| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Intersection of curves via iteration |
| Difficulty | Standard +0.3 This is a multi-part question covering standard A-level techniques: binomial expansion of f(1+h), derivative from first principles, setting up an equation from curve intersection, one iteration of Newton-Raphson with given formula and starting value, and a routine improper integral. Each part is straightforward recall/application with no novel insight required, making it slightly easier than average. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x1.09d Newton-Raphson method4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((1+h)^3 = 1 + 3h + 3h^2 + h^3\) | B1 | |
| \(f(1+h) = 1 + 5h + 4h^2 + h^3\) | M1A1\(\checkmark\) | PI; ft wrong coefficients |
| \(f(1+h) - f(1) = 5h + 4h^2 + h^3\) | A1\(\checkmark\) (4) | ft numerical errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Dividing by \(h\) | M1 | |
| \(f'(1) = 5\) | A1\(\checkmark\) (2) | ft numerical errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(x^2(x+1) = 1\), hence result | B1 (1) | Convincingly shown AG |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(x_2 = 1 - \frac{1}{5} = \frac{4}{5}\) | M1A1\(\checkmark\), A1\(\checkmark\) (3) | ft c's value of \(f'(1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Area \(= \int_1^{\infty} x^{-2}\, dx\) | M1 | |
| \(= \left[-x^{-1}\right]_1^{\infty}\) | M1 | Ignore limits here |
| \(= 0 - -1 = 1\) | A1 (3) |
## Question 8:
### Part (a)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $(1+h)^3 = 1 + 3h + 3h^2 + h^3$ | B1 | |
| $f(1+h) = 1 + 5h + 4h^2 + h^3$ | M1A1$\checkmark$ | PI; ft wrong coefficients |
| $f(1+h) - f(1) = 5h + 4h^2 + h^3$ | A1$\checkmark$ (4) | ft numerical errors |
### Part (a)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| Dividing by $h$ | M1 | |
| $f'(1) = 5$ | A1$\checkmark$ (2) | ft numerical errors |
### Part (b)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $x^2(x+1) = 1$, hence result | B1 (1) | Convincingly shown AG |
### Part (b)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $x_2 = 1 - \frac{1}{5} = \frac{4}{5}$ | M1A1$\checkmark$, A1$\checkmark$ (3) | ft c's value of $f'(1)$ |
### Part (c)
| Working | Marks | Guidance |
|---------|-------|----------|
| Area $= \int_1^{\infty} x^{-2}\, dx$ | M1 | |
| $= \left[-x^{-1}\right]_1^{\infty}$ | M1 | Ignore limits here |
| $= 0 - -1 = 1$ | A1 (3) | |
---8
\begin{enumerate}[label=(\alph*)]
\item The function f is defined for all real values of $x$ by
$$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
\begin{enumerate}[label=(\roman*)]
\item Express $\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )$ in the form
$$p h + q h ^ { 2 } + r h ^ { 3 }$$
where $p , q$ and $r$ are integers.
\item Use your answer to part (a)(i) to find the value of $f ^ { \prime } ( 1 )$.
\end{enumerate}\item The diagram shows the graphs of
$$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
\end{center}
The graphs intersect at the point $P$.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinate of $P$ satisfies the equation $\mathrm { f } ( x ) = 0$, where f is the function defined in part (a).
\item Taking $x _ { 1 } = 1$ as a first approximation to the root of the equation $\mathrm { f } ( x ) = 0$, use the Newton-Raphson method to find a second approximation $x _ { 2 }$ to the root.\\
(3 marks)
\end{enumerate}\item The region enclosed by the curve $y = \frac { 1 } { x ^ { 2 } }$, the line $x = 1$ and the $x$-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q8 [10]}}