Figure 1 shows a sketch of the probability density function $$\(\mathrm{f}(x)\)$$ of the random variable $$\(X\)$$. For $$\(1 \leqslant x \leqslant 2, \mathrm{f}(x)\)$$ is represented by a curve with equation $$\(\mathrm{f}(x)=k\left(\frac{1}{2} x^{3}-3 x^{2}+a x+1\right)\)$$ where $$\(k\)$$ and $$\(a\)$$ are constants. For all other values of $$\(x, \mathrm{f}(x)=0\)$$ (a) Use algebraic integration to show that $$\(k(12 a-33)=8\)$$ (4) Given that $$\(a=5\)$$ (b) calculate the mode of $$\(X\)$$. (4)

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