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2023 VCAA Maths Methods Exam 1
Adapted for Year 11. Unit 1 & 2.
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: 22
Reading time: 10 minutes
Writing time: 35 minutes
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.
Solve \( e^{2x} - 12 = 4e^x \) for \( x \in \mathbb{R} \). 3 marks
a. Sketch the graph of \( f(x) = 2-\frac{3}{x - 1} \) on the axes below, labelling all asymptotes with their equations and axial intercepts with their coordinates. 3 marks
b. Find the values of \( x \) for which \( f(x) \leq 1 \). 1 mark
The graph of \( y = x + \frac{1}{x} \) is shown over part of its domain.
Use two trapeziums of equal width to approximate the area between the curve, the x-axis and the lines \( x = 1 \) and \( x = 3 \). 2 marks
Consider \( f : (-\infty, 1] \rightarrow \mathbb{R} \), \( f(x) = x^2 - 2x \). Part of the graph of \( y = f(x) \) is shown below.
a. State the range of \( f \). 1 mark
b. Sketch the graph of the inverse function \( y = f^{-1}(x) \) on the axes above. Label any endpoints and axial intercepts with their coordinates. 2 marks
c. Determine the equation and the domain for the inverse function \( f^{-1} \). 2 marks
d. Calculate the area of the regions enclosed by the curves of \( f \), \( f^{-1} \) and \( y = -x \). 2 marks
The shapes of two walking tracks are shown below.
Track 1 is described by the function \( f(x) = a - x(x - 2)^2 \).
Track 2 is defined by the function \( g(x) = 12x + bx^2 \).
The unit of length is kilometres.
a. Given that \( f(0) = 12 \) and \( g(1) = 9 \), verify that \( a = 12 \) and \( b = -3 \). 1 mark
b. Verify that \( f(x) \) and \( g(x) \) both have a turning point at \( P \).
Give the coordinates of \( P \). 2 marks
c. A theme park is planned whose boundaries will form the triangle \( \triangle OAB \) where \( O \) is the origin, \( A \) is at \( (k, 0) \) and \( B \) is at \( (k, g(k)) \), as shown below, where \( k \in (0, 4) \).
Find the maximum possible area of the theme park, in km\(^2\). 3 marks
End of examination questions
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