2025 VCAA Maths Methods Exam 1
This is the full VCE Maths Methods Exam with worked solutions. You can also try Mini-Tests, which are official VCAA exams split into short tests you can do anytime.
Number of marks: 40
Reading time: 15 minutes
Writing time: 1 hours
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In all questions where a numerical answer is required, an exact value must be given unless
otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
a. Let \( y = x^2 \cos(x) \). Find \( \frac{dy}{dx} \). 1 mark
b. Let \( f(x) = 6\sqrt{x+1} + 5 \). Find the gradient of the tangent to \( y = f(x) \) at \( x = 8 \). 2 marks
Let \( g(x) \) be a function defined for \( x > -\frac{3}{2} \) so that \( g'(x) = \frac{1}{2x+3} \) and \( g(1) = 0 \). Find \( g(x) \). 2 marks
Let \( f : [0, 2\pi] \rightarrow \mathbb{R}, f(x) = 2 \cos(2x) + 1 \).
a. State the range of \( f \). 1 mark
b. Solve \( f(x) = 0 \) for \( x \). 3 marks
c. Sketch the graph of \( y = f(x) \) for \( x \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \) on the axes below. Label the endpoints with their coordinates. 2 marks
The probability distribution for the discrete random variable \( X \) is given in the table below, where \( k \) is a positive real number.
| \( X \) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \( \Pr(X=x) \) | \( \frac{4}{k} \) | \( \frac{2k}{75} \) | \( \frac{k}{75} \) | \( \frac{2}{k} \) |
a. Show that \( k = 10 \) or \( k = 15 \). 2 marks
b. Let \( k = 15 \).
i. Find \( \Pr(X > 1) \). 1 mark
ii. Find \( E(X) \). 1 mark
a. Solve \( e^{2x} - 8e^x + 7 = 0 \) for \( x \). 2 marks
b. Let \( g(x) = e^{2x} - 8e^x + 7 \), where \( x \in \mathbb{R} \). The function \( g(x) \) has exactly one stationary point, a local minimum. Find the largest value of \( a \) such that when \( g \) is restricted to the domain \( (-\infty, a] \) it has an inverse function. 2 marks
Consider the binomial random variable \( X \sim \text{Bi}\left(6, \frac{1}{4}\right) \).
a. Find \( \text{var}(X) \). 1 mark
b. Determine \( \Pr(X \ge 5) \). Give your answer in the form \( \frac{a}{2^b} \), where \( a, b \in \mathbb{Z} \). 2 marks
Let \( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^3 - x^2 - 16x - 20 \).
a. Verify that \( x = 5 \) is a solution of \( f(x) = 0 \). 1 mark
b. Express \( f(x) \) in the form \( (x+d)^2(x-5) \) where \( d \in \mathbb{R} \). 2 marks
c. Consider the graph of \( y = f(x) \), as shown below. Complete the coordinate pairs of all axial intercepts of \( y = f(x) \). 1 mark
d. Let \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = x + 2 \).
i. State the coordinates of the stationary point of inflection for the graph of \( y = f(x)g(x) \). 1 mark
ii. Write down the values of \( x \) for which \( f(x)g(x) \ge 0 \). 1 mark
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
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
End of examination questions
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