| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve transformed function equations |
| Difficulty | Moderate -0.3 This C1 question involves standard transformations of functions and basic differentiation. Part (a) requires substituting coordinates into transformed functions, part (b) is routine product rule differentiation, and part (c) involves solving a quadratic equation after setting derivatives equal. All techniques are straightforward applications of core methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k=(-5)^2\times3=75\) | M1A1 | M1: Attempts to find the \(y\)-intercept. Accept \((-5)^2\times3\) with or without bracket. If they expand \(f(x)\) to polynomial form they must select their constant term. May be implied by sight of 75 on diagram. A1: \(k=75\). Must be clearly identified as \(k\). Allow from incorrect/incomplete expansion as long as constant \(k=75\) is obtained. Do not isw if 75 followed by \(k=-75\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(c=\frac{5}{2}\) only | B1 | \(c=\frac{5}{2}\) oe (and no other values). Do not award just from diagram — must be stated as the value of \(c\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x)=(2x-5)^2(x+3)=(4x^2-20x+25)(x+3)=4x^3-8x^2-35x+75\) | M1 | Attempts \(f(x)\) as cubic polynomial by squaring first bracket and multiplying by linear bracket, or expands \((2x-5)(x+3)\) then multiplies by \(2x-5\). Must be seen or used in (b). Allow poor squaring e.g. \((2x-5)^2=4x^2\pm25\). |
| \(f'(x)=12x^2-16x-35\)* | M1A1* | M1: Reduces powers by 1 in all terms including any constant \(\to 0\). A1: Correct proof. Withhold if there have been any errors including missing brackets earlier e.g. \((2x-5)^2(x+3)=4x^2-20x+25(x+3)=\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(3)=12\times3^2-16\times3-35\) | M1 | Substitutes \(x=3\) into their \(f'(x)\) or the given \(f'(x)\). Must be a changed function, not \(f(x)\). |
| \(12x^2-16x-35=\text{'25'}\) | dM1 | Sets their \(f'(x)\) or the given \(f'(x)=\) their \(f'(3)\) with a consistent \(f'\). Dependent on previous method mark. |
| \(12x^2-16x-60=0\) | A1 cso | \(12x^2-16x-60=0\) or equivalent 3-term quadratic e.g. \(12x^2-16x=60\). Must come from correct work — they must be using the given \(f'(x)\). |
| \((x-3)(12x+20)=0 \Rightarrow x=\ldots\) | ddM1 | Solves 3-term quadratic by suitable method. Dependent on both previous method marks. |
| \(x=-\frac{5}{3}\) | A1 cso | \(x=-\frac{5}{3}\) oe clearly identified. If \(x=3\) also given and not rejected, mark is withheld. Allow \(-1.6\) recurring as long as it is clear (dot above the 6). Must come from correct work using the given \(f'(x)\). |
| Alt (b) Product rule: \(f(x)=(2x-5)^2(x+3)\Rightarrow f'(x)=(2x-5)^2\times1+(x+3)\times4(2x-5)\). M1: Attempts product rule to give form \(p(2x-5)^2+q(x+3)(2x-5)\). M1: Multiplies out and collects terms. A1: \(f'(x)=12x^2-16x-35\)* | M1, M1A1* |
## Question 10(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k=(-5)^2\times3=75$ | M1A1 | M1: Attempts to find the $y$-intercept. Accept $(-5)^2\times3$ with or without bracket. If they expand $f(x)$ to polynomial form they must select their constant term. May be implied by sight of 75 on diagram. A1: $k=75$. Must be clearly identified as $k$. Allow from incorrect/incomplete expansion as long as constant $k=75$ is obtained. Do not isw if 75 followed by $k=-75$. |
---
## Question 10(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $c=\frac{5}{2}$ only | B1 | $c=\frac{5}{2}$ oe (and no other values). Do not award just from diagram — must be stated as the value of $c$. |
---
## Question 10(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x)=(2x-5)^2(x+3)=(4x^2-20x+25)(x+3)=4x^3-8x^2-35x+75$ | M1 | Attempts $f(x)$ as cubic polynomial by squaring first bracket and multiplying by linear bracket, or expands $(2x-5)(x+3)$ then multiplies by $2x-5$. Must be seen or used in (b). Allow poor squaring e.g. $(2x-5)^2=4x^2\pm25$. |
| $f'(x)=12x^2-16x-35$* | M1A1* | M1: Reduces powers by 1 in **all** terms including any constant $\to 0$. A1: Correct proof. Withhold if there have been any errors including missing brackets earlier e.g. $(2x-5)^2(x+3)=4x^2-20x+25(x+3)=\ldots$ |
---
## Question 10(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(3)=12\times3^2-16\times3-35$ | M1 | Substitutes $x=3$ into their $f'(x)$ or the given $f'(x)$. Must be a changed function, not $f(x)$. |
| $12x^2-16x-35=\text{'25'}$ | dM1 | Sets their $f'(x)$ or the given $f'(x)=$ their $f'(3)$ with a consistent $f'$. Dependent on previous method mark. |
| $12x^2-16x-60=0$ | A1 cso | $12x^2-16x-60=0$ or equivalent 3-term quadratic e.g. $12x^2-16x=60$. Must come from correct work — they must be using the **given** $f'(x)$. |
| $(x-3)(12x+20)=0 \Rightarrow x=\ldots$ | ddM1 | Solves 3-term quadratic by suitable method. Dependent on both previous method marks. |
| $x=-\frac{5}{3}$ | A1 cso | $x=-\frac{5}{3}$ oe clearly identified. If $x=3$ also given and not rejected, mark is withheld. Allow $-1.6$ recurring as long as it is clear (dot above the 6). Must come from correct work using the **given** $f'(x)$. |
**Alt (b) Product rule:** $f(x)=(2x-5)^2(x+3)\Rightarrow f'(x)=(2x-5)^2\times1+(x+3)\times4(2x-5)$. M1: Attempts product rule to give form $p(2x-5)^2+q(x+3)(2x-5)$. M1: Multiplies out and collects terms. A1: $f'(x)=12x^2-16x-35$* | M1, M1A1* | |10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c1b0a49d-9def-4289-a4cd-288991f67caf-24_666_1195_260_370}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve $y = \mathrm { f } ( x ) , x \in \mathbb { R }$, where
$$f ( x ) = ( 2 x - 5 ) ^ { 2 } ( x + 3 )$$
\begin{enumerate}[label=(\alph*)]
\item Given that
\begin{enumerate}[label=(\roman*)]
\item the curve with equation $y = \mathrm { f } ( x ) - k , x \in \mathbb { R }$, passes through the origin, find the value of the constant $k$,
\item the curve with equation $y = \mathrm { f } ( x + c ) , x \in \mathbb { R }$, has a minimum point at the origin, find the value of the constant $c$.
\end{enumerate}\item Show that $\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 16 x - 35$
Points $A$ and $B$ are distinct points that lie on the curve $y = \mathrm { f } ( x )$.\\
The gradient of the curve at $A$ is equal to the gradient of the curve at $B$.\\
Given that point $A$ has $x$ coordinate 3
\item find the $x$ coordinate of point $B$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c1b0a49d-9def-4289-a4cd-288991f67caf-28_2630_1826_121_121}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2017 Q10 [11]}}