2017 VCAA Maths Methods Exam 2

Question 1 [2017 Exam 2 Section A Q1]

Let \(f: R \to R, f(x) = 5\sin(2x) - 1\).
The period and range of this function are respectively

• A. \(\pi\) and \([-1, 4]\)
• B. \(2\pi\) and \([-1, 5]\)
• C. \(\pi\) and \([-6, 4]\)
• D. \(2\pi\) and \([-6, 4]\)
• E. \(4\pi\) and \([-6, 4]\)

Question 2 [2017 Exam 2 Section A Q2]

Part of the graph of a cubic polynomial function \(f\) and the coordinates of its stationary points are shown below.

\(f'(x) < 0\) for the interval

• A. \((0, 3)\)
• B. \((-\infty, -5) \cup (0, 3)\)
• C. \((-\infty, -3) \cup (\frac{5}{3}, \infty)\)
• D. \((-3, \frac{5}{3})\)
• E. \((-\frac{400}{27}, 36)\)

Question 3 [2017 Exam 2 Section A Q3]

A box contains five red marbles and three yellow marbles. Two marbles are drawn at random from the box without replacement.
The probability that the marbles are of different colours is

• A. \(\frac{5}{8}\)
• B. \(\frac{3}{5}\)
• C. \(\frac{15}{28}\)
• D. \(\frac{15}{56}\)
• E. \(\frac{30}{28}\)

Question 4 [2017 Exam 2 Section A Q4]

Let \(f\) and \(g\) be functions such that \(f(2) = 5\), \(f(3) = 4\), \(g(2) = 5\), \(g(3) = 2\) and \(g(4) = 1\).
The value of \(f(g(3))\) is

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Question 5 [2017 Exam 2 Section A Q5]

The 95% confidence interval for the proportion of ferry tickets that are cancelled on the intended departure day is calculated from a large sample to be \((0.039, 0.121)\).
The sample proportion from which this interval was constructed is

• A. 0.080
• B. 0.041
• C. 0.100
• D. 0.062
• E. 0.059

Question 6 [2017 Exam 2 Section A Q6]

Part of the graph of the function \(f\) is shown below. The same scale has been used on both axes.

The corresponding part of the graph of the inverse function \(f^{-1}\) is best represented by

Question 7 [2017 Exam 2 Section A Q7]

The equation \((p - 1)x^2 + 4x = 5 - p\) has no real roots when

• A. \(p^2 - 6p + 6 < 0\)
• B. \(p^2 - 6p + 1 > 0\)
• C. \(p^2 - 6p - 6 < 0\)
• D. \(p^2 - 6p + 1 < 0\)
• E. \(p^2 - 6p + 6 > 0\)

Question 8 [2017 Exam 2 Section A Q8]

If \(y = a^{b-4x} + 2\), where \(a > 0\), then \(x\) is equal to

• A. \(\frac{1}{4}(b - \log_a(y-2))\)
• B. \(\frac{1}{4}(b - \log_a(y+2))\)
• C. \(b - \log_a(\frac{1}{4}(y+2))\)
• D. \(\frac{b}{4} - \log_a(y-2)\)
• E. \(\frac{1}{4}(b+2 - \log_a(y))\)

Question 9 [2017 Exam 2 Section A Q9]

The average rate of change of the function with the rule \(f(x) = x^2 - 2x\) over the interval \([1, a]\), where \(a > 1\), is 8.
The value of \(a\) is

• A. 9
• B. 8
• C. 7
• D. 4
• E. \(1 + \sqrt{2}\)

Question 10 [2017 Exam 2 Section A Q10]

A transformation \(T: R^2 \to R^2\) with rule \(T\left(\begin{bmatrix} x' \\ y' \end{bmatrix}\right) = \begin{bmatrix} 2 & 0 \\ 0 & \frac{1}{3} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\) maps the graph of \(y = 3\sin\left(2\left(x+\frac{\pi}{4}\right)\right)\) onto the graph of

• A. \(y = \sin(x+\pi)\)
• B. \(y = \sin(x-\frac{\pi}{2})\)
• C. \(y = \cos(x+\pi)\)
• D. \(y = \cos(x)\)
• E. \(y = \cos(x-\frac{\pi}{2})\)

