2018 VCAA Maths Methods Exam 1

Question 1 [2018 Exam 1 Q1]

a. If \( y = (-3x^3 + x^2 - 64)^3 \), find \( \frac{dy}{dx} \). 1 mark

b. Let \( f(x) = \frac{e^x}{\cos(x)} \).
Evaluate \( f'(\pi) \). 2 marks

Question 2 [2018 Exam 1 Q2]

The derivative with respect to \(x\) of the function \( f: (1, \infty) \rightarrow \mathbb{R} \) has the rule \( f'(x) = \frac{1}{2} - \frac{1}{(2x-2)} \).
Given that \( f(2) = 0 \), find \( f(x) \) in terms of \(x\). 3 marks

Question 3 [2018 Exam 1 Q3]

Let \( f: [0, 2\pi] \rightarrow \mathbb{R}, f(x) = 2\cos(x) + 1 \).

a. Solve the equation \( 2\cos(x) + 1 = 0 \) for \( 0 \le x \le 2\pi \). 2 marks

b. Sketch the graph of the function \(f\) on the axes below. Label the endpoints and local minimum point with their coordinates. 3 marks

Question 4 [2018 Exam 1 Q4]

Let \(X\) be a normally distributed random variable with a mean of 6 and a variance of 4. Let \(Z\) be a random variable with the standard normal distribution.

a. Find \( \Pr(X > 6) \). 1 mark

b. Find \(b\) such that \( \Pr(X > 7) = \Pr(Z < b) \). 1 mark

Question 5 [2018 Exam 1 Q5]

Let \( f: (2, \infty) \rightarrow \mathbb{R} \), where \( f(x) = \frac{1}{(x-2)^2} \).
State the rule and domain of \( f^{-1} \). 3 marks

Question 6 [2018 Exam 1 Q6]

Two boxes each contain four stones that differ only in colour.
Box 1 contains four black stones.
Box 2 contains two black stones and two white stones.
A box is chosen randomly and one stone is drawn randomly from it.
Each box is equally likely to be chosen, as is each stone.

a. What is the probability that the randomly drawn stone is black? 2 marks

b. It is not known from which box the stone has been drawn.
Given that the stone that is drawn is black, what is the probability that it was drawn from Box 1? 2 marks

Question 7 [2018 Exam 1 Q7]

Let \(P\) be a point on the straight line \(y = 2x - 4\) such that the length of \(OP\), the line segment from the origin \(O\) to \(P\), is a minimum.

a. Find the coordinates of \(P\). 3 marks

b. Find the distance \(OP\). Express your answer in the form \( \frac{a\sqrt{b}}{b} \), where \(a\) and \(b\) are positive integers. 2 marks

Question 8 [2018 Exam 1 Q8]

Let \( f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2e^{kx} \), where \(k\) is a positive real constant.

a. Show that \( f'(x) = xe^{kx}(kx + 2) \). 1 mark

b. Find the value of \(k\) for which the graphs of \( y = f(x) \) and \( y = f'(x) \) have exactly one point of intersection. 2 marks

Let \( g(x) = -\frac{2xe^{kx}}{k} \). The diagram below shows sections of the graphs of \(f\) and \(g\) for \(x \ge 0\).

Let \(A\) be the area of the region bounded by the curves \(y = f(x)\), \(y = g(x)\) and the line \(x = 2\).

c. Write down a definite integral that gives the value of \(A\). 1 mark

d. Using your result from part a., or otherwise, find the value of \(k\) such that \( A = \frac{16}{k} \). 3 marks

Question 9 [2018 Exam 1 Q9]

Consider a part of the graph of \(y = x\sin(x)\), as shown below.

a.

i. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive even integer or 0. Give your answer in simplest form. 2 marks

ii. Given that \( \int (x\sin(x))dx = \sin(x) - x\cos(x) + c \), evaluate \( \int_{n\pi}^{(n+1)\pi} (x\sin(x))dx \) when \(n\) is a positive odd integer. Give your answer in simplest form. 1 mark

b. Find the equation of the tangent to \( y = x\sin(x) \) at the point \( \left(-\frac{5\pi}{2}, \frac{5\pi}{2}\right) \). 2 marks

c. The translation \(T\) maps the graph of \( y = x\sin(x) \) onto the graph of \( y = (3\pi - x)\sin(x) \), where
\( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2, T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ 0 \end{bmatrix} \)
and \(a\) is a real constant.
State the value of \(a\). 1 mark

d. Let \( f: [0, 3\pi] \rightarrow \mathbb{R}, f(x) = (3\pi - x)\sin(x) \) and \( g: [0, 3\pi] \rightarrow \mathbb{R}, g(x) = (x - 3\pi)\sin(x) \).
The line \(l_1\) is the tangent to the graph of \(f\) at the point \( \left(\frac{\pi}{2}, \frac{5\pi}{2}\right) \) and the line \(l_2\) is the tangent to the graph of \(g\) at \( \left(\frac{\pi}{2}, -\frac{5\pi}{2}\right) \), as shown in the diagram below.

Find the total area of the shaded regions shown in the diagram above. 2 marks