| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths Pure question requiring algebraic division to convert an improper fraction, then integration using the resulting form. Part (a) is routine polynomial division, and part (b) is standard integration of polynomial plus logarithmic terms with simple substitution of limits. While it's a Further Maths topic, the techniques are mechanical with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Find \(P\) and show \(Q = 8\) where \(\frac{x^2 - 1}{x + 3} \equiv x + P + \frac{Q}{x + 3}\) | (2) | M1 Multiplies by \((x + 3)\) and attempts to find values for \(P\) and \(Q\), or attempts to divide \(x^2 - 1\) by \(x + 3\) and obtains linear quotient and constant remainder; A1 \(P = -3\), \(Q = 8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Find exact area of \(R\), writing answer in form \(a \ln 2\) | (4) | b. M1 Integrates expression of form \(x + P + \frac{Q}{x+3}\) to obtain \(\frac{x^2}{2} + Px + k \ln(x + 3)\); A1 Correct integration; M1 Substitutes both limits 1 and 5 into \(\frac{x^2}{2} + Px + k \ln(x + 3)\) and subtracts either way round with fully correct log work to combine two log terms leading to answer of form \(a \ln b\); A1 \(8 \ln 2\) |
**Part (a):**
Find $P$ and show $Q = 8$ where $\frac{x^2 - 1}{x + 3} \equiv x + P + \frac{Q}{x + 3}$ | (2) | M1 Multiplies by $(x + 3)$ and attempts to find values for $P$ and $Q$, or attempts to divide $x^2 - 1$ by $x + 3$ and obtains linear quotient and constant remainder; A1 $P = -3$, $Q = 8$
**Part (b):**
Find exact area of $R$, writing answer in form $a \ln 2$ | (4) | b. M1 Integrates expression of form $x + P + \frac{Q}{x+3}$ to obtain $\frac{x^2}{2} + Px + k \ln(x + 3)$; A1 Correct integration; M1 Substitutes both limits 1 and 5 into $\frac{x^2}{2} + Px + k \ln(x + 3)$ and subtracts either way round with fully correct log work to combine two log terms leading to answer of form $a \ln b$; A1 $8 \ln 2$
---5. a. Given that
$$\frac { x ^ { 2 } - 1 } { x + 3 } \equiv x + P + \frac { Q } { x + 3 }$$
find the value of the constant $P$ and show that $Q = 8$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-07_1082_1271_1363_415}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The curve $C$ has equation $y = \mathrm { g } ( x )$, where
$$\mathrm { g } ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 } \quad x > - 3$$
Figure 3 shows a sketch of the curve $C$.\\
The region $R$, shown shaded in Figure 4, is bounded by $C$, the $x$-axis and the line with equation $x = 5$.\\
b. Find the exact area of $R$, writing your answer in the form $a \ln 2$, where $a$ is constant to be found.\\
(4)\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q5 [6]}}