Problem

Figure 1 shows a sketch of the probability density function f(x) of the random variable X.

For 1 ≤ x ≤ 2, f(x) is represented by a curve with equation

$f(x)=k\left(\dfrac12 x^3-3x^2+ax+1\right)$

where k and a are constants.

For all other values of x, f(x)=0

(a) Use algebraic integration to show that k(12a−33)=8

Given that a = 5

(b) calculate the mode of X.

(Total for Question 1 is 8 marks)