11

11. [Maximum mark: 14]

A mathematics class of 15 students plays a game which requires three equal size teams.

(a) Find the total number of ways that the three teams can be chosen. [3]

The game involves the spinning of a top. The time, $T$, in minutes that the spinning top is in motion can be modelled by the probability density function $f$ where

$f(t) = \begin{cases} kte^{-3t}, & t \ge 0 \\ 0, & \text{otherwise} \end{cases}$

and $k \in \mathbb{Z}^+$.

(b) Show that $\displaystyle\int_0^a f(t) \, dt = \frac{k}{9} \bigl[1 - (3a + 1)e^{-3a}\bigr]$, where $a \in \mathbb{R}^+$. [4]

(c)
(i) Use l’Hôpital’s rule to find $\displaystyle\lim_{x \to \infty} (3x + 1)e^{-3x}$.

(ii) Hence, by considering $\displaystyle\lim_{a \to \infty} \int_0^a f(t) \, dt$, find the value of $k$. [5]

(d) Find the median length of time that a spinning top is in motion. [2]