| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Deduce inequality solutions from sketch |
| Difficulty | Standard +0.3 This is a multi-part question requiring standard techniques: reading inequalities from a sketch (routine), expanding a cubic (algebraic manipulation), and solving a cubic equation where one root is known. Part (c) requires solving f(x) = -16 and using the known root x = 2/3 to factor, then finding the distance between roots. All techniques are standard P1 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x>4\) | B1 (1) | Only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((3x-2)^2(x-4)=(9x^2-12x+4)(x-4)\) | M1 | Multiplies two brackets then result by third |
| \(=9x^3-48x^2+52x-16\) | A1 A1 (3) | A1: any two correct simplified terms; A1: fully correct (ignore \(=0\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(9x^3-48x^2+52x-16=-16 \Rightarrow 9x^3-48x^2+52x(=0)\) | B1ft | Follow through on their \(a\), \(b\), \(c\) |
| \(x(9x^2-48x+52)=0 \Rightarrow x=\frac{48\pm\sqrt{48^2-4\times9\times52}}{18}=\frac{16\pm4\sqrt{3}}{6}\) | M1 B1 | M1: quadratic seen or implied; B1: correct roots \(\frac{16\pm4\sqrt{3}}{6}\) (allow awrt 1.51, 3.82) |
| Distance \(PQ=\frac{16+4\sqrt{3}}{6}-\frac{16-4\sqrt{3}}{6}=\frac{4}{3}\sqrt{3}\) | M1 A1 (5)(9 marks) | M1: subtracts two non-zero roots; A1: \(\frac{4}{3}\sqrt{3}\) cso (or exact equivalent e.g. \(\frac{16}{12}\sqrt{3}\)); must have \(...\sqrt{3}\); do not allow \(1.33\sqrt{3}\) |
## Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x>4$ | B1 **(1)** | Only |
---
## Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(3x-2)^2(x-4)=(9x^2-12x+4)(x-4)$ | M1 | Multiplies two brackets then result by third |
| $=9x^3-48x^2+52x-16$ | A1 A1 **(3)** | A1: any two correct simplified terms; A1: fully correct (ignore $=0$) |
---
## Question 8(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $9x^3-48x^2+52x-16=-16 \Rightarrow 9x^3-48x^2+52x(=0)$ | B1ft | Follow through on their $a$, $b$, $c$ |
| $x(9x^2-48x+52)=0 \Rightarrow x=\frac{48\pm\sqrt{48^2-4\times9\times52}}{18}=\frac{16\pm4\sqrt{3}}{6}$ | M1 B1 | M1: quadratic seen or implied; B1: correct roots $\frac{16\pm4\sqrt{3}}{6}$ (allow awrt 1.51, 3.82) |
| Distance $PQ=\frac{16+4\sqrt{3}}{6}-\frac{16-4\sqrt{3}}{6}=\frac{4}{3}\sqrt{3}$ | M1 A1 **(5)(9 marks)** | M1: subtracts two non-zero roots; A1: $\frac{4}{3}\sqrt{3}$ cso (or exact equivalent e.g. $\frac{16}{12}\sqrt{3}$); must have $...\sqrt{3}$; do not allow $1.33\sqrt{3}$ |8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-26_718_1076_260_434}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of part of the curve $C$ with equation $y = \mathrm { f } ( x )$, where
$$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$
\begin{enumerate}[label=(\alph*)]
\item Deduce the values of $x$ for which $\mathrm { f } ( x ) > 0$
\item Expand f(x) to the form
$$a x ^ { 3 } + b x ^ { 2 } + c x + d$$
where $a$, $b$, $c$ and $d$ are integers to be found.
The line $l$, also shown in Figure 4, passes through the $y$ intercept of $C$ and is parallel to the $x$-axis.
The line $l$ cuts $C$ again at points $P$ and $Q$, also shown in Figure 4 .
\item Using algebra and showing your working, find the length of line $P Q$. Write your answer in the form $k \sqrt { 3 }$, where $k$ is a constant to be found.\\
(Solutions relying entirely on calculator technology are not acceptable.)
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q8 [9]}}