| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson with trigonometric or exponential functions |
| Difficulty | Standard +0.3 This is a straightforward application of Newton-Raphson with standard functions (ln and linear). Part (i) is basic transformation recall, part (ii) is routine curve sketching to show unique root, and part (iii) is mechanical iteration with clear starting value. The functions are simple to differentiate and no conceptual obstacles arise—slightly easier than average due to the guided structure and standard techniques. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.09d Newton-Raphson method |
**Question 7(i)**
State **translation** – NOT "shift" or "move" **B1**
… one unit to the left OR in the negative direction OR to the left by 1 OR 1 unit parallel to the $x$ axis OR by specifying the vector $\begin{pmatrix}-1\\0\end{pmatrix}$ **B1**
(Stating "in" OR "on" OR "along" the $x$-axis OR "a factor of $-1$" B0)
**Question 7(ii)**
Sketch $\ln$ graph with asymptote at $x = -1$ clearly indicated. **B1**
Sketch correct $y = 4 - x$ to show intersection points with the axes clearly. **B1**
State one intersection implies one root. **B1\***
**Total: 5 marks**7 (i) Describe the transformation which maps the graph of $y = \ln x$ onto the graph of $y = \ln ( 1 + x )$.\\
(ii) By sketching the curves $y = \ln ( 1 + x )$ and $y = 4 - x$ on a single diagram, show that the equation
$$\ln ( 1 + x ) = 4 - x$$
has exactly one root.\\
(iii) Use the Newton-Raphson method with $x _ { 0 } = 2$ to find the root of the equation $\ln ( 1 + x ) = 4 - x$ correct to 3 decimal places. Show the result of each iteration.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q7 [5]}}