2025 VCE Maths Methods Mini Test 4
Number of marks: 14
Reading time: 3 minutes
Writing time: 21 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In 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 \( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = \frac{x}{2}+7 \) and \( g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = A e^{kx} \), where \( A, k \in \mathbb{R} \).
The graphs of \( y = f(x) \) and \( y = g(x) \) intersect at the points \( (-12, 1) \) and \( (2, 8) \), as shown below.
a. Write down two simultaneous equations in terms of \( A \) and \( k \). Solve them, using algebra, to show that \(A = 2^{\frac{18}{7}}\) and \( k = \frac{3}{14} \log_e(2) \). 3 marks
b. Find the value of \( b \), where \( b \in \mathbb{R} \), such that \( g(x) \) can be expressed in the form \( g(x) = A \times 2^{bx} \). 1 mark
c. Use a definite integral to evaluate the area bounded by the graphs of \( y = f(x) \) and \( y = g(x) \), where \( x \in [-12, 2] \). Give the area correct to two decimal places. 2 marks
d. Let \( h(x) = f(x) - g(x) \).
i. Write down an expression for the derivative of \( h(x) \). 1 mark
d. ii. Find the maximum value of \( h(x) \), where \( x \in [-12, 2] \). Give your answer correct to two decimal places. (1 mark)
e. Let \(g^{-1}\) be the inverse of \(g\).
Find the points where the graph of \(y = g^{-1}(x)\) intersects with the graph of \(y = 2(x - 7)\). 2 marks
f. Let \(F\) be an anti-derivative of \(f\) that passes through \((0, c)\), where \(c \in \mathbb{R}\).
i. Show that it is not possible for the graph of \(y = F(x)\) to pass through both \((-12, 1)\) and \((2, 8)\). 2 marks
ii. The graph of \(y = F(x)\) can be dilated by a factor of \(m\) from the \(x\)-axis such that its image passes through both \((-12, 1)\) and \((2, 8)\).
Find the values of \(m\) and \(c\). 2 marks
End of examination questions
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