Question 11 [2017 Exam 2 Section A Q11]

The function \(f: R \to R, f(x) = x^3 + ax^2 + bx\) has a local maximum at \(x = -1\) and a local minimum at \(x = 3\).
The values of \(a\) and \(b\) are respectively

• A. -2 and -3
• B. 2 and 1
• C. 3 and -9
• D. -3 and -9
• E. -6 and -15

Question 12 [2017 Exam 2 Section A Q12]

The sum of the solutions of \(\sin(2x) = \frac{\sqrt{3}}{2}\) over the interval \([-\pi, d]\) is \(-\pi\).
The value of \(d\) could be

• A. 0
• B. \(\frac{\pi}{6}\)
• C. \(\frac{3\pi}{4}\)
• D. \(\frac{7\pi}{6}\)
• E. \(\frac{3\pi}{2}\)

Question 13 [2017 Exam 2 Section A Q13]

Let \(h: (-1, 1) \to R, h(x) = \frac{1}{x-1}\).
Which one of the following statements about \(h\) is not true?

• A. \(h(x)h(-x) = -h(x^2)\)
• B. \(h(x) + h(-x) = 2h(x^2)\)
• C. \(h(x) - h(0) = xh(x)\)
• D. \(h(x) - h(-x) = 2xh(x^2)\)
• E. \((h(x))^2 = h(x^2)\)

Question 14 [2017 Exam 2 Section A Q14]

The random variable \(X\) has the following probability distribution, where \(0 < p < \frac{1}{3}\).

\(x\)	-1	0	1
\(\Pr(X=x)\)	\(p\)	\(2p\)	\(1-3p\)

The variance of \(X\) is

• A. \(2p(1-3p)\)
• B. \(1-4p\)
• C. \((1-3p)^2\)
• D. \(6p - 16p^2\)
• E. \(p(5-9p)\)

Question 15 [2017 Exam 2 Section A Q15]

A rectangle \(ABCD\) has vertices \(A(0, 0)\), \(B(u, 0)\), \(C(u, v)\) and \(D(0, v)\), where \((u, v)\) lies on the graph of \(y = -x^3 + 8\), as shown below.

The maximum area of the rectangle is

• A. \(\sqrt[3]{2}\)
• B. \(6\sqrt[3]{2}\)
• C. 16
• D. 8
• E. \(3\sqrt[3]{2}\)

Question 16 [2017 Exam 2 Section A Q16]

For random samples of five Australians, \(\hat{P}\) is the random variable that represents the proportion who live in a capital city.
Given that \(\Pr(\hat{P}=0) = \frac{1}{243}\), then \(\Pr(\hat{P} > 0.6)\), correct to four decimal places, is

• A. 0.0453
• B. 0.3209
• C. 0.4609
• D. 0.5390
• E. 0.7901

Question 17 [2017 Exam 2 Section A Q17]

The graph of a function \(f\), where \(f(-x) = f(x)\), is shown below.

The graph has \(x\)-intercepts at \((a, 0)\), \((b, 0)\), \((c, 0)\) and \((d, 0)\) only.
The area bound by the curve and the \(x\)-axis on the interval \([a, d]\) is

• A. \(\int_a^d f(x)dx\)
• B. \(\int_a^b f(x)dx - \int_b^c f(x)dx + \int_c^d f(x)dx\)
• C. \(2\int_a^b f(x)dx + \int_b^c f(x)dx\)
• D. \(2\int_a^b f(x)dx - 2\int_b^{b+c} f(x)dx\)
• E. \(\int_a^b f(x)dx + \int_c^b f(x)dx + \int_d^c f(x)dx\)

Question 18 [2017 Exam 2 Section A Q18]

Let \(X\) be a discrete random variable with binomial distribution \(X \sim \text{Bi}(n, p)\). The mean and the standard deviation of this distribution are equal.
Given that \(0 < p < 1\), the smallest number of trials, \(n\), such that \(p \le 0.01\) is

• A. 37
• B. 49
• C. 98
• D. 99
• E. 101

Question 19 [2017 Exam 2 Section A Q19]

