1.
\begin{figure}[h]
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). For \(1 \leqslant x \leqslant 2 , \mathrm { f } ( x )\) is represented by a curve with equation \(\mathrm { f } ( x ) = k \left( \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + a x + 1 \right)\) where \(k\) and \(a\) are constants.
For all other values of \(x , \mathrm { f } ( x ) = 0\)
- Use algebraic integration to show that \(k ( 12 a - 33 ) = 8\)
Given that \(a = 5\)
- calculate the mode of \(X\).
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