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 3 of 26

1. Q11 · 2024 QCAA · Paper 1 · 6 marks
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2. Q12 · 2024 QCAA · Paper 1 · 6 marks
   Each day over a three-day long weekend, a family spins a pointer on a circular board to decide whether they will spend the day at the beach or bushwalking. The circular board consists of three equal sections.
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3. Q13 · 2024 QCAA · Paper 1 · 6 marks
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4. Q14 · 2024 QCAA · Paper 1 · 5 marks
   At a particular game at a local sporting venue, 60% of spectators support the home team and the remainder support the away team. A researcher asked six groups of 10 spectators which team they supported. Each spectator was recorded as either H (supports home team) or A (supports away team). The results were:

   Group 1: H A H H H A H H H H Group 2: A A H A A H…
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5. Q15 · 2024 QCAA · Paper 1 · 4 marks
   A survey was conducted to understand whether people support a new policy. Using a $z$-score of 2, the approximate confidence interval for the population proportion of people who support the policy was calculated as $\left(\frac3{10},\frac7{10}\right)$.
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6. Q16 · 2024 QCAA · Paper 1 · 4 marks
   The graph is of the form $y=\log_a(x+b)$. A point on the graph $(4,3)$ is labelled. The line $x=-4$ is an asymptote.

   There is a point $P(x_P,y_P)$ on the graph where $y_P$ is twice the value of the $y$-intercept of the curve. Determine the value of $x_P$.
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7. Q17 · 2024 QCAA · Paper 1 · 3 marks
   A community group that uses social media created a new post on the internet on a day when they had 1000 members. The rate of change in their number of members (members/day) is given by $f'(t)=3e^{0.5t}$, where $t$ represents days after the new post. Determine the time it will take for the community group to achieve seven times the initial number of members.…
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8. Q18 · 2024 QCAA · Paper 1 · 5 marks
   The diagram shows some dimensions of a large storage container that is a rectangular prism. The angle $ABC$ is $60^\circ$. A person requires a container that is at least 4 metres in height.

   Make a justified decision about whether this storage container meets the person’s requirements.
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9. Q19 · 2024 QCAA · Paper 1 · 6 marks
   A permanent ice glacier is in a valley in New Zealand. Due to the temperature changes of the seasons each year, the glacier expands for six months and recedes for six months. The changing distance of a point on the front edge of the glacier to a car park can be modelled by a sine function. During the colder months, when the glacier expands, the front edge of…
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10. Q1 · 2024 QCAA · Paper 2 · 1 mark
    The probability of hitting a target in a particular binomial experiment is 0.72. Determine the mean of the number of hits if this experiment is repeated eight times.
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11. Q2 · 2024 QCAA · Paper 2 · 1 mark
    Calculate the expected value of a continuous random variable $X$ with the probability density function $$p(x)=\begin{cases}\frac14x^2,&0\le x\le\sqrt[3]{12}\\0,&\text{otherwise}.\end{cases}$$
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12. Q3 · 2024 QCAA · Paper 2 · 1 mark
    The derivative of the function $f(x)$ is given by $f'(x)=\sin(2x)$. It is known that $f(\frac{\pi}{2})=4$. Determine $f(x)$.
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13. Q4 · 2024 QCAA · Paper 2 · 1 mark
    Consider the Bernoulli distribution where the outcomes for rolling a six-sided die are a four and not rolling a four. Determine the variance of the resulting Bernoulli distribution in this scenario.
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14. Q5 · 2024 QCAA · Paper 2 · 1 mark
    The mass (g) of adult kookaburras in a certain region is normally distributed with a mean of 300 g and a standard deviation of 13 g. Select the correct statement about the mass of adult kookaburras.
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15. Q6 · 2024 QCAA · Paper 2 · 1 mark
    Determine the derivative of $y=2x\cos(3x)$.
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16. Q7 · 2024 QCAA · Paper 2 · 1 mark
    Identify the possible values for $a$ in the triangle.
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17. Q8 · 2024 QCAA · Paper 2 · 1 mark
    Calculate the total enclosed area between the graph of $y=x^2-x-6$ and the $x$-axis from $x=1$ to $x=5$.
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18. Q9 · 2024 QCAA · Paper 2 · 1 mark
    It is known that $f'(x)=0$ and $f''(x)<0$ for one of the labelled points on the graph of $f(x)$. Which point matches this description?
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19. Q10 · 2024 QCAA · Paper 2 · 1 mark
    The velocity (m s$^{-1}$) at time $t$ (s) of an object is given by $v(t)=0.4t^2+3t$ for $t\ge0$. The change in displacement (m) of the object from four to five seconds is
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20. Q11 · 2024 QCAA · Paper 2 · 4 marks
    State the trapezoidal rule and use it with six strips to determine an approximate value of the definite integral for the curve of $f(x)=4(x-3)^2$ from $x=0$ to $x=3$. Show all substitutions made into the rule.
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21. Q12 · 2024 QCAA · Paper 2 · 5 marks
    The magnitude of an earthquake can be modelled by the logarithmic equation $M_A=\log_{10}\left(\frac{I_A}{I_0}\right)$, where $M_A$ is the magnitude at a location A, $I_A$ is the intensity of the earthquake at location A and $I_0$ is a constant. An earthquake at location P had a magnitude of 5.2. A different earthquake at location Q had a magnitude of 3.5.
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22. Q13 · 2024 QCAA · Paper 2 · 8 marks
    The number of termites in a particular nest can be modelled by $N(t)=\dfrac{A}{2+e^{-t}}$, where $A$ is a constant and $t$ represents time (months) since the nest first became a visible mound above ground level.

    It is estimated that when the mound first became visible, the population was $3\times10^5$ termites.
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23. Q14 · 2024 QCAA · Paper 2 · 6 marks
    A football coach offered a 12-day intensive training clinic. During the clinic, the height that each player could kick a football was monitored. One player’s kick heights could be modelled by $H(t)=\log_{10}(10t+10)+5$, $0\le t\le12$, where $H(t)$ is vertical height (m) and $t$ is the time (days) spent in training.
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24. Q15 · 2024 QCAA · Paper 2 · 4 marks
    The term extremely tall is used to describe any person whose height is three standard deviations or more above the mean height of the population. A person who just qualifies as extremely tall in a country where heights are normally distributed with a mean height of 180 cm and a standard deviation of 10 cm travels to another country. The person discovers they…
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