| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | Specimen |
| Marks | 16 |
| Topic | Differentiating Transcendental Functions |
| Type | Solve equation involving derivatives |
| Difficulty | Standard +0.3 This is a multi-part question involving product rule differentiation and curve fitting. Part (i) is routine calculus (finding stationary points). Parts (ii)-(iv) involve substituting values and solving simultaneous equations from given conditions. While it requires several techniques, each step is straightforward with no novel insights needed—slightly easier than average A-level. |
| Spec | 1.07q Product and quotient rules: differentiation4.08a Maclaurin series: find series for function |
**(i)**
- Attempt to use product rule [M1]
- $y' = ae^{ax}\cos bx - be^{ax}\sin bx$ [A1]
- Set $y' = 0$ and rearrange [M1]
- $\tan bx = \frac{a}{b}$ validly obtained [A1]
**(ii)**
- Correct method to solve $\tan 12x = -\frac{1}{12} \Rightarrow x = -0.006928$ [M1]
- Obtain $y = 1.00$ [A1]
- Correct method to solve $x + \frac{\pi}{12} = 0.2549$ [M1]
- Obtain $y = -0.772$ [A1]
- State $y = -0.664$ [B1]
**(iii)**
- Obtain $f + g = 1$ [B1]
- Obtain $-f + g = -0.8$ [B1]
- Attempt to solve their equations simultaneously [M1ft]
- Obtain $f = 0.9,\ g = 0.1$ [A1]
- Obtain $\lambda = 5\pi$ [B1]
**(iv)**
- State $y = 0.1$. This model only differs from the true $y$ value at $x = 0.3$ by $0.06$ [B1]
- First model not so good as $0.028$ error in calculating the minimum and error of $0.704$ in finding the value at $x = 0.3$ [B1]
**Total: 16 marks**11 A curve has equation
$$y = \mathrm { e } ^ { a x } \cos b x$$
where $a$ and $b$ are constants.\\
(i) Show that, at any stationary points on the curve, $\tan b x = \frac { a } { b }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571}
Values of related quantities $x$ and $y$ were measured in an experiment and plotted on a graph of $y$ against $x$, as shown in the diagram. Two of the points, labelled $A$ and $B$, have coordinates $( 0,1 )$ and $( 0.2 , - 0.8 )$ respectively. A third point labelled C has coordinates ( $0.3,0.04$ ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.\\
(ii) In the first model the equation is
$$y = \mathrm { e } ^ { - x } \cos 12 x$$
Show that this model has a maximum point close to $A$ and a minimum point close to $B$, and state the coordinates of these maximum and minimum points and also the $y$ value when $x = 0.3$.\\
(iii) In an alternative model the equation is
$$y = f \cos ( \lambda x ) + g$$
where the constants $f , \lambda$ and $g$ are chosen to give a maximum precisely at the point $A ( 0,1 )$ and a minimum precisely at the point $B ( 0.2 , - 0.8 )$. Find suitable values for $f , \lambda$ and $g$.\\
(iv) Using the alternative model, state the value of $y$ when $x = 0.3$ and hence comment on how accurate each model is in fitting the three given points.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q11 [16]}}