| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve transformed function equations |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on curve sketching and transformations. Part (a) requires basic factorization and plotting intercepts. Parts (b) and (c) involve standard function transformations (horizontal stretch and translation) that are routine P1 content. All steps follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Shape for positive cubic (one max, one min) | B1 | Any position, condone no axes, condone cusp-like minimum |
| Cuts \(x\)-axis at \(\left(\frac{3}{4}, 0\right)\) and meets at \((5, 0)\) | B1 | Graph should not stop or cross at \((5,0)\); allow just \(x\) values; condone slip of \(x\) and \(y\) wrong way round if sketch gives correct coordinates. Only if graph drawn |
| Crosses \(y\)-axis at \((0, -75)\) | B1 | Allow \(y\) value alone; do not condone 75 on negative \(y\)-axis. Only if graph drawn |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x =)\ 3,\ 20\) | B1ft | Follow through on \(4\times\) their \(x\) intercepts; allow \((3,0)\), \((20,0)\); ignore \((0,-75)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((p =)\ 75\) | B1ft | Follow through on their \(y\) intercept; allow if \((y=)\ f(x)+75\) seen; do not allow \(y=75\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(g(x) = \left(4(x+1)-3\right)(x+1-5)^2 = (4x+1)(x-4)^2\) | M1 A1 | M1: attempts \(g(x)=\left(4(x+1)-3\right)(x+1-5)^2\), condoning slips; award for sight of \(x+1\) embedded or form \((4x+a)(x-4)^2\). Alternatively expand and replace \(x\) with \(x+1\). A1: \((4x+1)(x-4)^2\) or simplified equivalent e.g. \(4x^3-31x^2+56x+16\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(16\) | B1 | Accept \((0,16)\); note \(f(1)=(4\times1-3)(1-5)^2=16\) |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Shape for positive cubic (one max, one min) | B1 | Any position, condone no axes, condone cusp-like minimum |
| Cuts $x$-axis at $\left(\frac{3}{4}, 0\right)$ and meets at $(5, 0)$ | B1 | Graph should not stop or cross at $(5,0)$; allow just $x$ values; condone slip of $x$ and $y$ wrong way round if sketch gives correct coordinates. **Only if graph drawn** |
| Crosses $y$-axis at $(0, -75)$ | B1 | Allow $y$ value alone; do not condone 75 on negative $y$-axis. **Only if graph drawn** |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x =)\ 3,\ 20$ | B1ft | Follow through on $4\times$ their $x$ intercepts; allow $(3,0)$, $(20,0)$; ignore $(0,-75)$ |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(p =)\ 75$ | B1ft | Follow through on their $y$ intercept; allow if $(y=)\ f(x)+75$ seen; do not allow $y=75$ |
### Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $g(x) = \left(4(x+1)-3\right)(x+1-5)^2 = (4x+1)(x-4)^2$ | M1 A1 | M1: attempts $g(x)=\left(4(x+1)-3\right)(x+1-5)^2$, condoning slips; award for sight of $x+1$ embedded or form $(4x+a)(x-4)^2$. Alternatively expand and replace $x$ with $x+1$. A1: $(4x+1)(x-4)^2$ or simplified equivalent e.g. $4x^3-31x^2+56x+16$ |
### Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $16$ | B1 | Accept $(0,16)$; note $f(1)=(4\times1-3)(1-5)^2=16$ |
---10. The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$, where
$$f ( x ) = ( 4 x - 3 ) ( x - 5 ) ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Sketch $C _ { 1 }$ showing the coordinates of any point where the curve touches or crosses the coordinate axes.
\item Hence or otherwise
\begin{enumerate}[label=(\roman*)]
\item find the values of $x$ for which $\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0$
\item find the value of the constant $p$ such that the curve with equation $y = \mathrm { f } ( x ) + p$ passes through the origin.
A second curve $C _ { 2 }$ has equation $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find, in simplest form, $\mathrm { g } ( x )$. You may leave your answer in a factorised form.
\item Hence, or otherwise, find the $y$ intercept of curve $C _ { 2 }$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2020 Q10 [8]}}