| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 12 |
| 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 (PDF properties, numerical methods), the combination of calculus, numerical solving, and interpretation across multiple parts makes it moderately challenging but still within typical Further Maths scope. |
| Spec | 1.09d Newton-Raphson method5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (iv) \(P( | X - \mu | < \sigma) = \frac{4}{27}\int_{1.2}^{2.4}(3x^2 - x^3)\,dx\) (Limits required) M1 |
From Question 6 in the mark scheme:
**(i)** Above $x$-axis between $(0, 0)$ to $(3, 0)$ **B1**
Correct concavity. (Do not condone parabolas) **B1**
**(ii)** $\mu = \frac{4}{27}\int_0^3 (3x^3 - x^4)\,dx$ (Limits required) **M1**
$= \frac{4}{27}\left[\frac{3x^4}{4} - \frac{x^5}{5}\right]_0^3 = 1.8$ **A1 A1**
$f'(x) = \frac{4}{27}(6x - 3x^2) = 0$ **M1 A1**
$\Rightarrow x = 0, 2$ $\therefore$ Mode $= 2$ **A1**
**(iii)** Mean less than mode in **(ii)** matches negative skew in sketch. **B1**
**(iv)** $P(|X - \mu| < \sigma) = \frac{4}{27}\int_{1.2}^{2.4}(3x^2 - x^3)\,dx$ (Limits required) **M1**
$= \frac{4}{27}\left[x^3 - \frac{x^4}{4}\right]_{1.2}^{2.4} = 0.64$ **A1 A1**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
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\
0 & \text { otherwise }
\end{array} \right.$$
(i) Draw a sketch of this probability density function.\\
(ii) Calculate the mean and the mode of $X$.\\
(iii) Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).\\
(iv) 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.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q6 [12]}}