| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Finding quadratic from vertex information |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard techniques: (a) sketching from given information, (b) finding a quadratic from vertex form using three known points, and (c) solving simultaneous equations. All parts use routine methods with no novel insight required, making it easier than average for A-level. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct \(\cap\) shape passing through \((0,0)\), maximum on rhs of \(y\)-axis | B1 | \(\cap\) shaped quadratic through origin with maximum on rhs of \(y\)-axis; appears in quadrants 3, 1 and 4; ignore extra graphs superimposed |
| Intersection at \((4, 0)\) | B1 | Allow 4 marked on positive \(x\)-axis; \((0,4)\) is B0; condone graphs that just meet \(x\)-axis at 4 |
| (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts form of equation e.g. \(y = Ax(x-4)\) or \(y = 20 \pm C(x-2)^2\) | M1 | Condone \(A,C=1\); also allow \(y=ax^2+bx+c\) with attempt to use all three coordinates |
| Full attempt to find equation e.g. \(20 = A\times2(2-4) \Rightarrow A = \ldots\) or \(0 = 20+C(4-2)^2 \Rightarrow C=\ldots\) | dM1 | Full attempt at finding values of \(A\) or \(C\) or \(a,b,c\) |
| \(y = -5x(x-4)\), \(y = 20-5(x-2)^2\) o.e. | A1 | Allow \(f(x)\leftrightarrow y\); allow \(5x(4-x)\); ISW after correct answer |
| (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(x(x^2-4) = ``{-5x(x-4)}''\) | M1 | Form of quadratic must be correct (pass through \((0,0)\)); condoning slips |
| \(x^3+5x^2-24x=0 \Rightarrow x\left(x^2+5x-24\right)=0\) | dM1 | Multiplies out to form \(px^3+qx^2+rx=0\), factorises/cancels \(x\) |
| \((x+8)(x-3)=0 \Rightarrow x\) coordinate of \(P\) is \(-8\) | ddM1, A1 | Solves resulting quadratic; chooses \(-8\); reject or ignore \(x=0,3\) |
| \(P = (-8, -480)\) | A1 | May be given separately as \(x=\ldots, y=\ldots\) |
| (5 marks) | ||
| (10 marks total) | All stages of working must be shown |
# Question 9:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct $\cap$ shape passing through $(0,0)$, maximum on rhs of $y$-axis | B1 | $\cap$ shaped quadratic through origin with maximum on rhs of $y$-axis; appears in quadrants 3, 1 and 4; ignore extra graphs superimposed |
| Intersection at $(4, 0)$ | B1 | Allow 4 marked on positive $x$-axis; $(0,4)$ is B0; condone graphs that just meet $x$-axis at 4 |
| **(2 marks)** | | |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts form of equation e.g. $y = Ax(x-4)$ or $y = 20 \pm C(x-2)^2$ | M1 | Condone $A,C=1$; also allow $y=ax^2+bx+c$ with attempt to use all three coordinates |
| Full attempt to find equation e.g. $20 = A\times2(2-4) \Rightarrow A = \ldots$ or $0 = 20+C(4-2)^2 \Rightarrow C=\ldots$ | dM1 | Full attempt at finding values of $A$ or $C$ or $a,b,c$ |
| $y = -5x(x-4)$, $y = 20-5(x-2)^2$ o.e. | A1 | Allow $f(x)\leftrightarrow y$; allow $5x(4-x)$; ISW after correct answer |
| **(3 marks)** | | |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $x(x^2-4) = ``{-5x(x-4)}''$ | M1 | Form of quadratic must be correct (pass through $(0,0)$); condoning slips |
| $x^3+5x^2-24x=0 \Rightarrow x\left(x^2+5x-24\right)=0$ | dM1 | Multiplies out to form $px^3+qx^2+rx=0$, factorises/cancels $x$ |
| $(x+8)(x-3)=0 \Rightarrow x$ coordinate of $P$ is $-8$ | ddM1, A1 | Solves resulting quadratic; **chooses** $-8$; reject or ignore $x=0,3$ |
| $P = (-8, -480)$ | A1 | May be given separately as $x=\ldots, y=\ldots$ |
| **(5 marks)** | | |
| **(10 marks total)** | | All stages of working must be shown |\begin{enumerate}
\item The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$.
\end{enumerate}
Given that
\begin{itemize}
\item $\mathrm { f } ( x )$ is a quadratic expression
\item $C _ { 1 }$ has a maximum turning point at $( 2,20 )$
\item $C _ { 1 }$ passes through the origin\\
(a) sketch a graph of $C _ { 1 }$ showing the coordinates of any points where $C _ { 1 }$ cuts the coordinate axes,\\
(b) find an expression for $\mathrm { f } ( x )$.
\end{itemize}
The curve $C _ { 2 }$ has equation $y = x \left( x ^ { 2 } - 4 \right)$\\
Curve $C _ { 1 }$ and $C _ { 2 }$ meet at the origin, and at the points $P$ and $Q$\\
Given that the $x$ coordinate of the point $P$ is negative,\\
(c) using algebra and showing all stages of your working, find the coordinates of $P$
\hfill \mbox{\textit{Edexcel P1 2024 Q9 [10]}}