A probability density function \(f\) is given by

\( f(x) = \begin{cases} \cos(x)+1 & k < x < k+1 \\ 0 & \text{elsewhere} \end{cases} \)

where \(0 < k < 2\).
The value of \(k\) is

• A. 1
• B. \(\frac{3\pi-1}{2}\)
• C. \(\pi-1\)
• D. \(\frac{\pi-1}{2}\)
• E. \(\frac{\pi}{2}\)

Question 20 [2017 Exam 2 Section A Q20]

The graphs of \(f: [0, \frac{\pi}{2}] \to R, f(x) = \cos(x)\) and \(g: [0, \frac{\pi}{2}] \to R, g(x) = \sqrt{3}\sin(x)\) are shown below.
The graphs intersect at \(B\).

The ratio of the area of the shaded region to the area of triangle \(OAB\) is

• A. \(9:8\)
• B. \(\sqrt{3}-1 : \frac{\sqrt{3}\pi}{8}\)
• C. \(8\sqrt{3}-3 : 3\pi\)
• D. \(\sqrt{3}-1 : \frac{\sqrt{3}\pi}{4}\)
• E. \(1 : \frac{\sqrt{3}\pi}{8}\)

Question 1 [2017 Exam 2 Section B Q1]

Let \(f: \mathbb{R} \to \mathbb{R}, f(x) = x^3 - 5x\). Part of the graph of \(f\) is shown below.

a. Find the coordinates of the turning points. 2 marks

b. \(A(-1, f(-1))\) and \(B(1, f(1))\) are two points on the graph of \(f\).

i. Find the equation of the straight line through \(A\) and \(B\). 2 marks

ii. Find the distance \(AB\). 1 mark

c. Let \(g: \mathbb{R} \to \mathbb{R}, g(x) = x^3 - kx, k \in \mathbb{R}^+\).
Let \(C(-1, g(-1))\) and \(D(1, g(1))\) be two points on the graph of \(g\).

i. Find the distance \(CD\) in terms of \(k\). 2 marks

ii. Find the values of \(k\) such that the distance \(CD\) is equal to \(k + 1\). 1 mark

d. The diagram below shows part of the graphs of \(g\) and \(y = x\). These graphs intersect at the points with the coordinates \((0, 0)\) and \((a, a)\).

i. Find the value of \(a\) in terms of \(k\). 1 mark

ii. Find the area of the shaded region in terms of \(k\). 2 marks

Question 2 [2017 Exam 2 Section B Q2]

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point \(P\). The height of \(P\) above the ground, \(h\), is modelled by \(h(t) = 65 - 55\cos\left(\frac{\pi t}{15}\right)\), where \(t\) is the time in minutes after Sammy enters the capsule and \(h\) is measured in metres.
Sammy exits the capsule after one complete rotation of the Ferris wheel.

a. State the minimum and maximum heights of \(P\) above the ground. 1 mark

b. For how much time is Sammy in the capsule? 1 mark

c. Find the rate of change of \(h\) with respect to \(t\) and, hence, state the value of \(t\) at which the rate of change of \(h\) is at its maximum. 2 marks

d. As the Ferris wheel rotates, a stationary boat at \(B\), on a nearby river, first becomes visible at point \(P_1\). \(B\) is 500 m horizontally from the vertical axis through the centre \(C\) of the Ferris wheel and angle \(CBO = \theta\), as shown below.

Find \(\theta\) in degrees, correct to two decimal places. 1 mark

Part of the path of \(P\) is given by \(y = \sqrt{3025 - x^2} + 65, x \in [-55, 55]\), where \(x\) and \(y\) are in metres.

e. Find \(\frac{dy}{dx}\). 1 mark

As the Ferris wheel continues to rotate, the boat at \(B\) is no longer visible from the point \(P_2(u, v)\) onwards. The line through \(B\) and \(P_2\) is tangent to the path of \(P\), where angle \(OBP_2 = \alpha\).

f. Find the gradient of the line segment \(P_2B\) in terms of \(u\) and, hence, find the coordinates of \(P_2\), correct to two decimal places. 3 marks

g. Find \(\alpha\) in degrees, correct to two decimal places. 1 mark

h. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at \(B\) is visible. 2 marks

