2017 VCAA Maths Methods Exam 1

Question 1 [2017 Exam 1 Q1]

a. Let \(f: (-2, \infty) \to R, f(x) = \frac{x}{x+2}\). Differentiate \(f\) with respect to \(x\). 2 marks

b. Let \(g(x) = (2 - x^3)^3\). Evaluate \(g'(1)\). 2 marks

Question 2 [2017 Exam 1 Q2]

Let \(y = x \log_e(3x)\).

a. Find \(\frac{dy}{dx}\). 2 marks

b. Hence, calculate \(\int_1^2 (\log_e(3x)+1)dx\). Express your answer in the form \(\log_e(a)\), where \(a\) is a positive integer. 2 marks

Question 3 [2017 Exam 1 Q3]

Let \(f: [-3, 0] \to R, f(x) = (x+2)^2(x-1)\).

a. Show that \((x+2)^2(x-1) = x^3 + 3x^2 - 4\). 1 mark

b. Sketch the graph of \(f\) on the axes below. Label the axis intercepts and any stationary points with their coordinates. 3 marks

Question 4 [2017 Exam 1 Q4]

In a large population of fish, the proportion of angel fish is \(\frac{1}{4}\).
Let \(\hat{P}\) be the random variable that represents the sample proportion of angel fish for samples of size \(n\) drawn from the population.
Find the smallest integer value of \(n\) such that the standard deviation of \(\hat{P}\) is less than or equal to \(\frac{1}{100}\). 2 marks

Question 5 [2017 Exam 1 Q5]

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is \(\frac{2}{5}\). Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

a. What is the probability that Jac does not log on to the computer successfully? 1 mark

b. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are positive integers. 1 mark

c. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form \(\frac{c}{d}\), where \(c\) and \(d\) are positive integers. 2 marks

Question 6 [2017 Exam 1 Q6]

Let \((\tan(\theta)-1)(\sin(\theta)-\sqrt{3}\cos(\theta))(\sin(\theta)+\sqrt{3}\cos(\theta)) = 0\).

a. State all possible values of \(\tan(\theta)\). 1 mark

b. Hence, find all possible solutions for \((\tan(\theta)-1)(\sin^2(\theta)-3\cos^2(\theta)) = 0\), where \(0 \le \theta \le \pi\). 2 marks

Question 7 [2017 Exam 1 Q7]

Let \(f: [0, \infty) \to R, f(x) = \sqrt{x+1}\).

a. State the range of \(f\). 1 mark

b. Let \(g: (-\infty, c] \to R, g(x) = x^2+4x+3\), where \(c < 0\).

i. Find the largest possible value of \(c\) such that the range of \(g\) is a subset of the domain of \(f\). 2 marks

ii. For the value of \(c\) found in part b.i., state the range of \(f(g(x))\). 1 mark

c. Let \(h: R \to R, h(x) = x^2+3\).
State the range of \(f(h(x))\). 1 mark

Question 8 [2017 Exam 1 Q8]

For events \(A\) and \(B\) from a sample space, \(\Pr(A|B) = \frac{1}{5}\) and \(\Pr(B|A) = \frac{1}{4}\). Let \(\Pr(A \cap B) = p\).

a. Find \(\Pr(A)\) in terms of \(p\). 1 mark

b. Find \(\Pr(A' \cap B')\) in terms of \(p\). 2 marks

c. Given that \(\Pr(A \cup B) \le \frac{1}{5}\), state the largest possible interval for \(p\). 2 marks

Question 9 [2017 Exam 1 Q9]

The graph of \(f: [0, 1] \to R, f(x) = \sqrt{x}(1-x)\) is shown below.

a. Calculate the area between the graph of \(f\) and the \(x\)-axis. 2 marks

b. For \(x\) in the interval \((0, 1)\), show that the gradient of the tangent to the graph of \(f\) is \(\frac{1-3x}{2\sqrt{x}}\). 1 mark

The edges of the right-angled triangle \(ABC\) are the line segments \(AC\) and \(BC\), which are tangent to the graph of \(f\), and the line segment \(AB\), which is part of the horizontal axis, as shown below. Let \(\theta\) be the angle that \(AC\) makes with the positive direction of the horizontal axis, where \(45^\circ \le \theta \le 90^\circ\).

c. Find the equation of the line through \(B\) and \(C\) in the form \(y=mx+c\), for \(\theta = 45^\circ\). 2 marks

d. Find the coordinates of \(C\) when \(\theta = 45^\circ\). 4 marks