| Date | November 2015 | Marks available | 2 | Reference code | 15N.2.hl.TZ0.8 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 8 | Adapted from | N/A |
Question
The continuous random variable \(X\) has the probability distribution function \(f(x) = A\sin \left( {\ln (x)} \right),{\text{ }}1 \le x \le 5\).
Find the value of \(A\) to three decimal places.
Find the mode of \(X\).
Find the value \({\text{P}}(X \le 3|X \ge 2)\).
Markscheme
\(A\int_1^5 {\sin (\ln x){\text{d}}x = 1} \) (M1)
\(A = 0.323{\text{ (3 dp only)}}\) A1
[2 marks]
either a graphical approach or \(f'(x) = \frac{{\cos (\ln x)}}{x} = 0\) (M1)
\(x = 4.81\;\;\;\left( { = {{\text{e}}^{\frac{\pi }{2}}}} \right)\) A1
Note: Do not award A1FT for a candidate working in degrees.
[2 marks]
\({\text{P}}(X \le 3|X \ge 2) = \frac{{{\text{P}}(2 \le X \le 3)}}{{{\text{P}}(X \ge 2)}}\;\;\;\left( { = \frac{{\int_2^3 {\sin \left( {\ln (x)} \right){\text{d}}x} }}{{\int_2^5 {\sin \left( {\ln (x)} \right){\text{d}}x} }}} \right)\) (M1)
\( = 0.288\) A1
Note: Do not award A1FT for a candidate working in degrees.
[2 marks]
Total [6 marks]
