| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2019 |
| Session | Specimen |
| Marks | 2 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring integration for the mean, differentiation for the mode, solving a quartic equation numerically, and interpreting results. While the techniques are standard Further Maths content, the combination of calculus, numerical methods, and interpretation across multiple parts elevates it above average difficulty. |
| Spec | 1.09d Newton-Raphson method5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
**Question 6(a)**
Above $x$-axis between $(0, 0)$ to $(3, 0)$ [B1]
Correct concavity. (Do not condone parabolas) [B1]
**Total: 2 marks**
**Question 6(b)**
$\mu = \dfrac{4}{27}\int_0^3 (3x^3 - x^4)\,dx$ (Limits required) [M1]
$= \dfrac{4}{27}\left[\dfrac{3x^4}{4} - \dfrac{x^5}{5}\right]_0^3 = 1.8$ [A1A1]
$f'(x) = \dfrac{4}{27}(6x - 3x^2) = 0$ [M1A1]
$\Rightarrow x = 0, 2$ $\therefore$ Mode $= 2$ [A1]
**Total: 6 marks**
**Question 6(c)**
Mean less than mode in **(b)** matches negative skew in sketch. [B1]
**Total: 1 mark**
**Question 6(d)**
$\left[\dfrac{4}{27}\left(x^3 - \dfrac{x^4}{4}\right)\right]_0^l = \dfrac{1}{4} \Rightarrow Q_1^4 - 4Q_1^3 + 6.75 = 0$ **AG** [B1]
Use e.g. $Q_{1_{n+1}} = Q_{1_n} - \dfrac{Q_{1_n}^4 - 4Q_{1_n}^3 + 6.75}{4Q_{1_n}^3 - 12Q_{1_n}^2}$ or decimal search etc. [M1A1]
Convincingly obtain $1.37$ [A1]
**Total: 4 marks**6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function $\mathrm { f } ( x )$, where
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 , \\
0 & \text { otherwise } .
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Draw a sketch of this probability density function.
\item Calculate the mean and the mode of $X$.
\item Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
\item Show that the lower quartile $\mathrm { Q } _ { 1 }$ of $X$ satisfies the equation $\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0$, and use an appropriate numerical method to find the value of $\mathrm { Q } _ { 1 }$ correct to 2 decimal places, showing full details of your method.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2019 Q6 [2]}}