| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2019 |
| Session | Specimen |
| Marks | 12 |
| Topic | Differentiating Transcendental Functions |
| Type | Solve equation involving derivatives |
| Difficulty | Standard +0.8 Part (a) is a standard product rule differentiation with transcendental functions requiring setting dy/dx=0 and algebraic manipulation—routine A-level technique. Part (b) requires evaluating two models against experimental data by substituting coordinates and checking consistency, involving some calculation but straightforward application. The multi-part structure and model evaluation adds modest complexity beyond typical textbook exercises, but no novel mathematical insight is required. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.06a Exponential function: a^x and e^x graphs and properties1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
**(a)**
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** [4]
**(b)**
**Model 1:** Correct method to solve $\tan 15x = -\frac{1}{15} \Rightarrow x = -0.00444...$ — **M1**
Obtain $y = 1.0022$ — **A1**
Correct method to solve $x + \frac{\pi}{15} = 0.20499$ — **M1**
Obtain $y = -0.81284$ — **A1**
State when $x = 0.3$, $y = -0.156$ — **B1**
**Model 2:** Obtain $f + g = 1$ — **B1**
Obtain $-f + g = -0.8$ — **B1**
Attempt to solve their equations simultaneously — **M1**
Obtain $f = 0.9$, $g = 0.1$ — **A1**
Obtain $\lambda = 5\pi$ — **B1**
State when $x = 0.3$, $y = 0.1$ — **B1**
Relevant comment that model 2 matches experimental data more closely — **B1** [12]10 A curve has equation
$$y = \mathrm { e } ^ { a x } \cos b x$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Show that, at any stationary points on the curve, $\tan b x = \frac { a } { b }$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-4_620_894_1064_342}
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.
In the first model the equation is $y = \mathrm { e } ^ { - x } \cos 15 x$.
In the second 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 $B ( 0.2 , - 0.8 )$.
By calculating suitable values evaluate the suitability of the two models.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2019 Q10 [12]}}