Mathematical Methods questions and solutions

Browse original practice and official past-paper questions. Each question has its own link, with its source and worked solution.

610 questions · Page 10 of 26

1. Q5 · 2020 QCAA · Paper 2 · 1 mark
   An object moves in a straight line with a velocity $v$ given by $$v(t)=40(e^{-t}-e^{-2t})\;\mathrm{m\,s^{-1}},\quad t\ge0.$$ The object is at the origin initially. The displacement–time graph in the first 6 seconds is
   Question and worked solution
2. Q6 · 2020 QCAA · Paper 2 · 1 mark
   Oil is leaking from a tanker at the rate of $r(t)=9000e^{-0.2t}$ litres per hour, where $t$ is in hours. Determine how much oil leaks from the tanker (to the nearest litre) from time $t=0$ to time $t=10$.
   Question and worked solution
3. Q7 · 2020 QCAA · Paper 2 · 1 mark
   The records of a shoe manufacturer show that 10% of shoes made are defective. Assuming independence, the probability of getting 2 defective shoes in a batch of 20 is
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4. Q8 · 2020 QCAA · Paper 2 · 1 mark
   Determine the size of angle $A$ in the triangle.
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5. Q9 · 2020 QCAA · Paper 2 · 1 mark
   The displacement of a particle (in metres) at time $t$ (in seconds) is represented by the function $$s(t)=t\ln(t)-t,\quad 0<t<4.$$ Determine the approximate acceleration of the particle at time $t=3$.
   Question and worked solution
6. Q10 · 2020 QCAA · Paper 2 · 1 mark
   The approximate value of $x$ where the graph of the function $y=x^3+6x^2+7x-2\cos(x)$ changes concavity is
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7. Q11 · 2020 QCAA · Paper 2 · 4 marks
   A sugar company samples the packets of sugar it produces and finds that 5% of packets are underweight. Consider a batch of 20 packets.
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8. Q12 · 2020 QCAA · Paper 2 · 7 marks
   The rates of change in population for two cities are given by $$\text{City A: }A'(t)=\frac{45}{t+1},$$ $$\text{City B: }B'(t)=105e^{0.03t},$$ where $t$ is the number of years since 2018 and both $A'(t)$ and $B'(t)$ are measured in people per year. At the beginning of 2018, City A had a population of 5000, and City B had a population of 3500.
   Question and worked solution
9. Q13 · 2020 QCAA · Paper 2 · 6 marks
   An online retailer claims that 90% of all orders are shipped within 12 hours of being received. On a particular day, 121 orders were received and 102 orders were shipped within 12 hours. The distribution of the sample proportion of all orders that are shipped within 12 hours of being received on any day is approximately normal.
   Question and worked solution
10. Q14 · 2020 QCAA · Paper 2 · 6 marks
    Let $X$ denote the time in minutes between the arrival of trains at a station. The cumulative distribution function of $X$ is defined by $$F(x)=\begin{cases}2-\frac{10}{x},&5\le x\le10,\\0,&\text{otherwise.}\end{cases}$$
    Question and worked solution
11. Q15 · 2020 QCAA · Paper 2 · 3 marks
    A field is divided into five sections as shown. The width of each section is 1 metre. The perpendicular height, in metres, of each section is given in the diagram. The area of the field was approximated using the trapezoidal rule and found to be $11.12\;\mathrm{m^2}$.
    Question and worked solution
12. Q16 · 2020 QCAA · Paper 2 · 4 marks
    Bottles of soft drink should contain a volume with a mean of 591 mL, but some variation is expected. Any bottle at or below the 20th percentile of the volume distribution is rejected. A percentile is a measure in statistics that shows the values below which a given percentage of observations occur. Thirty-five per cent of the bottles contain 593 mL or more o…
    Question and worked solution
13. Q17 · 2020 QCAA · Paper 2 · 4 marks
    In a survey of 326 lecturers, 303 said that on at least one occasion a mobile phone had rung in a lecture they were giving. Determine the sample size required to conduct a follow-up survey that provides 95% confidence that this one-point estimate is correct to within $\pm0.02$ of the population proportion.
    Question and worked solution
14. Q18 · 2020 QCAA · Paper 2 · 4 marks
    The diagram shows the quadrilateral $ABCD$. Determine the perimeter of the quadrilateral.
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15. Q19 · 2020 QCAA · Paper 2 · 7 marks
    Consider the following information when completing this question. The length of a curve $y=f(x)$ over the interval $[a,b]$ is $$\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx.$$ You are driving along a road with a vertical distance above sea level $D$ (in metres) given by the function $$D(x)=300+\ln(x^2-3x+e),$$ where $x$ is the horizontal distance from a…
    Question and worked solution
16. Q20 · 2020 QCAA · Paper 2 · 5 marks
    Assuming the approximate normality of sample proportions $(\hat p_1$ and $\hat p_2)$ and based on two independent samples, the approximate confidence interval for the difference of two proportions is given by $$\left(\hat p_1-\hat p_2-z\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}},\;\hat p_1-\hat p_2+z\sqrt{\frac{\hat p_1(1-\hat p_…
    Question and worked solution
17. Q1 · Original practice · 1 mark
    If $f(x)=e^{2x}$, then $f'(x)$ is
    Differentiation of exponential and logarithmic functions
18. Q2 · Original practice · 1 mark
    Determine $\dfrac{d}{dx}[\tan(3x)]$.
    Differentiation of trigonometric functions and differentiation rules
19. Q3 · Original practice · 1 mark
    The graph shown is the graph of $f'(x)$. At which value of $x$ does $f$ have a local maximum?
    Further applications of differentiation
20. Q4 · Original practice · 1 mark
    An antiderivative of $6x^2-4x+1$ is
    Introduction to integration
21. Q5 · Original practice · 1 mark
    A discrete random variable $X$ has $P(X=x)=k(x+1)$ for $x=0,1,2$. Determine $k$.
    Discrete random variables
22. Q6 · Original practice · 1 mark
    Evaluate $\displaystyle\int_0^2 3x^2\,dx$.
    Further integration
23. Q7 · Original practice · 1 mark
    In triangle $ABC$, $AB=10$, $AC=7$ and $\angle A=60^\circ$. Determine $BC$.
    Trigonometry
24. Q8 · Original practice · 1 mark
    A continuous random variable has density $f(x)=2x$ for $0\le x\le1$. Determine $P(X>0.6)$.
    Continuous random variables and the normal distribution