| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.3 This is a standard multi-part P2 question combining factor/remainder theorem with differentiation. Part (a) is direct application of remainder theorem, part (b) involves solving simultaneous equations, parts (c-d) are routine differentiation and finding stationary points. All techniques are straightforward with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a(-4)^3-(-4)^2+b(-4)+4=-108\) | M1 | Attempts \(f(-4)=-108\). Score when "–4" embedded in equation or 2 correct terms (excluding "+4") on lhs |
| \(-64a-16-4b+4=-108 \Rightarrow 16a+b=24\) | A1* | Correct equation with no errors and at least one line of intermediate working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a\left(\frac{1}{2}\right)^3-\left(\frac{1}{2}\right)^2+b\left(\frac{1}{2}\right)+4=0\) | M1 | Attempts \(f\!\left(\frac{1}{2}\right)=0\). Score when "\(\frac{1}{2}\)" embedded or 2 correct terms (excluding "+4") on lhs |
| Solve \(16a+b=24\), \(a+4b=-30\) simultaneously | M1 | Attempts to solve simultaneously. May be implied by values of \(a\) and \(b\) |
| \(a=2,\ b=-8\) | A1 | Correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x)=2x^3-x^2-8x+4 \Rightarrow f'(x)=6x^2-2x-8\) | B1ft | Correct derivative (follow through their \(a\) and \(b\)). Allow unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6x^2-2x-8=0 \Rightarrow (3x-4)(x+1)=0 \Rightarrow x=...\) | M1 | Sets \(f'(x)=0\) and solves 3-term quadratic |
| \(x=\frac{4}{3},\ -1 \Rightarrow y=...\) | M1 | Uses at least one \(x\) value to find \(y\) using \(f(x)\), from attempt to solve \(f'(x)=0\) |
| \(\left(\frac{4}{3},-\frac{100}{27}\right)\) or \((-1, 9)\) | A1 | One correct point. Fractional coordinates must be exact |
| \(\left(\frac{4}{3},-\frac{100}{27}\right)\) and \((-1, 9)\) | A1 | Both correct points. Depends on both previous M marks. Fully correct answers with no working score 4/4 following correct part (c) |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a(-4)^3-(-4)^2+b(-4)+4=-108$ | M1 | Attempts $f(-4)=-108$. Score when "–4" embedded in equation or 2 correct terms (excluding "+4") on lhs |
| $-64a-16-4b+4=-108 \Rightarrow 16a+b=24$ | A1* | Correct equation with no errors and at least one line of intermediate working |
**(2 marks)**
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a\left(\frac{1}{2}\right)^3-\left(\frac{1}{2}\right)^2+b\left(\frac{1}{2}\right)+4=0$ | M1 | Attempts $f\!\left(\frac{1}{2}\right)=0$. Score when "$\frac{1}{2}$" embedded or 2 correct terms (excluding "+4") on lhs |
| Solve $16a+b=24$, $a+4b=-30$ simultaneously | M1 | Attempts to solve simultaneously. May be implied by values of $a$ and $b$ |
| $a=2,\ b=-8$ | A1 | Correct values |
**(3 marks)**
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x)=2x^3-x^2-8x+4 \Rightarrow f'(x)=6x^2-2x-8$ | B1ft | Correct derivative (follow through their $a$ and $b$). Allow unsimplified |
**(1 mark)**
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6x^2-2x-8=0 \Rightarrow (3x-4)(x+1)=0 \Rightarrow x=...$ | M1 | Sets $f'(x)=0$ and solves 3-term quadratic |
| $x=\frac{4}{3},\ -1 \Rightarrow y=...$ | M1 | Uses at least one $x$ value to find $y$ using $f(x)$, from attempt to solve $f'(x)=0$ |
| $\left(\frac{4}{3},-\frac{100}{27}\right)$ **or** $(-1, 9)$ | A1 | One correct point. Fractional coordinates must be exact |
| $\left(\frac{4}{3},-\frac{100}{27}\right)$ **and** $(-1, 9)$ | A1 | Both correct points. Depends on both previous M marks. Fully correct answers with no working score 4/4 following correct part (c) |
**(4 marks) — Total 10**
---3.
$$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$
where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x + 4$ ), the remainder is - 108
\begin{enumerate}[label=(\alph*)]
\item Use the remainder theorem to show that
$$16 a + b = 24$$
Given also that ( $2 x - 1$ ) is a factor of $\mathrm { f } ( x )$,
\item find the value of $a$ and the value of $b$.
\item Find $\mathrm { f } ^ { \prime } ( x )$.
\item Hence find the exact coordinates of the stationary points of the curve with equation $y = \mathrm { f } ( x )$.
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VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
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\hfill \mbox{\textit{Edexcel P2 2020 Q3 [10]}}