 5 shows part of the curve $C$ with equation $y = \mathrm { f } ( x )$ where
$$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$
\begin{enumerate}[label=(\alph*)]
\item Write $2 x ^ { 2 } - 12 x + 14$ in the form
$$a ( x + b ) ^ { 2 } + c$$
where $a$, $b$ and $c$ are constants to be found.
Given that $C$ has a minimum at the point $P$
\item state the coordinates of $P$
The line $l$ intersects $C$ at $( - 1,28 )$ and at $P$ as shown in Figure 5.
\item Find the equation of $l$ giving your answer in the form $y = m x + c$ where $m$ and $c$ are constants to be found.
The finite region $R$, shown shaded in Figure 5, is bounded by the $x$-axis, $l$, the $y$-axis, and $C$.
\item Use inequalities to define the region $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2023 Q11 [10]}}