| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Expand from factored form |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing routine algebraic manipulation and curve sketching skills. Part (a) requires identifying roots and basic shape from factored form, part (b) is mechanical expansion of brackets, and part (c) applies standard differentiation and tangent line formula. All techniques are standard P1 content with no problem-solving insight required, making it easier than average but not trivial due to the algebraic work involved. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Positive cubic shape with 1 maximum and 1 minimum | M1 | Correct shape for \(y = +x^3\) graph. Condone no axes, condone cusp-like turning points |
| Positive cubic shape reaching \(x\)-axis at \(x = -1\) with minimum on \(x\)-axis at \(x = 3\) | A1 | Must not stop or cross at \(x = 3\) |
| \(y\)-intercept at 18 | B1 | Must correspond with sketch |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2x+2)(x^2 - 6x + 9) = \ldots\) | M1 | Attempt to multiply out. Look for attempt to square \((x-3)\) to obtain \(x^2 \pm 6x \pm 9\) then multiply |
| \(= 2x^3 - 10x^2 + 6x + 18\) | A1 A1 | First A1 for two correct terms, second A1 for fully correct. Ignore spurious "\(= 0\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x) = 6x^2 - 20x + 6\) | B1ft | Correctly differentiates their \(2x^3 - 10x^2 + 6x + 18\). Follow through from 4-term cubic only |
| \(f'\!\left(\frac{1}{3}\right) = 6\!\left(\frac{1}{3}\right)^2 - 20\!\left(\frac{1}{3}\right) + 6\) | M1 | Attempts \(f'\!\left(\frac{1}{3}\right)\) |
| \(f'\!\left(\frac{1}{3}\right) = 0\) | A1 | Must follow correct derivative. Also accept solving \(f'(x)=0\) to get \(x=\frac{1}{3}\) as acceptable alternative |
| \(y = \frac{512}{27}\) | A1 | All previous marks in (c) must have been scored |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Positive cubic shape with 1 maximum and 1 minimum | M1 | Correct shape for $y = +x^3$ graph. Condone no axes, condone cusp-like turning points |
| Positive cubic shape reaching $x$-axis at $x = -1$ with minimum on $x$-axis at $x = 3$ | A1 | Must not stop or cross at $x = 3$ |
| $y$-intercept at 18 | B1 | Must correspond with sketch |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2x+2)(x^2 - 6x + 9) = \ldots$ | M1 | Attempt to multiply out. Look for attempt to square $(x-3)$ to obtain $x^2 \pm 6x \pm 9$ then multiply |
| $= 2x^3 - 10x^2 + 6x + 18$ | A1 A1 | First A1 for two correct terms, second A1 for fully correct. Ignore spurious "$= 0$" |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x) = 6x^2 - 20x + 6$ | B1ft | Correctly differentiates their $2x^3 - 10x^2 + 6x + 18$. Follow through from 4-term cubic only |
| $f'\!\left(\frac{1}{3}\right) = 6\!\left(\frac{1}{3}\right)^2 - 20\!\left(\frac{1}{3}\right) + 6$ | M1 | Attempts $f'\!\left(\frac{1}{3}\right)$ |
| $f'\!\left(\frac{1}{3}\right) = 0$ | A1 | Must follow correct derivative. Also accept solving $f'(x)=0$ to get $x=\frac{1}{3}$ as acceptable alternative |
| $y = \frac{512}{27}$ | A1 | All previous marks in (c) must have been scored |
---6. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
A curve $C$ has equation $y = \mathrm { f } ( x )$ where
$$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Sketch a graph of $C$.
Show on your graph the coordinates of the points where $C$ cuts or meets the coordinate axes.
\item Write $\mathrm { f } ( x )$ in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d$, where $a , b , c$ and $d$ are constants to be found.
\item Hence, find the equation of the tangent to $C$ at the point where $x = \frac { 1 } { 3 }$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q6 [10]}}