| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2016 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions in differential equations |
| Difficulty | Challenging +1.2 This question combines partial fractions with separable differential equations, requiring multiple techniques but following a standard template. Part (a) is routine partial fractions with a repeated linear factor. Part (b) requires recognizing the separable form, integrating using the partial fractions result, and manipulating to the required form—all standard C4 material with no novel insight needed, though the multi-step nature and algebraic manipulation elevate it slightly above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{3x^2-4}{x^2(3x-2)} \equiv \dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{3x-2}\) | ||
| \(\dfrac{2}{x^2}\), \(\dfrac{-6}{3x-2}\) (\(B=2\), \(C=-6\)) | B1, B1 | For either \(+\dfrac{2}{x^2}\) or \(\dfrac{-6}{3x-2}\) being one partial fraction; then for both |
| \(3x^2-4 \equiv Ax(3x-2)+B(3x-2)+Cx^2 \Rightarrow A=..\) | M1 | Need three terms; correct method comparing coefficients or substituting |
| \(\dfrac{3}{x}\) (\(A=3\)) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\dfrac{1}{y}\,dy = \int\dfrac{3x^2-4}{x^2(3x-2)}\,dx\) | B1 | Separates variables correctly; no integral signs needed |
| \(\ln y = \int\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{3x-2}\,dx\) | M1 | Integrates LHS to give \(\ln y\); uses partial fractions from (a) |
| \(= A\ln x - \dfrac{B}{x}+\dfrac{C}{3}\ln(3x-2)\) \((+k)\) | M1A1ft | Two ln terms and one reciprocal term; condone missing bracket on \(\ln(3x-2)\) |
| \(y = e^{A\ln x - \frac{B}{x}+\frac{C}{3}\ln(3x-2)+D}\) or \(y=De^{A\ln x-\frac{B}{x}+\frac{C}{3}\ln(3x-2)}\) | M1 | Undoing logs correctly; need constant of integration |
| \(y = Kx^3(3x-2)^{-2}e^{-\frac{2}{x}}\) or \(\dfrac{Kx^3e^{-\frac{2}{x}}}{(3x-2)^2}\) or \(\dfrac{e^k x^3 e^{-\frac{2}{x}}}{(3x-2)^2}\) | A1cso |
# Question 9(a) - Partial Fractions
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{3x^2-4}{x^2(3x-2)} \equiv \dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{3x-2}$ | | |
| $\dfrac{2}{x^2}$, $\dfrac{-6}{3x-2}$ ($B=2$, $C=-6$) | B1, B1 | For either $+\dfrac{2}{x^2}$ or $\dfrac{-6}{3x-2}$ being one partial fraction; then for both |
| $3x^2-4 \equiv Ax(3x-2)+B(3x-2)+Cx^2 \Rightarrow A=..$ | M1 | Need three terms; correct method comparing coefficients or substituting |
| $\dfrac{3}{x}$ ($A=3$) | A1 | |
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# Question 9(b) - Differential Equation
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\dfrac{1}{y}\,dy = \int\dfrac{3x^2-4}{x^2(3x-2)}\,dx$ | B1 | Separates variables correctly; no integral signs needed |
| $\ln y = \int\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{3x-2}\,dx$ | M1 | Integrates LHS to give $\ln y$; uses partial fractions from (a) |
| $= A\ln x - \dfrac{B}{x}+\dfrac{C}{3}\ln(3x-2)$ $(+k)$ | M1A1ft | Two ln terms and one reciprocal term; condone missing bracket on $\ln(3x-2)$ |
| $y = e^{A\ln x - \frac{B}{x}+\frac{C}{3}\ln(3x-2)+D}$ or $y=De^{A\ln x-\frac{B}{x}+\frac{C}{3}\ln(3x-2)}$ | M1 | Undoing logs correctly; need constant of integration |
| $y = Kx^3(3x-2)^{-2}e^{-\frac{2}{x}}$ or $\dfrac{Kx^3e^{-\frac{2}{x}}}{(3x-2)^2}$ or $\dfrac{e^k x^3 e^{-\frac{2}{x}}}{(3x-2)^2}$ | A1cso | |
**Special case (two partial fractions):** B1B1M0A0 in (a); in (b) scores B1,M1,M0,A0,M1,A0 = 5/10
---\begin{enumerate}
\item (a) Express $\frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x - 2 ) }$ in partial fractions.\\
(b) Given that $x > \frac { 2 } { 3 }$, find the general solution of the differential equation
\end{enumerate}
$$x ^ { 2 } ( 3 x - 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \left( 3 x ^ { 2 } - 4 \right)$$
Give your answer in the form $y = \mathrm { f } ( x )$.\\
\hfill \mbox{\textit{Edexcel C34 2016 Q9 [10]}}