| Date | November 2011 | Marks available | 5 | Reference code | 11N.1.hl.TZ0.10 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find | Question number | 10 | Adapted from | N/A |
Question
A continuous random variable X has the probability density function
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
{k\sin x,}&{0 \leqslant x \leqslant \frac{\pi }{2}} \\
{0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]
Find the value of k.
Find \({\text{E}}(X)\).
Find the median of X.
Markscheme
\(k\int_0^{\frac{\pi }{2}} {\sin x{\text{d}}x = 1} \) M1
\(k[ - \cos x]_0^{\frac{\pi }{2}} = 1\)
k = 1 A1
[2 marks]
\({\text{E}}(X) = \int_0^{\frac{\pi }{2}} {x\sin x{\text{d}}x} \) M1
integration by parts M1
\([ - x\cos x]_0^{\frac{\pi }{2}} + \int_0^{\frac{\pi }{2}} {\cos x{\text{d}}x} \) A1A1
= 1 A1
[5 marks]
\(\int_0^M {\sin x{\text{d}}x} = \frac{1}{2}\) M1
\([ - \cos x]_0^M = \frac{1}{2}\) A1
\(\cos M = \frac{1}{2}\)
\(M = \frac{\pi }{3}\) A1
Note: accept \(\arccos \frac{1}{2}\)
[3 marks]
Examiners report
Most candidates scored maximum marks on this question. A few candidates found k = –1.
Most candidates scored maximum marks on this question. A few candidates found k = –1.
Most candidates scored maximum marks on this question. A few candidates found k = –1.
