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 7 of 26

1. Q12 · 2022 QCAA · Paper 2 · 4 marks
   Suppose that the distance travelled by vehicles in a year can be modelled by a normal distribution. In 2021, vehicles travelled a mean of 13 700 km with a standard deviation of 3400 km.
   Question and worked solution
2. Q13 · 2022 QCAA · Paper 2 · 4 marks
   A sandy beach has a fence on one side and ocean on the other. The width of the beach is the distance (in metres) from the fence to the water’s edge. The width, $w(t)$, at a certain point is given by $$w(t)=a+b\sin\left(\frac{\pi}{6}t-\frac{\pi}{3}\right),\qquad0\le t\le24,$$ where $t$ is time (in hours) since 6 am. The width of the beach is 8 metres at 8 am…
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3. Q14 · 2022 QCAA · Paper 2 · 8 marks
   Ravi randomly sampled 200 different pet owners in Brisbane and found that 50 celebrate their pet’s birthday.
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4. Q15 · 2022 QCAA · Paper 2 · 7 marks
   A hiker begins her journey at a youth hostel ($H$) and walks for 8 km on a bearing of $052^\circ$T to her lunch stop ($L$). She then walks on a bearing of $210^\circ$T for 5.2 km until she reaches a campsite ($C$). Determine the direction she would need to walk in a straight line to return directly to the youth hostel.
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5. Q16 · 2022 QCAA · Paper 2 · 4 marks
   The time spent waiting in a queue at a certain supermarket is given by $(X+11)$ minutes, where $X$ is a random variable with the probability density function $$f(x)=\begin{cases}\dfrac{a(4-x^2)}{32},&-2\le x\le2,\\0,&\text{otherwise.}\end{cases}$$ Determine the probability of waiting between 10 and 12 minutes in a queue at this supermarket.
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6. Q17 · 2022 QCAA · Paper 2 · 4 marks
   A snail is travelling along a straight path from point $A$. The snail’s velocity (cm min$^{-1}$) is modelled by $v(t)=1.4\ln(1+t^2)$, where $t$ is time (in minutes) for $0\le t\le15$. An ant passes point $A$ 12 minutes after the snail and follows the snail’s path. The ant moves with a constant acceleration of 2 cm min$^{-2}$ and passes the snail at $t=15$ mi…
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7. Q18 · 2022 QCAA · Paper 2 · 3 marks
   The intelligence quotient (IQ) of individuals in a population is normally distributed, with a mean of 100 and a standard deviation of 16. Nine individuals are chosen at random from the population. Determine the probability that no more than two of the individuals have an IQ of at least 120.
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8. Q19 · 2022 QCAA · Paper 2 · 4 marks
   Flying foxes enter and leave a fruit-growing region every evening. The rate at which the flying foxes enter the region is modelled by $$A(t)=42\sin\left(0.03t-\frac{\pi}{3}\right)+71,\qquad0\le t\le240.$$ The rate at which the flying foxes leave the region is modelled by $$L(t)=42\sin\left(0.04t-\frac{\pi}{3}\right)+42,\qquad0\le t\le240.$$ Both $A(t)$ and…
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9. Q1 · 2021 QCAA · Paper 1 · 1 mark
   $2\log_{10}(x)-\log_{10}(3x)$ is equal to
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10. Q2 · 2021 QCAA · Paper 1 · 1 mark
    The table shows the time a technician has spent servicing photocopiers.

    Time (in minutes)	Frequency
    $0\le t<5$	10
    $5\le t<10$	20
    $10\le t<15$	30
    $15\le t<20$	40
    $20\le t<25$	100

    What is the probability that a given service required at least 10 minutes but less than 20 minutes?
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11. Q3 · 2021 QCAA · Paper 1 · 1 mark
    Determine $$\int 10e^{4x}\,dx.$$
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12. Q4 · 2021 QCAA · Paper 1 · 1 mark
    The second derivative of the function $f(x)$ is given by $$f''(x)=\frac{2x}{1+x^2}.$$ The interval on which the graph of $f(x)$ is concave up is
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13. Q5 · 2021 QCAA · Paper 1 · 1 mark
    The graph of $f''(x)$ is shown. Which of the following could be the graph of $f'(x)$?
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14. Q6 · 2021 QCAA · Paper 1 · 1 mark
    A random variable $X$ is the number of successes in a Bernoulli experiment with $n$ trials, each with a probability of success $p$ and a probability of failure $q$. The probability distribution table of $X$ is shown.

    $k$	$P(X=k)$
    0	$1/81$
    1	$8/81$
    2	$24/81$
    3	$32/81$
    4	$16/81$

    Which values of $n$, $p$ and $q$…
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15. Q7 · 2021 QCAA · Paper 1 · 1 mark
    Determine $$\int_1^3(2x+3)\,dx.$$
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16. Q8 · 2021 QCAA · Paper 1 · 1 mark
    The continuous random variable $X$ has the probability density function $$f(x)=\begin{cases}\dfrac{3}{x^2},&1\le x\le\dfrac32,\\0,&\text{otherwise.}\end{cases}$$ The mean of $X$ is
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17. Q9 · 2021 QCAA · Paper 1 · 1 mark
    A basket contains 10 green apples and 30 red apples. Three apples are drawn at random from the basket with replacement. Determine the probability that exactly two of the three apples are green.
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18. Q10 · 2021 QCAA · Paper 1 · 1 mark
    Handspans of teenagers are approximately normally distributed, with a mean of 15 cm and a standard deviation of 2 cm. Which of the following groups is expected to be the largest?
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19. Q11 · 2021 QCAA · Paper 1 · 5 marks
    Determine the derivative with respect to $x$ of the following functions.
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20. Q12 · 2021 QCAA · Paper 1 · 5 marks
    Solve for $x$ in the following.
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21. Q13 · 2021 QCAA · Paper 1 · 5 marks
    Consider the functions $f(x)=x^2$ and $g(x)=4x$.
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22. Q14 · 2021 QCAA · Paper 1 · 4 marks
    Consider the function $f(x)=\ln(3x+4)$, for $x>-\dfrac43$.
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23. Q15 · 2021 QCAA · Paper 1 · 4 marks
    In the isosceles triangle $ABC$, angle $C$ is $120^\circ$ and side $a$ is 4 cm.
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24. Q16 · 2021 QCAA · Paper 1 · 4 marks
    A tangent is drawn at the point $(1,e)$ on the graph of the function $y=e^{2-x}$ as shown. Determine the area of the shaded triangle.
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