| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions then differentiate |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring standard partial fractions decomposition followed by term-by-term differentiation. Part (ii) requires showing all terms in f'(x) are negative, which is routine algebraic verification. Slightly easier than average as it follows a predictable template with no conceptual surprises. |
| Spec | 1.02y Partial fractions: decompose rational functions1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{A}{x+2} + \frac{B}{x-3}\) | M1 | s.o.i. in answer |
| \(A=1\) and \(B=2\) | A1 | 2 for both |
| \(-A(x+2)^2 - B(x-3)^2\) | M1 | f.t. |
| Convincing statement that each denom \(> 0\) State whole exp \(< 0\) | AG | |
| Accept \(\geq 0\). Do not accept \(x^2 > 0\). Dep on previous 4 marks. |
$\frac{A}{x+2} + \frac{B}{x-3}$ | M1 | s.o.i. in answer
$A=1$ and $B=2$ | A1 | 2 for both
$-A(x+2)^2 - B(x-3)^2$ | M1 | f.t.
Convincing statement that each denom $> 0$ State whole exp $< 0$ | AG |
| | | Accept $\geq 0$. Do not accept $x^2 > 0$. Dep on previous 4 marks.
---1 The equation of a curve is $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence find $\mathrm { f } ^ { \prime } ( x )$ and deduce that the gradient of the curve is negative at all points on the curve.
\hfill \mbox{\textit{OCR C4 2007 Q1 [5]}}