| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions for differential equations |
| Difficulty | Standard +0.3 This is a standard two-part question combining routine partial fractions decomposition (with a repeated linear factor) and separable differential equations. Part (i) is textbook practice, and part (ii) requires separation of variables, integration using the partial fractions result, and applying an initial condition—all well-rehearsed A-level techniques with no novel problem-solving required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{A}{(x+1)} + \frac{B}{(x-2)} + \frac{C}{(x-2)^2}\) | B1 | AO 3.1a |
| \(x^2 - 8x + 9 = A(x-2)^2 + B(x+1)(x-2) + C(x+1)\) | M1 | AO 2.1 |
| \(A = 2\); \(B = -1\); \(C = -1\) giving \(\frac{2}{(x+1)} - \frac{1}{(x-2)} - \frac{1}{(x-2)^2}\) | A1, A1, A1 [5] | AO 1.1, 1.1, 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\frac{dy}{y} = \int\frac{x^2-8x+9}{(x+1)(x-2)^2}\,dx\) soi | M1* | AO 3.1a |
| use of their partial fractions in integration | M1* | AO 2.1 |
| \(\ln | y | = 2\ln |
| A1 | AO 1.1 | A1 for \(\frac{1}{x-2}\) FT their \(\frac{k}{(x-2)^2}\) |
| substitution of \(y=16\) and \(x=3\); expression must include \(+c\) and must include at least one natural log term; may be awarded after exponentiating; NB \(c=-1\) | M1dep* | AO 1.1 |
| correctly exponentiate both sides of their equation | M1 | AO 1.1 |
| \(y = \frac{(x+1)^2}{x-2}e^{\frac{1}{x-2}}\) oe eg \(\frac{(x+1)^2}{x-2}e^{\frac{1}{x-2}}e^{-1}\) | A1 [7] | AO 2.1 |
## Question 17(i):
$\frac{A}{(x+1)} + \frac{B}{(x-2)} + \frac{C}{(x-2)^2}$ | B1 | AO 3.1a | may be seen later
$x^2 - 8x + 9 = A(x-2)^2 + B(x+1)(x-2) + C(x+1)$ | M1 | AO 2.1 |
$A = 2$; $B = -1$; $C = -1$ giving $\frac{2}{(x+1)} - \frac{1}{(x-2)} - \frac{1}{(x-2)^2}$ | A1, A1, A1 [5] | AO 1.1, 1.1, 1.1 |
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## Question 17(ii):
$\int\frac{dy}{y} = \int\frac{x^2-8x+9}{(x+1)(x-2)^2}\,dx$ soi | M1* | AO 3.1a | allow omission of integral signs and/or omission of $dy$ and/or $dx$
use of their partial fractions in integration | M1* | AO 2.1 | allow one sign error and/or one coefficient error; condone use of brackets instead of modulus signs; these two A marks are only available following the award of **both** M marks
$\ln|y| = 2\ln|x+1| - \ln|x-2| + \frac{1}{x-2} + c$ | A1 | AO 1.1 | **A1** for any correct natural log integral on RHS FT their $\frac{2}{x+1}$ or their $\frac{-1}{x-2}$
| A1 | AO 1.1 | **A1** for $\frac{1}{x-2}$ FT their $\frac{k}{(x-2)^2}$
substitution of $y=16$ and $x=3$; expression must include $+c$ and must include at least one natural log term; may be awarded after exponentiating; NB $c=-1$ | M1dep* | AO 1.1 |
correctly exponentiate both sides of their equation | M1 | AO 1.1 |
$y = \frac{(x+1)^2}{x-2}e^{\frac{1}{x-2}}$ oe eg $\frac{(x+1)^2}{x-2}e^{\frac{1}{x-2}}e^{-1}$ | A1 [7] | AO 2.1 |
The image provided appears to be only the **back/contact page** of an OCR mark scheme document — it contains only OCR's address, contact details, and legal information. There is **no mark scheme content** (no questions, answers, mark allocations, or guidance notes) visible on this page.
To extract mark scheme content, please upload the **interior pages** of the mark scheme document that contain the actual question-by-question marking guidance.17 (i) Express $\frac { \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } }$ in partial fractions.\\
(ii) Express $y$ in terms of $x$ given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \text { and } y = 16 \text { when } x = 3 .$$
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR MEI Paper 2 2018 Q17 [12]}}