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 4 of 26

1. Q16 · 2024 QCAA · Paper 2 · 4 marks
   At council meetings in a particular town, new proposals are only discussed if more than 80% of the community are in favour of the proposal. To discover community opinion on a new bus route proposal, the council conducted several surveys, each with a sample size of 120. The distribution of the sample proportions from the surveys had a standard deviation of 0.…
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2. Q17 · 2024 QCAA · Paper 2 · 3 marks
   At a particular orchard, 3% of fruit is bruised during picking. After picking, the fruit is packed into boxes, each containing four pieces of fruit. A grocery shop orders 140 boxes of fruit to sell to their customers. Determine the expected number of boxes that will contain bruised fruit.
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3. Q18 · 2024 QCAA · Paper 2 · 5 marks
   An object experiencing straight-line motion along a path has an acceleration (m s$^{-2}$) defined by the function $a(t)=3\sin(2t)$ where $t$ is time (s) since the object begins moving, $t\ge0$. When $t=0$, both displacement and velocity are zero. On the path is a motion sensor that is able to detect motion up to 2 metres away. The object passes directly by t…
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4. Q19 · 2024 QCAA · Paper 2 · 6 marks
   The normal distribution probability density function is $$p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2},$$ with the parameters mean, $\mu$, and standard deviation, $\sigma$.

   The speeds of electric scooter (e-scooter) riders on a particular section of a bike path are approximately normally distributed with a mean of 18 km/h.…
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5. Q1 · 2023 QCAA · Paper 1 · 1 mark
   For $x>0$, $e^{\ln(x)}$ is equivalent to
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6. Q2 · 2023 QCAA · Paper 1 · 1 mark
   If $f(x)=e^{6-2x}$, determine the value of $f'(2)$.
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7. Q3 · 2023 QCAA · Paper 1 · 1 mark
   A bag contains 10 buttons of the same shape and size in different colours: 5 blue, 3 green and 2 red. If 3 buttons are randomly drawn from the bag, which probability can be calculated using the binomial distribution?
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8. Q4 · 2023 QCAA · Paper 1 · 1 mark
   Given $f'(x)=\frac{20}{x^3}$, an antiderivative is
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9. Q5 · 2023 QCAA · Paper 1 · 1 mark
   Determine $\displaystyle\int_1^3\frac1{2x}\,dx$.
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10. Q6 · 2023 QCAA · Paper 1 · 1 mark
    Which sequence of $h$-values is suitable for estimating $\displaystyle\lim_{h\to0}\frac{a^h-1}{h}$?
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11. Q7 · 2023 QCAA · Paper 1 · 1 mark
    A random variable has probability density function $f(x)=\frac{x}{8}$ for $0\le x\le4$. Its mean is
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12. Q8 · 2023 QCAA · Paper 1 · 1 mark
    For a 95% confidence interval with margin of error $E=0.03$, $z=1.96$ and estimated proportion $0.6$, which expression gives the required sample size?
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13. Q9 · 2023 QCAA · Paper 1 · 1 mark
    Determine $\displaystyle\int_0^3 \pi\sin\left(\frac{\pi x}{3}\right)\,dx$.
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14. Q10 · 2023 QCAA · Paper 1 · 1 mark
    A random variable has probability density function $f(y)=1+y$ for $0\le y\le\sqrt3-1$. Determine $P(0\le Y\le\frac12)$.
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15. Q11 · 2023 QCAA · Paper 1 · 4 marks
    Two random samples (A and B) were obtained using two different Bernoulli experiments. Each Bernoulli trial in the random samples was recorded as 1 (for success) or 0 (for failure). The results are shown.

    | Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | A | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |…
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16. Q12 · 2023 QCAA · Paper 1 · 4 marks
    The region bounded by the $x$-axis and the curve of $f(x)=x(4-x)$ represents the plan of a garden bed. All measurements are in metres.
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17. Q13 · 2023 QCAA · Paper 1 · 5 marks
    At a certain airport, the departure of one in five international flights is delayed every day. The status of any flight is independent of other flights.

    One international flight is selected at random each day for three days. Each selection is recorded as either ‘delayed’ or ‘not delayed’.
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18. Q14 · 2023 QCAA · Paper 1 · 6 marks
    The rate of change in the number of bacteria in a science experiment is represented by $$\frac{dP}{dt}=e^{2t},\qquad t\ge0,$$ where $t$ represents the time (hours) since starting the experiment and $P$ represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present, i.e. $P(0)=60$.
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19. Q15 · 2023 QCAA · Paper 1 · 4 marks
    In a certain game, players throw one water balloon at a target. There is a one in four chance of hitting the target.
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20. Q16 · 2023 QCAA · Paper 1 · 5 marks
    Solve for $x$ in the equation $$4+7e^{-2x}=3e^{2x}.$$
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21. Q17 · 2023 QCAA · Paper 1 · 6 marks
    A chemical is added to the water in a swimming pool at 10:00 am to prevent algae. The amount of chemical absorbed into the water over time $t$ (hours) is represented by $$A=10t^2-4t^3,\qquad0\le t\le1\frac23.$$ Determine the time of day when the rate of absorption of the chemical is at its maximum. Use calculus techniques to verify that your time corresponds…
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22. Q18 · 2023 QCAA · Paper 1 · 4 marks
    A person enters the lowest carriage of a miniature Ferris wheel with a six-metre diameter. The bottom carriage is one metre off the ground. When top speed is reached, it takes three seconds for a carriage to travel from the lowest to the highest point of the ride. It is claimed that:

    **The vertical motion of the Ferris wheel produces a maximum vertical acce…
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23. Q19 · 2023 QCAA · Paper 1 · 7 marks
    Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days.

    	Day 1	Day 2
    Jaxon’s customers	40	30
    Shari’s customers	10	20
    Total	50	50

    The number of daily customers for each shop can be modelled by the equation $y=A\ln(Bx)$, where $x$ is the day and $y$…
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24. Q1 · 2023 QCAA · Paper 2 · 1 mark
    For $f(x)=\sin(3x)$, $f'(\frac\pi8)$ is closest to
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