| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 15 |
| Topic | Differentiating Transcendental Functions |
| Type | Solve equation involving derivatives |
| Difficulty | Standard +0.8 This is a multi-part question requiring product rule differentiation, solving transcendental equations, and model evaluation. Part (i) is standard A-level calculus, but part (ii) requires systematic evaluation of two models against experimental data, involving substitution of multiple points and comparative analysis. The combination of techniques and the model-fitting context elevates this above routine exercises, though it remains accessible with methodical work. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
(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) **Model 1** Correct method to solve $\tan 15x = -\frac{1}{15} \Rightarrow x = -0.00444\ldots$ [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 [M1ft]
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]10 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, $\tan b x = \frac { a } { b }$.\\
(ii)\\
\includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333}
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 $y = f \cos ( \lambda x ) + \mathrm { g }$, 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.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q10 [15]}}