| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring partial fractions (standard technique), asymptote identification, and curve sketching of y²=f(x). While part (iv) requires careful analysis of where f(x)≥0 and consideration of ±√f(x), the techniques are all within FP2 syllabus and follow a structured progression. The improper fraction adds mild complexity but is routine for this level. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{x^2 - 25}{(x-1)(x+2)} = A + \frac{B}{(x-1)} + \frac{C}{(x+2)}\) | M1 | Splitting in correct way to give partial fractions (may be seen anywhere) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{x^2-25}{(x-1)(x+2)} = 1 - \frac{8}{(x-1)} + \frac{7}{(x+2)}\) | B1 | For \(A\) |
| A1 | For \(B\) | |
| A1 | For \(C\) |
Total: 4
| Answer | Marks |
|---|---|
| \(x = 1,\ x = -2\) | B1 |
| \(y = 1\) | B1 |
Total: 2
| Answer | Marks |
|---|---|
| \(y = 1 \Rightarrow (x-1)(x+2) = x^2 - 25\) | M1 |
| \(x^2 + x - 2 = x^2 - 25 \Rightarrow x = -23\) | A1 |
Total: 2
| Answer | Marks | Guidance |
|---|---|---|
| 4 bits as shown, roughly symmetric about axes, approaching asymptotes | B1 | |
| Lh side crosses asymptotes and upper section approaches from above and lower section approaches from below | B1 | Ignore any graph of \(y = f(x)\) |
Total: 2
## Question 7:
### Part (i):
$\frac{x^2 - 25}{(x-1)(x+2)} = A + \frac{B}{(x-1)} + \frac{C}{(x+2)}$ | **M1** | Splitting in correct way to give partial fractions (may be seen anywhere)
$x^2 - 25 = A(x-1)(x+2) + B(x+2) + C(x-1)$
3 processes of equating coefficients or substituting:
$x = 1 \Rightarrow -24 = 3B \Rightarrow B = -8$
$x = -2 \Rightarrow -21 = -3C \Rightarrow C = 7$
coeff of $x^2$: $A = 1$
$\frac{x^2-25}{(x-1)(x+2)} = 1 - \frac{8}{(x-1)} + \frac{7}{(x+2)}$ | **B1** | For $A$
| **A1** | For $B$
| **A1** | For $C$
Total: **4**
### Part (ii):
$x = 1,\ x = -2$ | **B1** |
$y = 1$ | **B1** |
Total: **2**
### Part (iii):
$y = 1 \Rightarrow (x-1)(x+2) = x^2 - 25$ | **M1** |
$x^2 + x - 2 = x^2 - 25 \Rightarrow x = -23$ | **A1** |
Total: **2**
### Part (iv):
4 bits as shown, roughly symmetric about axes, approaching asymptotes | **B1** |
Lh side crosses asymptotes and upper section approaches from above and lower section approaches from below | **B1** | Ignore any graph of $y = f(x)$
Total: **2**
---7 It is given that $\mathrm { f } ( x ) = \frac { x ^ { 2 } - 25 } { ( x - 1 ) ( x + 2 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Write down the equations of the asymptotes of the curve $y = \mathrm { f } ( x )$.\\
(iii) Find the value of $x$ where the graph of $y = \mathrm { f } ( x )$ cuts the horizontal asymptote.\\
(iv) Sketch the graph of $y ^ { 2 } = \mathrm { f } ( x )$.
\hfill \mbox{\textit{OCR FP2 2015 Q7 [10]}}