S2 June 2016 Q3
3. The random variable \(R\) has a continuous uniform distribution over the interval [5, 9]
The random variable \(A\) is the area of a circle radius \(R\) cm.
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(r) = \begin{cases} \dfrac{1}{4} & 5 \leqslant r \leqslant 9 \\ 0 & \text{otherwise} \end{cases}\) | B1 |
| (1) |
Notes
B1: Allow \(r \lt 5\) and \(r \gt 9\) instead of 0 otherwise
Allow < instead of \(\leqslant\) signs.
Any letter may be used - condone mixed letters
Must have f(r) – condone F(r)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(7 \lt R \lt 10) = 2 \times \dfrac{1}{4}\) \(= \dfrac{1}{2}\) | B1 |
| (1) |
Notes
B1: oe
| Scheme | Marks |
|---|---|
| \(\left[\mathrm{E}(A) = \mathrm{E}(\pi R^2)\right]\) \(\mathrm{E}(R^2) = \mathrm{Var}(R) + [\mathrm{E}(R)]^2\) or \(\displaystyle\int_5^9 \frac{r^2}{4}\,\mathrm{d}r\) | M1 |
| \(\mathrm{E}(R) = 7, \quad \mathrm{Var}(R) = \dfrac{4}{3}\) or \(\left[\dfrac{r^3}{12}\right]_5^9\) | B1 |
| \(= 50\tfrac{1}{3}\) | A1 |
| \(\mathrm{E}(A) = 50\tfrac{1}{3}\pi\) oe | A1 |
| (4) | |
| (6 marks) |
Notes
M1: Using correct formula for \(\mathrm{E}(R^2)\). This may be in any order or written in words
B1: Var \((R) = \dfrac{4}{3}\) or awrt 1.33 and \(\mathrm{E}(R) = 7\) or \(\left[\dfrac{r^3}{12}\right]_5^9\). These may be implied by a correct answer
A1: Allow awrt 50.3
A1: Allow exact multiple of \(\pi\) eg \(50.\dot{3}\pi\) or awrt 158
Do Not allow \(50.3\pi\)
NB If both \(\mathrm{E}(R)^2\) and \([\mathrm{E}(R)]^2\) are both worked out and neither is selected they lose the final A marks. The best they can get is M1 B1 A1A0