| 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 Part (i) is a standard product rule differentiation with transcendental functions requiring setting dy/dx=0 and algebraic manipulation—routine A-level work. Part (ii) requires evaluating two models against data points: the first model needs simple substitution, while the second requires using the maximum/minimum conditions to find three unknowns from three constraints, then checking fit at point C. This involves more problem-solving and multi-step reasoning than typical questions, but remains accessible with standard techniques. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates2.02i Select/critique data presentation |
**(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]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_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]}}