2025 VCE Maths Methods Mini Test 10

Question 1 [2025 Exam 1 Q8]

Consider \( f(x) = \begin{cases} \frac{3}{8}(4-3x) & 0 \le x \le \frac{4}{3} \\ 0 & \text{otherwise} \end{cases} \).

a. The continuous random variable \( X \) has probability density function \( f(x) \). Find \( k \) such that \( \Pr(X > k) = \frac{9}{16} \). 3 marks

b. The function \( h(x) \) is a transformation of \( f(x) \) such that \( h(x) = m f(x) + n \) where \( m \) and \( n \) are real numbers. Find \( \int_{0}^{\frac{4}{3}} h(x) \, dx \) in terms of \( m \) and \( n \). 2 marks

Question 2 [2025 Exam 1 Q9]

Consider the functions \( f : \mathbb{R} \setminus \{1\} \rightarrow \mathbb{R}, f(x) = \frac{w^2}{(x-1)^2}\) and \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = (x-w)^2 \), where \( w \in \mathbb{R} \).

a. If \( w =- 3 \), find the four solutions to \( f(x) = g(x) \). 3 marks

b. Consider the case where \( w > 0 \).

i. Find, in terms of \( w \), the coordinates of the minimum point of the graph of \( y = (x-1)(x-w) \). 2 marks

ii. Hence, or otherwise, find the positive values of \( w \) for which \( f(x) = g(x) \) has exactly three solutions. 2 marks