WACE Maths Methods ATAR Paper 1 Topic Tests

Question 1 (8 marks) [2024 Section 1 Q5]

The function \(f(x) = \log_a(x)\) is plotted below, where \(a\) and \(p\) are constants.

(a) Express \(\log_a(0.5)\) in terms of \(p\). (2 marks)

(b) Evaluate \(a^{5p}\). (2 marks)

(c) Solve \(\log_a(x-3) = 3p\) for \(x\). (2 marks)

The function \(f(x) = \log_a(x)\) has been transformed to give two other logarithmic functions with base \(a\) on the axes as shown below.

(d) Determine an equation for each of the two functions, A and B. (2 marks)

Question 1 (8 marks) [2023 Section 1 Q4]

An internet search engine uses a logarithmic scale to rank the importance of internet websites. If a website has \(S\) visits each week, the site rank, \(R\), is given by \[ R = 2\log_{10}\left(\frac{S}{S_0}\right) \] where \(S_0\) is the reference value (the same for all websites). The reference value is the minimum number of visits per week required for a website to register on the site rank scale.

(a) Determine the site rank for a website whose weekly visits are one hundred times the reference value. (2 marks)

(b) Given that a site rank of 12 is assigned to a website with 1.5 billion (\(1.5 \times 10^9\)) visits per week, determine the value of \(S_0\). (3 marks)

(c) The plot of \(y = \log_{10}(x)\) is shown below. If a website has a site rank of 3.2, use the plot and your answer from part (b) to approximate the website's number of weekly visits. (3 marks)

Question 1 (12 marks) [2022 Section 1 Q4]

The graph of the function \(f(x) = \log_2(x)\) is shown below.

(a) Using the graph:

(i) solve \( \log_2(x-5) = 3 \). (2 marks)

(ii) determine \( \sqrt{7} \), correct to one decimal place. (Hint: let \(x = \sqrt{7}\).) (3 marks)

(b) The function \(f(x) = \log_2(x)\) is translated to give the new function \(g(x)\), which is shown in the graph below.

Determine the equation for \(g(x)\). (2 marks)

(c)

(i) Show that \( \log_2 \left(\frac{1}{x-1}\right) = -\log_2(x-1) \). (2 marks)

(ii) Hence sketch the graph of \( h(x) = \log_2 \left(\frac{1}{x-1}\right) \) on the axes below. (3 marks)

Question 1 (6 marks) [2019 Section 1 Q4]

Consider the graph of \(y=\ln(x)\) shown below.

(a) Use the graph to estimate the value of \(p\) in each of the following.

(i) \(1.4 = \ln(p)\) (1 mark)

(ii) \(e^{p+1}-3=0\) (2 marks)

(b) On the axes below, sketch the graph of \(y = \ln(x-2)+1\). (3 marks)

Question 2 (7 marks) [2020 Section 1 Q6]

Consider the function \(f(x) = \ln(x)\). The function \(g(x) = f(x) + a\) is a vertical translation of \(f\) by \(a\) units.

(a) Express the function \(g(x) = \ln(4x)\) in terms of a vertical translation of \(f\) (i.e. in the form \(g(x) = f(x) + a\)), stating the number of units that \(f\) is translated. (2 marks)

The function \(h(x) = cf(x)\) is a vertical dilation of \(f\) by a scale factor of \(c\).

(b) Express the function \(h(x) = \ln(\sqrt{x})\) in terms of a vertical dilation of \(f\), stating the scale factor. (2 marks)

The function \(p(x) = f(bx)\) is a horizontal dilation of \(f\) by a scale factor of \( \frac{1}{b} \).

(c) Express the function \(p(x) = \ln(x) + 4\) in terms of a horizontal dilation of \(f\), stating the scale factor. (3 marks)

Question 1 (4 marks) [2017 Section 1 Q3]

Solve \(4e^{2x} = 81 - 5e^{2x}\) exactly for \(x\).

Question 2 (6 marks) [2017 Section 1 Q7]

Given that \(\log_{10}2 = x\) and \(\log_{10}7 = y\)

(a) express \(\log_{10}14\) in terms of \(x\) and \(y\). (2 marks)

(b) show that \(\log_{10}17.5 = y - 2x + 1\). (2 marks)

(c) evaluate \(10^{y-x}\). (2 marks)