| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with trigonometric or exponential functions |
| Difficulty | Standard +0.3 This question combines routine sketching of standard functions with a straightforward application of the Newton-Raphson method. Part (i) requires recognizing that exponential and linear graphs intersect once (standard curve sketching), while part (ii) is a direct algorithmic application of Newton-Raphson with a given starting value—no conceptual difficulty or problem-solving insight required beyond following the standard procedure. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (i) A positive exponential graph | B1 | |
| Straight line with positive slope | B1 | |
| Show or state two intersections or roots | B1* | |
| (ii) \(x_{n+1} = x_n - \frac{e^{0.2x} - x}{0.2e^{0.2x} - 1}\) | M1, B1 | Use correct NR formula. Correct derivative |
| M1* | Starts at 0 and states at least two iterates. States 1.296 | |
| \(0, 1.25, 1.2958, 1.2959\) | A1 | [7] |
(i) A positive exponential graph | B1 |
Straight line with positive slope | B1 |
Show or state two intersections or roots | B1* |
(ii) $x_{n+1} = x_n - \frac{e^{0.2x} - x}{0.2e^{0.2x} - 1}$ | M1, B1 | Use correct NR formula. Correct derivative
| M1* | Starts at 0 and states at least two iterates. States 1.296
$0, 1.25, 1.2958, 1.2959$ | A1 | [7] |\begin{enumerate}[label=(\roman*)]
\item Sketch, on a single diagram, the graphs of $y = e^{3x}$ and $y = x$ and state the number of roots of the equation $e^{3x} = x$. [3]
\item Use the Newton-Raphson method with $x_0 = 0$ to determine the value of a root of the equation $e^{3x} = x$ correct to 3 decimal places. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q6 [7]}}