| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Topic | Differential equations |
| Type | Separable variables - partial fractions |
| Difficulty | Standard +0.3 This is a standard separable differential equations question with routine preliminary parts. Part (i) is basic partial fractions with a repeated linear factor, part (ii) is polynomial division, and part (iii) combines these results in a separable DE requiring integration of both sides. While it requires multiple techniques and careful algebra across 10 marks total, each component is a textbook exercise with no novel insight required—slightly above average difficulty due to the multi-step nature and potential for algebraic errors. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Either using \(x - 1 = x + 1 - 2\) and then dividing by \((x+1)^2\) or rationalising and equating coefficients, leading to \(\frac{1}{x+1} - \frac{2}{(x+1)^2}\) | M1 A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| A correct process of division, leading to \(2y - 2 + \frac{3}{y+1}\), or equivalent statement | M1 A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at separating the variables in the form \(\int f(y) \, dy = \int g(x) \, dx\) or un-integrated equivalent form | M1 | |
| Correctly using the results of parts (i) and (ii) to obtain: \(\int\left(2y - 2 + \frac{3}{y+1}\right) dy = \int\left(\frac{1}{x+1} - \frac{2}{(x+1)^2}\right) dx\) | A1 | |
| Perform indefinite integrations: \(y^2 - 2y + 3\ln(y+1) = \ln(x+1) + \frac{2}{x+1} + (C)\) | A1 A1 | |
| *Note: The first A1 is for at least one ln term.* | ||
| Correct substitution of \(x = 0\) and \(y = 2\): | M1 | |
| Obtain \(C = 3\ln 3 - 2\) | A1 | [6] |
### (i)
Either using $x - 1 = x + 1 - 2$ and then dividing by $(x+1)^2$ or rationalising and equating coefficients, leading to $\frac{1}{x+1} - \frac{2}{(x+1)^2}$ | M1 A1 | [2]
### (ii)
A correct process of division, leading to $2y - 2 + \frac{3}{y+1}$, or equivalent statement | M1 A1 | [2]
### (iii)
Attempt at separating the variables in the form $\int f(y) \, dy = \int g(x) \, dx$ or un-integrated equivalent form | M1 |
Correctly using the results of parts (i) and (ii) to obtain: $\int\left(2y - 2 + \frac{3}{y+1}\right) dy = \int\left(\frac{1}{x+1} - \frac{2}{(x+1)^2}\right) dx$ | A1 |
Perform indefinite integrations: $y^2 - 2y + 3\ln(y+1) = \ln(x+1) + \frac{2}{x+1} + (C)$ | A1 A1 |
*Note: The first A1 is for at least one ln term.* |
Correct substitution of $x = 0$ and $y = 2$: | M1 |
Obtain $C = 3\ln 3 - 2$ | A1 | [6]\begin{enumerate}[label=(\roman*)]
\item Express $\frac{x-1}{x^2+2x+1}$ in the form $\frac{A}{x+1} + \frac{B}{(x+1)^2}$, where $A$ and $B$ are integers. [2]
\item Find the quotient and remainder when $2y^2 + 1$ is divided by $y + 1$. [2]
\item A curve in the $x$-$y$ plane passes through the point $(0, 2)$ and satisfies the differential equation
$$(2y^2 + 1)(x^2 + 2x + 1)\frac{dy}{dx} = (x - 1)(y + 1).$$
By solving the differential equation find the equation of the curve in implicit form. [6]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2010 Q6 [10]}}