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 2 of 26

1. Q6 · 2025 QCAA · Paper 2 · 1 mark
   Solve the equation $\cos(\theta)=-0.6$ for $0^\circ\le\theta\le360^\circ$.
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2. Q7 · 2025 QCAA · Paper 2 · 1 mark
   An object starts from rest at the origin with acceleration (m s$^{-2}$) given by $a(t)=16-t^2+t^3$ for $t\ge0$, where $t$ represents time (s). Determine the displacement from the origin of the object 1.5 seconds after it starts moving.
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3. Q8 · 2025 QCAA · Paper 2 · 1 mark
   The table shows the probability distribution for a random variable $X$ in a Bernoulli experiment. The random variable has only two possible values: 0 represents failure and 1 represents success.

   $x$	0	1
   $P(X=x)$	0.6	0.4

   If six Bernoulli experiments are conducted, determine the probability of getting exactly two successes.
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4. Q9 · 2025 QCAA · Paper 2 · 1 mark
   Determine the gradient of the line perpendicular to the tangent to the graph of $y=\ln(4x)$ at the point $(0.8,1.163)$.
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5. Q10 · 2025 QCAA · Paper 2 · 1 mark
   The outside air temperature, $T$ (°C), on a particular day at a certain location is modelled by the function $$T=25-7\cos\left(\frac{\pi t}{12}\right),\qquad0\le t\le24,$$ where $t$ is the time (hours) since 6:00 am. The rate of change of temperature at 11:30 am is
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6. Q11 · 2025 QCAA · Paper 2 · 6 marks
   View question and marking material
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7. Q12 · 2025 QCAA · Paper 2 · 7 marks
   A cockroach population is modelled by the function $P(t)=P_0e^{kt}$, where $P$ is the population after $t$ weeks and $k$ is a population constant. Initially, 100 cockroaches were counted. After three weeks, there were 120.
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8. Q13 · 2025 QCAA · Paper 2 · 4 marks
   A school investigated how many hours students sleep per night. To obtain data, a random sample of students was surveyed. The results are shown.
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9. Q14 · 2025 QCAA · Paper 2 · 6 marks
   The number of tourists visiting a country at any given time is modelled by $$N(t)=18000\sin\left(\frac{\pi}{6}t+6\right)+22000,\qquad0\le t\le12,$$ where $t$ is the time (months) from the start of the year.
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10. Q15 · 2025 QCAA · Paper 2 · 3 marks
    A tour operator offers day cruises off the Queensland coast. They advertise that on any given day, customers have a 65% probability of seeing at least one whale. The tour operator conducts cruises for 40 consecutive days. Let $X$ be the binomial random variable representing the outcome — success or failure — over the 40 days. A success is customers seeing at…
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11. Q16 · 2025 QCAA · Paper 2 · 4 marks
    A species of fish is being raised in a fish pond. The number of fish, $M$, in the pond can be modelled by a function $M=100b^t$, where $t$ is the time (days) since the fish were initially introduced into the pond and $b$ is a constant to be determined. After seven days, there are 150 fish in the pond. Find the rate of population growth when the pond contains…
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12. Q17 · 2025 QCAA · Paper 2 · 4 marks
    The probability density function approximates the time (s) that people spend viewing a particular photo on a social media page. $$p(t)=\begin{cases}ke^{-7t/50},&0\le t\le20,\\0,&\text{otherwise.}\end{cases}$$ Determine the mean time people spend viewing the photo.
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13. Q18 · 2025 QCAA · Paper 2 · 5 marks
    The results of an employee satisfaction survey of 500 employees at a large company are presented to board members. The results include a 95% confidence interval for the proportion of satisfied employees. The lower end of the confidence interval is 0.648. A board member would like to use the survey results to make the claim that the proportion of the satisfie…
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14. Q19 · 2025 QCAA · Paper 2 · 6 marks
    A scientist is gathering data on two species of horned beetle, species A and B. Horn length is a method of distinguishing the species. Species A horn lengths are normally distributed with a mean of 20 mm and a standard deviation of 2 mm. It is known that 14.6% of species B beetles have horns shorter than 18 mm. In the particular population the scientist is s…
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15. Q1 · 2024 QCAA · Paper 1 · 1 mark
    Determine $\displaystyle\int x^4\,dx$.
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16. Q2 · 2024 QCAA · Paper 1 · 1 mark
    If $y=e^{\sin x}$, then $\frac{dy}{dx}$ is
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17. Q3 · 2024 QCAA · Paper 1 · 1 mark
    If the confidence level of a confidence interval is increased while the sample data are unchanged, what happens to the $z$-value and the margin of error?
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18. Q4 · 2024 QCAA · Paper 1 · 1 mark
    Simplify $y=2\ln(e^x)$.
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19. Q5 · 2024 QCAA · Paper 1 · 1 mark
    Determine $\displaystyle\int_a^b 2\cos(x)\,dx$, where $a=\frac{\pi}{3}$ and $b=\frac{\pi}{2}$.
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20. Q6 · 2024 QCAA · Paper 1 · 1 mark
    Differentiate $y=\ln(x)\cos(x)$ with respect to $x$.
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21. Q7 · 2024 QCAA · Paper 1 · 1 mark
    Twenty families are selected to participate in a lifestyle study related to family size. The number of children in these families is uniformly distributed as shown.

    A random sample of five families is chosen from this group, without replacement. A possible mean number of children in the sample is
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22. Q8 · 2024 QCAA · Paper 1 · 1 mark
    The graph of $f(x)$ is shown. Identify the graph of the second derivative $f''(x)$.
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23. Q9 · 2024 QCAA · Paper 1 · 1 mark
    At a certain location, the temperature (°C) can be modelled by the function $T=5\sin\left(\frac{\pi}{12}x\right)+23$, where $x$ is the number of hours after sunrise. Determine the rate of change of temperature (°C/hour) when $x=4$.
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24. Q10 · 2024 QCAA · Paper 1 · 1 mark
    Given that $\log_{10}6=0.778$, determine the value of $\log_{10}600$.
    Question and worked solution