Question 3 [2017 Exam 2 Section B Q3]

The time Jennifer spends on her homework each day varies, but she does some homework every day.
The continuous random variable \(T\), which models the time, \(t\), in minutes, that Jennifer spends each day on her homework, has a probability density function \(f\), where \( f(t) = \begin{cases} \frac{1}{625}(t-20) & 20 \le t < 45 \\ \frac{1}{625}(70-t) & 45 \le t \le 70 \\ 0 & \text{elsewhere} \end{cases} \)

a. Sketch the graph of \(f\) on the axes provided below. 3 marks

b. Find \(\Pr(25 \le T \le 55)\). 2 marks

c. Find \(\Pr(T \le 25 | T \le 55)\). 2 marks

d. Find \(a\) such that \(\Pr(T > a) = 0.7\), correct to four decimal places. 2 marks

e. The probability that Jennifer spends more than 50 minutes on her homework on any given day is \(\frac{8}{25}\). Assume that the amount of time spent on her homework on any day is independent of the time spent on her homework on any other day.

i. Find the probability that Jennifer spends more than 50 minutes on her homework on more than three of seven randomly chosen days, correct to four decimal places. 2 marks

ii. Find the probability that Jennifer spends more than 50 minutes on her homework on at least two of seven randomly chosen days, given that she spends more than 50 minutes on her homework on at least one of those days, correct to four decimal places. 2 marks

f. Let \(p\) be the probability that on any given day Jennifer spends more than \(d\) minutes on her homework.
Let \(q\) be the probability that on two or three days out of seven randomly chosen days she spends more than \(d\) minutes on her homework.
Express \(q\) as a polynomial in terms of \(p\). 2 marks

g.

i. Find the maximum value of \(q\), correct to four decimal places, and the value of \(p\) for which this maximum occurs, correct to four decimal places. 2 marks

ii. Find the value of \(d\) for which the maximum found in part g.i. occurs, correct to the nearest minute. 2 marks

Question 4 [2017 Exam 2 Section B Q4]

Let \(f: \mathbb{R} \to \mathbb{R} : f(x) = 2^{x+1} - 2\). Part of the graph of \(f\) is shown below.

a. The transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2, T\left(\begin{bmatrix} x' \\ y' \end{bmatrix}\right) = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} c \\ d \end{bmatrix}\) maps the graph of \(y = 2^x\) onto the graph of \(f\).
State the values of \(c\) and \(d\). 2 marks

b. Find the rule and domain for \(f^{-1}\), the inverse function of \(f\). 2 marks

c. Find the area bounded by the graphs of \(f\) and \(f^{-1}\). 3 marks

d. Part of the graphs of \(f\) and \(f^{-1}\) are shown below.

Find the gradient of \(f\) and the gradient of \(f^{-1}\) at \(x = 0\). 2 marks

The functions of \(g_k\), where \(k \in \mathbb{R}^+\), are defined with domain \(\mathbb{R}\) such that \(g_k(x) = 2e^{kx} - 2\).

e. Find the value of \(k\) such that \(g_1(x) = f(x)\). 1 mark

f. Find the rule for the inverse functions \(g_k^{-1}\) of \(g_k\), where \(k \in \mathbb{R}^+\). 1 mark

g.

i. Describe the transformation that maps the graph of \(g_1\) onto the graph of \(g_k\). 1 mark

ii. Describe the transformation that maps the graph of \(g_1^{-1}\) onto the graph of \(g_k^{-1}\). 1 mark

h. The lines \(L_1\) and \(L_2\) are the tangents at the origin to the graphs of \(g_k\) and \(g_k^{-1}\) respectively.
Find the value(s) of \(k\) for which the angle between \(L_1\) and \(L_2\) is 30°. 2 marks

i. Let \(p\) be the value of \(k\) for which \(g_k(x) = g_k^{-1}(x)\) has only one solution.

i. Find \(p\). 2 marks

ii. Let \(A(k)\) be the area bounded by the graphs of \(g_k\) and \(g_k^{-1}\) for all \(k > p\).
State the smallest value of \(b\) such that \(A(k) < b\). 1 mark