| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Partial fractions then inverse trig integration |
| Difficulty | Standard +0.8 This FP2 question requires understanding of asymptotes, solving quadratic inequalities in y (involving discriminant analysis), and integration combining logarithmic and arctangent forms. Part (ii) is the most demanding, requiring students to rearrange as a quadratic in x and apply discriminant conditions—a technique that's non-routine for many students. The integration in (iii) is standard FP2 fare but requires careful algebraic manipulation. Overall, this is moderately challenging for Further Maths, above average difficulty but not exceptional. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Require denom. = 0. Explain why denom. \(\neq 0\) | B1, B1 | Attempt to solve, explain always \(> 0\) etc. |
| (ii) Set up quadratic in \(x\). Get \(2yx^2-4x+(2a^2y+3a) = 0\). Use \(b^2 \geq 4ac\) for real \(x\). Attempt to solve their inequality. Get \(y > \frac{1}{2a}\) and \(y < \frac{2}{a}\) | M1, A1, M1, M1, A1 | Produce quadratic inequality in \(y\) from their quad.; allow use of = or <; Factors or formula; Justified from graph; SC Attempt diff. by quot./product rule. Solve \(dy/dx = 0\) for two values of \(x\). Get \(x=2a\) and \(x=-a/2\). Attempt to find two \(y\) values. Get correct inequalities (graph used to justify them) |
| (iii) Split into two separate integrals. Get \(k \ln(x^2+a^2)\). Get \(k_1 \tan^{-1}(x/a)\). Use limits and attempt to simplify. Get \(\ln 2.5 - 1.5 \tan^{-1}2 + 3\pi/8\) | M1, A1, A1, M1, A1 | Or \(p\ln(2x^2+2a^2)\); \(k_1\) not involving \(a\); AEEF; SC Sub. \(x = a\tan\theta\) and \(dx = a\sec^2\theta \, d\theta\). Reduce to \(\int p\tan\theta - p_1 \, d\theta\) (ignore limits here). Integrate to \(p\ln(\sec\theta)-p_1\theta\). Use limits (old or new) and attempt to simplify. Get answer above |
(i) Require denom. = 0. Explain why denom. $\neq 0$ | B1, B1 | Attempt to solve, explain always $> 0$ etc.
(ii) Set up quadratic in $x$. Get $2yx^2-4x+(2a^2y+3a) = 0$. Use $b^2 \geq 4ac$ for real $x$. Attempt to solve their inequality. Get $y > \frac{1}{2a}$ and $y < \frac{2}{a}$ | M1, A1, M1, M1, A1 | Produce quadratic inequality in $y$ from their quad.; allow use of = or <; Factors or formula; Justified from graph; SC Attempt diff. by quot./product rule. Solve $dy/dx = 0$ for two values of $x$. Get $x=2a$ and $x=-a/2$. Attempt to find two $y$ values. Get correct inequalities (graph used to justify them)
(iii) Split into two separate integrals. Get $k \ln(x^2+a^2)$. Get $k_1 \tan^{-1}(x/a)$. Use limits and attempt to simplify. Get $\ln 2.5 - 1.5 \tan^{-1}2 + 3\pi/8$ | M1, A1, A1, M1, A1 | Or $p\ln(2x^2+2a^2)$; $k_1$ not involving $a$; AEEF; SC Sub. $x = a\tan\theta$ and $dx = a\sec^2\theta \, d\theta$. Reduce to $\int p\tan\theta - p_1 \, d\theta$ (ignore limits here). Integrate to $p\ln(\sec\theta)-p_1\theta$. Use limits (old or new) and attempt to simplify. Get answer above | M1, A1, A1, M1, A1A curve has equation
$$y = \frac{4x - 3a}{2(x^2 + a^2)},$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Explain why the curve has no asymptotes parallel to the $y$-axis. [2]
\item Find, in terms of $a$, the set of values of $y$ for which there are no points on the curve. [5]
\item Find the exact value of $\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx$, showing that it is independent of $a$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q9 [12]}}