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2021 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: 29
Reading time: 10 minutes
Writing time: 45 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.
Let \( f'(x) = x^3 + x \).
Find \( f(x) \) given that \( f(1) = 2 \). 2 marks
Consider the function \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = 2 \sin(2x) \).
a. State the range of \(g\). 1 mark
b. State the period of \(g\). 1 mark
c. Solve \( 2 \sin(2x) = \sqrt{3} \) for \( x \in \mathbb{R} \). 3 marks
a. Sketch the graph of \( y = 1 - \frac{2}{x-2} \) on the axes below. Label asymptotes with their equations and axis intercepts with their coordinates. 3 marks
b. Find the values of \(x\) for which \( 1 - \frac{2}{x-2} \ge 3 \). 1 mark
Let \( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2 - 4 \) and \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = 4(x-1)^2 - 4 \).
a. The graphs of \(f\) and \(g\) have a common horizontal axis intercept at \((2, 0)\).
Find the coordinates of the other horizontal axis intercept of the graph of \(g\). 2 marks
b. Let the graph of \(h\) be a transformation of the graph of \(f\) where the transformations have been applied in the following order:
- • dilation by a factor of \( \frac{1}{2} \) from the vertical axis (parallel to the horizontal axis)
- • translation by two units to the right (in the direction of the positive horizontal axis)
An online shopping site sells boxes of doughnuts.
A box contains 20 doughnuts. There are only four types of doughnuts in the box. They are:
- • glazed, with custard
- • glazed, with no custard
- • not glazed, with custard
- • not glazed, with no custard.
- • \( \frac{1}{2} \) of the doughnuts are with custard
- • \( \frac{7}{10} \) of the doughnuts are not glazed
- • \( \frac{1}{10} \) of the doughnuts are glazed, with custard.
a. A doughnut is chosen at random from the box.
Find the probability that it is not glazed, with custard. 1 mark
b. The 20 doughnuts in the box are randomly allocated to two new boxes, Box A and Box B. Each new box contains 10 doughnuts.
One of the two new boxes is chosen at random and then a doughnut from that box is chosen at random.
Let \(g\) be the number of glazed doughnuts in Box A.
Find the probability, in terms of \(g\), that the doughnut comes from Box B given that it is glazed. 2 marks
c. Not applicable for Year 11 students. 3 marks
The gradient of a function is given by \( \frac{dy}{dx} = \sqrt{x+6} - \frac{x}{2} - \frac{3}{2} \).
The graph of the function has a single stationary point at \( \left(3, \frac{29}{4}\right) \).
a. Find the rule of the function. 3 marks
b. Determine the nature of the stationary point. 2 marks
Consider the unit circle \( x^2 + y^2 = 1 \) and the tangent to the circle at the point \(P\), shown in the diagram below.
a. Show that the equation of the line that passes through the points \(A\) and \(P\) is given by \( y = -\frac{x}{\sqrt{3}} + \frac{2}{\sqrt{3}} \). 2 marks
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2, T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix} 1 & 0 \\ 0 & q \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \), where \( q \in \mathbb{R}\setminus\{0\} \), and let the graph of the function \(h\) be the transformation of the line that passes through the points \(A\) and \(P\) under \(T\).
b.
i. Find the values of \(q\) for which the graph of \(h\) intersects with the unit circle at least once. 1 mark
ii. Let the graph of \(h\) intersect the unit circle twice.
Find the values of \(q\) for which the coordinates of the points of intersection have only positive values. 1 mark
c. For \( 0 < q \le 1 \), let \(P'\) be the point of intersection of the graph of \(h\) with the unit circle, where \(P'\) is always the point of intersection that is closest to \(A\), as shown in the diagram below.
Let \(g\) be the function that gives the area of triangle \(OAP'\) in terms of \( \theta \).
i. Define the function \(g\). 2 marks
ii. Determine the maximum possible area of the triangle \(OAP'\). 2 marks
End of examination questions
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