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 5 of 26

1. Q2 · 2023 QCAA · Paper 2 · 1 mark
   The probability of hitting a bullseye is $\frac1{1250}$. Which expression gives the probability of exactly one bullseye in 10 independent attempts?
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2. Q3 · 2023 QCAA · Paper 2 · 1 mark
   A normal distribution has 95% of values approximately between $50.32$ and $113.68$, with mean $82$. Its standard deviation is closest to
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3. Q4 · 2023 QCAA · Paper 2 · 1 mark
   An object has displacement $d=e^{0.5t}-1$. Its acceleration at $t=4$ is closest to
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4. Q5 · 2023 QCAA · Paper 2 · 1 mark
   Solve $\ln x+\ln(3.70)=\ln(9.25)$.
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5. Q6 · 2023 QCAA · Paper 2 · 1 mark
   For $a\ne0$, determine $\displaystyle\int_a^{5a}\frac1{x+a}\,dx$.
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6. Q7 · 2023 QCAA · Paper 2 · 1 mark
   The sampling distribution of a sample proportion has mean $0.70$ and standard deviation $0.02$. The sample size is
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7. Q8 · 2023 QCAA · Paper 2 · 1 mark
   The number of organisms is modelled by $N=15\ln(7t+1)$, $t\ge1$. The rate of change at $t=3$ is closest to
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8. Q9 · 2023 QCAA · Paper 2 · 1 mark
   For $f(x)=e^{3x}(x+1)^2$, $f'(x)=a e^{3x}(x+1)$. The expression for $a$ is
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9. Q10 · 2023 QCAA · Paper 2 · 1 mark
   A student is trying to determine which subject they performed best in compared to other students. Results from recent tests in four subjects (A to D) are shown. Assume student results in each subject are normally distributed. In which subject did the student perform best compared to other students?

   Class standard Student’s Class mean deviation result
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10. Q11 · 2023 QCAA · Paper 2 · 4 marks
    A researcher found that 17 out of 50 randomly selected people had used public transport in the past week.
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11. Q12 · 2023 QCAA · Paper 2 · 4 marks
    The graph shows the water level under a bridge over a 12-hour period.
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12. Q13 · 2023 QCAA · Paper 2 · 4 marks
    The curved lines represent graphs of the equations $y=x^2-4x+8$ and $y=10\cos(x+10)$.
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13. Q14 · 2023 QCAA · Paper 2 · 7 marks
    A fence divides a paddock into two triangular sections as shown.
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14. Q15 · 2023 QCAA · Paper 2 · 4 marks
    Determine the derivative of $f(x)=\ln x^2+\ln(x-5)^3$. Express the derivative as a single fraction in its simplest and factorised form.
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15. Q16 · 2023 QCAA · Paper 2 · 6 marks
    A particle is moving in a straight line. The velocity ($\text{m s}^{-1}$) of the particle is given by $$v(t)=\frac{20\sin(2t)}{6-5\cos(2t)},\qquad t\ge0,$$ where $t$ is time (s) after moving from its initial position. The initial position of the particle is $+6.0$ m from the origin.
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16. Q17 · 2023 QCAA · Paper 2 · 5 marks
    Model bridges were constructed for a competition. The models that could support the heaviest loads before collapsing were given awards.

    The load results of the competition were normally distributed, with a mean of 1.36 kg and a standard deviation of 0.12 kg.

    Three award categories were used: honours for the top 15% of load results; distinction for the next…
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17. Q18 · 2023 QCAA · Paper 2 · 5 marks
    A company makes windows using glass that has a mass of 5.6 kg per square metre. A customer orders an unusual window in a partial parabolic shape, as shown. Determine the mass of the window.
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18. Q19 · 2023 QCAA · Paper 2 · 6 marks
    Over a suitable domain, a hill has a cross-sectional area given by $$\int h(x)\,dx=\frac{a}{b}e^{bx}+c,$$ where: - $a$, $b$ and $c$ are constants, $b\ne0$ - $h(x)$ represents vertical distance (m), $x$ represents horizontal distance (m).

    It is known that $h(0)=1.22$ and $h(40)=25$.

    Where the gradient of the hill is 0.86 there is a tree stump. A second tree…
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19. Q1 · 2022 QCAA · Paper 1 · 1 mark
    Consider the graph of $f'(x)$ for $a\le x\le b$. Which statement describes all the local maxima and minima of the graph of $f(x)$ over $a\le x\le b$?
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20. Q2 · 2022 QCAA · Paper 1 · 1 mark
    A binomial random variable arises from the number of successes in $n$ independent Bernoulli trials. A context not suitable for modelling using a binomial random variable is recording the number of
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21. Q3 · 2022 QCAA · Paper 1 · 1 mark
    The area between the curve $y=9-x^2$ and the $x$-axis is
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22. Q4 · 2022 QCAA · Paper 1 · 1 mark
    The weekly amount of money a company spends on repairs is normally distributed, with a mean of $\char36 1200$and a standard deviation of $\char36 100.$Given that $P(Z\le-2.5)=0.0062$ and $P(Z>1)=0.1587$, where $Z$ is a standard normal random variable, determine the probability that the weekly repair costs will be between $\char36 950$and $\char36 1300.$
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23. Q5 · 2022 QCAA · Paper 1 · 1 mark
    Which normal distribution curve best represents a normal distribution with a mean of $1$ and a standard deviation of $0.5$?
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24. Q6 · 2022 QCAA · Paper 1 · 1 mark
    Which graph represents the function $f(x)=-3-\ln(x+3)$?
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