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 1 of 26

1. Q1 · 2025 QCAA · Paper 1 · 1 mark
   Determine the value of $\displaystyle\int_1^2 3x^2\,dx$.
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2. Q2 · 2025 QCAA · Paper 1 · 1 mark
   State the domain of the function $y=\ln(x)$.
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3. Q3 · 2025 QCAA · Paper 1 · 1 mark
   A council wants to survey residents about a new dog park. Which sampling method would best minimise bias in the survey?
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4. Q4 · 2025 QCAA · Paper 1 · 1 mark
   Solve for $x$ in the equation $\ln(x)+\ln(2)=\ln(8)$.
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5. Q5 · 2025 QCAA · Paper 1 · 1 mark
   A box contains 100 balls, each of which is either green, blue or red. A teacher asks students to select a ball at random, record its colour and return it to the box. This process is repeated until each student has selected and recorded the colours of a sample of balls. Each student chooses their own sample size. The sample size, $n$, and sample proportion of…
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6. Q6 · 2025 QCAA · Paper 1 · 1 mark
   Calculate the area (m$^2$) of the triangle.
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7. Q7 · 2025 QCAA · Paper 1 · 1 mark
   Determine $\dfrac{d}{dx}e^{4x+3}$.
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8. Q8 · 2025 QCAA · Paper 1 · 1 mark
   Determine the area (units$^2$) between the graph of $y=4x^3$ and the $x$-axis from $x=-1$ to $x=2$.
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9. Q9 · 2025 QCAA · Paper 1 · 1 mark
   Determine $\displaystyle\int\frac{1}{7x-4}\,dx$, where $x>\frac47$.
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10. Q10 · 2025 QCAA · Paper 1 · 1 mark
    Researchers asked 100 customers at a cafe about their daily coffee consumption. The results are shown.

    Determine the probability that a randomly selected customer from this group drinks more than three coffees a day.
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11. Q11 · 2025 QCAA · Paper 1 · 5 marks
    Determine the following integrals.
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12. Q12 · 2025 QCAA · Paper 1 · 4 marks
    A teacher conducted a survey about student involvement in school activity groups. They found that of a random sample of 25 students, five students were members of the environment group. Determine the approximate confidence interval for the proportion of students at the school who are members of the environment group, using $z=1$. Express your answer using fr…
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13. Q13 · 2025 QCAA · Paper 1 · 4 marks
    Use $f''(x)=2x-6$ to demonstrate that the function $y=f(x)$ has a point of inflection at $x=3$.
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14. Q14 · 2025 QCAA · Paper 1 · 4 marks
    Determine all solutions to $2\cos(\theta)-1=0$, where $0^\circ\le\theta\le720^\circ$.
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15. Q15 · 2025 QCAA · Paper 1 · 6 marks
    Determine all possible solutions for the equation $(\log_2(x))^2=\log_2(x^2)+8$. Evaluate the reasonableness of one solution.
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16. Q16 · 2025 QCAA · Paper 1 · 5 marks
    The energy produced by solar panels on a house is stored in a battery. Energy consumed in the house comes out of the battery.

    The rate of energy produced by the solar panels during daylight hours on a particular day is approximately modelled by $$R(t)=2.5\sin\left(\frac{\pi}{12}(t-6)\right),\qquad 6\le t\le18,$$ where $t$ is the number of hours after midnig…
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17. Q17 · 2025 QCAA · Paper 1 · 6 marks
    In a computer fishing game, a player repeatedly casts their hook into either a blue pond or a red pond. Each fish they catch scores points. In the blue pond, the probability of catching a fish on a cast is $\frac23$ and each fish caught scores 10 points. In the red pond, the probability of catching a fish on a cast is $\frac13$ and each fish caught scores 15…
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18. Q18 · 2025 QCAA · Paper 1 · 6 marks
    Two objects are launched simultaneously from different positions and travel along the same straight-line path. The objects are launched towards each other with the same initial speed. The first object’s displacement (m) from the origin is given by $$d_1=\frac13t^3-\frac12t^2+kt,$$ where $t$ is the time (s) since the objects were launched and $k$ is a constan…
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19. Q19 · 2025 QCAA · Paper 1 · 5 marks
    A farmer wishes to construct the shortest possible fence to enclose a triangular area in the corner of a paddock. There are two existing fences, the western fence and the southern fence. The new fence must pass through the old homestead gate, as shown.

    Determine the length of the shortest possible fence. Verifying that the fence length is a minimum is not r…
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20. Q1 · 2025 QCAA · Paper 2 · 1 mark
    Determine $\displaystyle\int10.4x^3\,dx$.
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21. Q2 · 2025 QCAA · Paper 2 · 1 mark
    Determine the approximate area (units$^2$) between the curve and the $x$-axis for $0\le x\le8$ by calculating the area of the shaded rectangles.
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22. Q3 · 2025 QCAA · Paper 2 · 1 mark
    A streaming service surveyed a random sample of 100 customers, and 60 customers said that they prefer to watch movies rather than miniseries. Based on these results, the 95% confidence interval for the proportion of customers who prefer to watch movies was $(0.5,0.7)$. Which statement shows the correct interpretation of this outcome?
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23. Q4 · 2025 QCAA · Paper 2 · 1 mark
    An object moves in a straight line such that its velocity (m s$^{-1}$) is given by $v(t)=t^2-e^{-t}$ for $0\le t\le5$, where $t$ represents time (s). Determine the time, $t$, when the acceleration of the object is $2$ m s$^{-2}$.
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24. Q5 · 2025 QCAA · Paper 2 · 1 mark
    Which statement best describes a feature of the graph of the exponential function $y=e^x$, $x\in\mathbb R$?
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