| Date | None Specimen | Marks available | 3 | Reference code | SPNone.1.hl.TZ0.4 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Determine | Question number | 4 | Adapted from | N/A |
Question
The continuous variable X has probability density function
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
{12{x^2}(1 - x),}&{0 \leqslant x \leqslant 1} \\
{0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]
Determine \({\text{E}}(X)\) .
[3]
a.
Determine the mode of X .
[3]
b.
Markscheme
\({\text{E}}(X) = \int_0^1 {12{x^3}(1 - x){\text{d}}x} \) M1
\( = 12\left[ {\frac{{{x^4}}}{4} - \frac{{{x^5}}}{5}} \right]_0^1\) A1
\( = \frac{3}{5}\) A1
[3 marks]
a.
\(f'(x) = 12(2x - 3{x^2})\) A1
at the mode \(f'(x) = 12(2x - 3{x^2}) = 0\) M1
therefore the mode \( = \frac{2}{3}\) A1
[3 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.
