| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Polynomial Identity Matching |
| Difficulty | Moderate -0.8 Part (a) is straightforward polynomial expansion requiring basic algebraic manipulation to match coefficients. Part (b) involves routine integration after dividing each term by 5√x and applying standard power rule - no problem-solving insight needed, just mechanical application of techniques. This is easier than average A-level questions which typically require more conceptual understanding or multi-step reasoning. |
| Spec | 1.03c Straight line models: in variety of contexts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 2\) | B1 | Which may be embedded. |
| \(b = -3\) | B1 | Which may be embedded. If contradiction between stated and embedded values, embedded takes precedence. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any two terms of \(\int \frac{2x^3 - 3x^2 - 32x - 15}{5\sqrt{x}}\,dx = \int \frac{2}{5}x^{\frac{5}{2}} - \frac{3}{5}x^{\frac{3}{2}} - \frac{32}{5}x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}\,dx\) | M1A1 | M1 for attempt to write as sum of terms — award for any term with correct index from correct working. A1 for any two correct unsimplified or simplified terms (indices must be processed). |
| \(x^n \to x^{n+1}\) | M1 | Increases power of any non-integer term by 1. Cannot be awarded for just increasing power in numerator or denominator only. Index does not need to be processed. |
| \(\frac{4}{35}x^{\frac{7}{2}} - \frac{6}{25}x^{\frac{5}{2}} - \frac{64}{15}x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + c\) | A1A1 | First A1: any two terms correct (unsimplified or simplified). Second A1: all terms correct and simplified on one line including \(+c\). Allow exact equivalents but not rounded decimals (\(\frac{4}{35}\) must be a fraction). Accept e.g. \(\frac{4}{35}x^3\sqrt{x}\). Withhold if integral sign and \(dx\) remain or spurious notation present. |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 2$ | B1 | Which may be embedded. |
| $b = -3$ | B1 | Which may be embedded. If contradiction between stated and embedded values, embedded takes precedence. |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any two terms of $\int \frac{2x^3 - 3x^2 - 32x - 15}{5\sqrt{x}}\,dx = \int \frac{2}{5}x^{\frac{5}{2}} - \frac{3}{5}x^{\frac{3}{2}} - \frac{32}{5}x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}\,dx$ | M1A1 | M1 for attempt to write as sum of terms — award for any term with correct index from correct working. A1 for any two correct unsimplified or simplified terms (indices must be processed). |
| $x^n \to x^{n+1}$ | M1 | Increases power of any non-integer term by 1. Cannot be awarded for just increasing power in numerator or denominator only. Index does not need to be processed. |
| $\frac{4}{35}x^{\frac{7}{2}} - \frac{6}{25}x^{\frac{5}{2}} - \frac{64}{15}x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + c$ | A1A1 | First A1: any two terms correct (unsimplified or simplified). Second A1: all terms correct and simplified on one line including $+c$. Allow exact equivalents but not rounded decimals ($\frac{4}{35}$ must be a fraction). Accept e.g. $\frac{4}{35}x^3\sqrt{x}$. Withhold if integral sign and $dx$ remain or spurious notation present. |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$
where $a$ and $b$ are constants,\\
(a) find the value of $a$ and the value of $b$.\\
(b) Hence find
$$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$
writing each term in simplest form.
\hfill \mbox{\textit{Edexcel P1 2022 Q2 [7]}}