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 8 of 26

1. Q17 · 2021 QCAA · Paper 1 · 3 marks
   In any five-day working week Leonardo either catches a bus to work or uses another form of transportation. On average, he catches the bus to work on three of the five days. His decision on any given day is independent of his decision on any other day. Determine the probability that Leonardo catches a bus to work on exactly one day in a given five-day working…
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2. Q18 · 2021 QCAA · Paper 1 · 4 marks
   The graph of $y=f(x)$, where $f(x)$ is the quadratic function $f(x)=ax^2+bx+4$, is shown. Two regions of the area between the graph of $y=f(x)$ and the $x$-axis are shaded. Region $P$ has an area of $\dfrac{13}{6}$ units$^2$ and Region $Q$ has an area of $\dfrac{43}{6}$ units$^2$. Determine the values of $a$ and $b$.
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3. Q19 · 2021 QCAA · Paper 1 · 4 marks
   A firm aims to have 95% confidence in estimating the proportion of office workers who respond to an email in less than an hour to within $\pm0.05$. A survey has never been undertaken before, so no past data is available. The firm believes that if the proportion is 0.5, then this will result in the largest variability in the sample proportion. Based on this,…
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4. Q20 · 2021 QCAA · Paper 1 · 7 marks
   The population of rabbits $(P)$ on an island, in hundreds, is given by $P(t)=t^2\ln(3t)+6$, $t>0$, where $t$ is time in years. Determine the intervals of time when the population is increasing and the intervals when it is decreasing.
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5. Q1 · 2021 QCAA · Paper 2 · 1 mark
   The scores obtained on a test can be assumed to be normally distributed with a mean of 102 and a standard deviation of 19. What proportion of scores are over 113?
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6. Q2 · 2021 QCAA · Paper 2 · 1 mark
   A substance is being heated such that its temperature $T$ in $^\circ$C after $t$ minutes is given by the function $T=2e^{0.5t}$. The first integer value of $t$ for which the instantaneous rate of change of temperature is greater than $100\ ^\circ$C per minute is
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7. Q3 · 2021 QCAA · Paper 2 · 1 mark
   A random sample of people were surveyed about the most important factor when deciding where to shop. The results appear in the table.

   Factor	Percentage (%)
   Price	40
   Quality of merchandise	30
   Service	15
   Shopping environment	15

   If the sample size was 1200, the approximate 95% confidence interval for the proportion of…
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8. Q4 · 2021 QCAA · Paper 2 · 1 mark
   Using the trapezoidal rule with an interval size of 1, the approximate value of the integral $$\int_0^3 0.5^x\,dx$$ is
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9. Q5 · 2021 QCAA · Paper 2 · 1 mark
   Solve for $x$ given that $\log_3(x-1)=2$.
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10. Q6 · 2021 QCAA · Paper 2 · 1 mark
    When seeds of a certain variety of flower are planted, the probability of each seed germinating is 0.8. If eight seeds are planted, what is the probability that at least six seeds will germinate?
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11. Q7 · 2021 QCAA · Paper 2 · 1 mark
    Determine $f(x)$, given $f'(x)=6x^2+\dfrac1{x^2}+\dfrac1x$ and $f(1)=5$.
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12. Q8 · 2021 QCAA · Paper 2 · 1 mark
    The displacement (in metres) of a particle is given by $s(t)=-3\cos(t)+2\sin(t)$, where $t$ is in seconds. The instantaneous velocity of the particle at time $t=\dfrac\pi2$ seconds is
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13. Q9 · 2021 QCAA · Paper 2 · 1 mark
    The graphs of the functions $f(x)=2e^x+5$ and $g(x)=\dfrac3{e^x}$ intersect at point A. Determine the coordinates of point A.
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14. Q10 · 2021 QCAA · Paper 2 · 1 mark
    An object travels in a straight line so that its velocity at time $t$ seconds is given by $v(t)=2t+\sin(2t)$. Determine the expression for acceleration as a function of time.
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15. Q11 · 2021 QCAA · Paper 2 · 5 marks
    Consider the function $f(x)=e^x\sin(x)$, $0\le x\le2\pi$.
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16. Q12 · 2021 QCAA · Paper 2 · 4 marks
    The velocity function of an object in $\text{m s}^{-1}$ is given by $v(t)=\cos\left(6t+\dfrac\pi2\right)+2$, $0\le t\le5$. Initially, the object is at the origin.
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17. Q13 · 2021 QCAA · Paper 2 · 7 marks
    The amount of gravel (in tonnes) sold by a construction company in a given week is a continuous random variable $X$ and has a probability density function defined by $$f(x)=\begin{cases}c(1-x^2),&0\le x\le1,\\0,&\text{otherwise.}\end{cases}$$
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18. Q14 · 2021 QCAA · Paper 2 · 7 marks
    The heights of students at School A are normally distributed with a mean of 165 cm and a standard deviation of 15 cm.
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19. Q15 · 2021 QCAA · Paper 2 · 4 marks
    A new internet search engine gives a ranking $R$ to each website based on the function $R=\log_{10}(50h^2)$, where $h$ is the number of hits (visits) the website has received. If a website currently has 100 hits, determine how many more hits they need to increase their ranking by 1.
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20. Q16 · 2021 QCAA · Paper 2 · 4 marks
    In the diagram DC represents a 60 metre vertical tower. A and B are two points in the same horizontal plane as the foot C of the tower. The angle above the horizontal from A to D is $28^\circ$ and the angle above the horizontal from B to D is $35^\circ$. The bearing of C from A is $050^\circ$T and the bearing of C from B is $300^\circ$T. Determine the distan…
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21. Q17 · 2021 QCAA · Paper 2 · 4 marks
    Rabbits and foxes are among two species of mammals that live on an isolated island. Rabbits represent a significant food source for the foxes. The populations of rabbits and foxes were monitored each month for two years. The graph shows the population of foxes (in thousands) and the population of rabbits (in thousands), at any time $t$ (in months) over the t…
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22. Q18 · 2021 QCAA · Paper 2 · 3 marks
    The number of animals in a population (in thousands) is modelled by the function $P$ such that $$P(t)=\frac{100}{1+4e^{-t}},$$ where $t$ is in years. Determine the number of animals in the population when the population is growing the fastest.
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23. Q19 · 2021 QCAA · Paper 2 · 4 marks
    A random variable $X$, defined over the interval $a\le x\le b$, is uniformly distributed if its probability density function is defined by $$f(x)=\begin{cases}\dfrac{1}{b-a},&a\le x\le b,\\0,&\text{otherwise.}\end{cases}$$ The expected value and variance of a uniform random variable $X$ are $$E(X)=\frac{a+b}{2},\qquad \operatorname{Var}(X)=\frac{(b-a)^2}{12}…
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24. Q20 · 2021 QCAA · Paper 2 · 3 marks
    The random variable $B$ is normally distributed with a mean of 0 and a standard deviation of 1. Determine the probability that the quadratic equation $x^2+3x+2B=0$ has real roots.
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