| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Derive equation from calculus condition |
| Difficulty | Moderate -0.3 This is a structured multi-part question that guides students through standard calculus techniques: finding coordinates by substitution, computing chord gradient using difference quotient, taking the limit as h→0 to find tangent gradient, and applying the Newton-Raphson formula. While it requires careful algebraic manipulation and understanding of first principles, each part follows a predictable template with clear scaffolding, making it slightly easier than average for an FP1 question. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07a Derivative as gradient: of tangent to curve1.07g Differentiation from first principles: for small positive integer powers of x1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((-1+h)^3 = -1 + 3h - 3h^2 + h^3\) | B1 | PI |
| \(y_B = (-1+3h-3h^2+h^3)+1-h+1\) | B1F | ft numerical error |
| \(= 1 + 2h - 3h^2 + h^3\) | B1 | convincingly shown (AG) |
| Subtotal | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Subtraction of 1 and division by \(h\) | M1M1 | |
| Gradient of chord \(= 2 - 3h + h^2\) | A1 | |
| Subtotal | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| As \(h \to 0\), gr(chord) \(\to\) gr(tgt) \(= 2\) | E1B1F | E0 if '\(h=0\)' used; ft wrong value of \(p\) |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x_2 = -1 - \frac{1}{2} = -1.5\) | M1 | |
| A1F | ft wrong gradient | |
| Subtotal | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Tangent at \(A\) drawn | M1 | |
| \(\alpha\) and \(x_2\) shown correctly | A1 | dep't only on the last M1 |
| Subtotal | 2 | |
| Total | 12 |
## Question 7(a)(i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(-1+h)^3 = -1 + 3h - 3h^2 + h^3$ | B1 | PI |
| $y_B = (-1+3h-3h^2+h^3)+1-h+1$ | B1F | ft numerical error |
| $= 1 + 2h - 3h^2 + h^3$ | B1 | convincingly shown (AG) |
| **Subtotal** | **3** | |
## Question 7(a)(ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Subtraction of 1 and division by $h$ | M1M1 | |
| Gradient of chord $= 2 - 3h + h^2$ | A1 | |
| **Subtotal** | **3** | |
## Question 7(a)(iii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| As $h \to 0$, gr(chord) $\to$ gr(tgt) $= 2$ | E1B1F | E0 if '$h=0$' used; ft wrong value of $p$ |
| **Subtotal** | **2** | |
## Question 7(b)(i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x_2 = -1 - \frac{1}{2} = -1.5$ | M1 | |
| | A1F | ft wrong gradient |
| **Subtotal** | **2** | |
## Question 7(b)(ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Tangent at $A$ drawn | M1 | |
| $\alpha$ and $x_2$ shown correctly | A1 | dep't only on the last M1 |
| **Subtotal** | **2** | |
| **Total** | **12** | |
---7 [Figure 1, printed on the insert, is provided for use in this question.]\\
The diagram shows the curve
$$y = x ^ { 3 } - x + 1$$
The points $A$ and $B$ on the curve have $x$-coordinates - 1 and $- 1 + h$ respectively.\\
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the $y$-coordinate of the point $B$ is
$$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
\item Find the gradient of the chord $A B$ in the form
$$p + q h + r h ^ { 2 }$$
where $p , q$ and $r$ are integers.
\item Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at $A$. State the value of this gradient.
\end{enumerate}\item The equation $x ^ { 3 } - x + 1 = 0$ has one real root, $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Taking $x _ { 1 } = - 1$ as a first approximation to $\alpha$, use the Newton-Raphson method to find a second approximation, $x _ { 2 }$, to $\alpha$.
\item On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points $\left( x _ { 2 } , 0 \right)$ and $( \alpha , 0 )$ on your diagram.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q7 [12]}}