| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Partial Fraction Form via Division |
| Difficulty | Standard +0.3 This is a straightforward C3 question combining polynomial long division (a standard technique) with differentiation and finding a normal line. Part (a) is routine division with coefficient matching, and part (b) is a 'show that' requiring derivative evaluation and normal gradient calculation—both standard procedures with no novel insight required. Slightly easier than average due to the structured nature and 'show that' format. |
| Spec | 1.02y Partial fractions: decompose rational functions1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation |
| Answer | Marks |
|---|---|
| M1 | |
| \(\therefore f(x) = x^2 + 4x - 4 + \frac{2x - 5}{x^2 - 3x + 3}\), where \(A = 4, B = -4, C = 2, D = -5\) | A3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = 2x + 4 + \frac{2x(x^2 - 3x + 3) - (2x - 5)(2x - 3)}{(x^2 - 3x + 3)^2}\) | M1 A2 | |
| At \(x = 1 \Rightarrow y = -2\), grad \(= 5\) | M1 | |
| \(\therefore\) grad of normal \(= -\frac{1}{5}\) | M1 | |
| \(\therefore y + 2 = -\frac{1}{5}(x - 1)\) | M1 | |
| \(5y + 10 = -x + 1\) | ||
| \(x + 5y + 9 = 0\) | A1 | (10 marks) |
**(a)**
$x^2 - 3x + 3 \enclose{longdiv}{x^4 + x^3 - 13x^2 + 26x - 17}$
Working shown:
- $x^4 - 3x^3 + 3x^2$
- $4x^3 - 16x^2 + 26x$
- $4x^3 - 12x^2 + 12x$
- $-4x^2 + 14x - 17$
- $-4x^2 + 12x - 12$
- $2x - 5$
| M1 |
$\therefore f(x) = x^2 + 4x - 4 + \frac{2x - 5}{x^2 - 3x + 3}$, where $A = 4, B = -4, C = 2, D = -5$ | A3 |
**(b)**
$f'(x) = 2x + 4 + \frac{2x(x^2 - 3x + 3) - (2x - 5)(2x - 3)}{(x^2 - 3x + 3)^2}$ | M1 A2 |
At $x = 1 \Rightarrow y = -2$, grad $= 5$ | M1 |
$\therefore$ grad of normal $= -\frac{1}{5}$ | M1 |
$\therefore y + 2 = -\frac{1}{5}(x - 1)$ | M1 |
$5y + 10 = -x + 1$ | |
$x + 5y + 9 = 0$ | A1 | (10 marks)\begin{enumerate}
\item $f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }$.\\
(a) Find the values of the constants $A$, $B$, $C$ and $D$ such that
\end{enumerate}
$$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$
The point $P$ on the curve $y = \mathrm { f } ( x )$ has $x$-coordinate 1.\\
(b) Show that the normal to the curve $y = \mathrm { f } ( x )$ at $P$ has the equation
$$x + 5 y + 9 = 0$$
\hfill \mbox{\textit{Edexcel C3 Q3 [10]}}