2016 VCAA Maths Methods Exam 1

Question 1 [2016 Exam 1 Q1]

a. Let \(y = \frac{\cos(x)}{x^2+2}\). Find \(\frac{dy}{dx}\). 2 marks

b. Let \(f(x) = x^2e^{5x}\). Evaluate \(f'(1)\). 2 marks

Question 2 [2016 Exam 1 Q2]

Let \(f: (-\infty, \frac{1}{2}] \to R\), where \(f(x) = \sqrt{1-2x}\).

a. Find \(f'(x)\). 1 mark

b. Find the angle \(\theta\) from the positive direction of the x-axis to the tangent to the graph of \(f\) at \(x = -1\), measured in the anticlockwise direction. 2 marks

Question 3 [2016 Exam 1 Q3]

Let \(f: R \setminus \{1\} \to R\), where \(f(x) = 2 + \frac{3}{x-1}\).

a. Sketch the graph of \(f\). Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation. 3 marks

b. Find the area enclosed by the graph of \(f\), the lines \(x=2\) and \(x=4\), and the x-axis. 2 marks

Question 4 [2016 Exam 1 Q4]

A paddock contains 10 tagged sheep and 20 untagged sheep. Four times each day, one sheep is selected at random from the paddock, placed in an observation area and studied, and then returned to the paddock.

a. What is the probability that the number of tagged sheep selected on a given day is zero? 1 mark

b. What is the probability that at least one tagged sheep is selected on a given day? 1 mark

c. What is the probability that no tagged sheep are selected on each of six consecutive days? Express your answer in the form \(\left(\frac{a}{b}\right)^c\), where \(a\), \(b\) and \(c\) are positive integers. 1 mark

Question 5 [2016 Exam 1 Q5]

Let \(f: (0, \infty) \to R\), where \(f(x) = \log_e(x)\) and \(g: R \to R\), where \(g(x) = x^2+1\).

a.

i. Find the rule for \(h\), where \(h(x) = f(g(x))\). 1 mark

ii. State the domain and range of \(h\). 2 marks

iii. Show that \(h(x) + h(-x) = f((g(x))^2)\). 2 marks

iv. Find the coordinates of the stationary point of \(h\) and state its nature. 2 marks

b. Let \(k: (-\infty, 0] \to R\), where \(k(x) = \log_e(x^2+1)\).

i. Find the rule for \(k^{-1}\). 2 marks

ii. State the domain and range of \(k^{-1}\). 2 marks

Question 6 [2016 Exam 1 Q6]

Let \(f: [-\pi, \pi] \to R\), where \(f(x) = 2\sin(2x) - 1\).

a. Calculate the average rate of change of \(f\) between \(x = -\frac{\pi}{3}\) and \(x = \frac{\pi}{6}\). 2 marks

b. Calculate the average value of \(f\) over the interval \(-\frac{\pi}{3} \le x \le \frac{\pi}{6}\). 3 marks

Question 7 [2016 Exam 1 Q7]

A company produces motors for refrigerators. There are two assembly lines, Line A and Line B. 5% of the motors assembled on Line A are faulty and 8% of the motors assembled on Line B are faulty. In one hour, 40 motors are produced from Line A and 50 motors are produced from Line B. At the end of an hour, one motor is selected at random from all the motors that have been produced during that hour.

a. What is the probability that the selected motor is faulty? Express your answer in the form \(\frac{1}{b}\), where \(b\) is a positive integer. 2 marks

b. The selected motor is found to be faulty. What is the probability that it was assembled on Line A? Express your answer in the form \(\frac{1}{c}\), where \(c\) is a positive integer. 1 mark

Question 8 [2016 Exam 1 Q8]

Let \(X\) be a continuous random variable with probability density function

\(f(x) = \begin{cases} -4x\log_e(x) & 0 < x \le 1 \\ 0 & \text{elsewhere} \end{cases} \)

Part of the graph of \(f\) is shown below. The graph has a turning point at \(x = \frac{1}{e}\).

a. Show by differentiation that \(\frac{x^k}{k^2}(k\log_e(x)-1)\) is an antiderivative of \(x^{k-1}\log_e(x)\), where \(k\) is a positive real number. 2 marks

b.

i. Calculate \(\Pr(X > \frac{1}{e})\). 2 marks

ii. Hence, explain whether the median of \(X\) is greater than or less than \(\frac{1}{e}\), given that \(e > \frac{5}{2}\). 2 marks