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 6 of 26

1. Q7 · 2022 QCAA · Paper 1 · 1 mark
   A circle with radius $r$ and internal angle $\theta$ has a shaded segment as shown. If $\theta$ is in radians, the area of the shaded segment is
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2. Q8 · 2022 QCAA · Paper 1 · 1 mark
   In a survey, 80 respondents exercised daily, while 120 did not. When calculating the approximate 95% confidence interval for the proportion of people who exercise daily, the margin of error is
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3. Q9 · 2022 QCAA · Paper 1 · 1 mark
   The approximate area under the curve $f(x)=\sqrt{2x+1}$ between $x=0$ and $x=4$ using the trapezoidal rule with four strips is
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4. Q10 · 2022 QCAA · Paper 1 · 1 mark
   A survey plans to draw conclusions based on a random sample of 1% of Queensland’s adult population. To be regarded as a random sample, every
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5. Q11 · 2022 QCAA · Paper 1 · 5 marks
   Solve for $x$ in the following.
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6. Q12 · 2022 QCAA · Paper 1 · 3 marks
   The probability that a debating team wins a debate can be modelled as a Bernoulli distribution. Given that the probability of winning a debate is $\frac45$.
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7. Q13 · 2022 QCAA · Paper 1 · 9 marks
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8. Q14 · 2022 QCAA · Paper 1 · 6 marks
   The rate that water fills an empty vessel is given by $\dfrac{dV}{dt}=0.25e^{0.25t}$ (in litres per hour), $0\le t\le8\ln(6)$, where $t$ is time (in hours).
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9. Q15 · 2022 QCAA · Paper 1 · 4 marks
   The derivative of a function is given by $f'(x)=e^x(x-4)$. Determine the interval on which the graph of $f(x)$ is both decreasing and concave up.
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10. Q16 · 2022 QCAA · Paper 1 · 3 marks
    A section of the graphs of the first and second derivatives of a function are shown. Sketch a possible graph of the function on the same axes over the domain $0\le x\le2\pi$. Explain all reasoning used to produce the sketch.
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11. Q17 · 2022 QCAA · Paper 1 · 4 marks
    Determine the value of $b$ given $\int_a^b3x^2\,dx=117$ and $\int_a^{b-1}3x^2\,dx=56$ for $b>1$.
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12. Q18 · 2022 QCAA · Paper 1 · 4 marks
    A percentile is a measure in statistics showing the value below which a given percentage of observations occur. The continuous random variable $X$ has the probability density function $$f(x)=\begin{cases}2x-2,&1\le x\le2,\\0,&\text{otherwise.}\end{cases}$$ Determine the 36th percentile of $X$.
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13. Q19 · 2022 QCAA · Paper 1 · 7 marks
    Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion, e.g. if $\triangle UVW$ is similar to $\triangle XYZ$ then $$\angle U=\angle X,\;\angle V=\angle Y\text{ and }\angle W=\angle Z\quad\text{and}\quad\frac{UV}{XY}=\frac{VW}{YZ}=\frac{UW}{XZ}.$$ Two parallel walls $AB$ and $CD$, where…
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14. Q1 · 2022 QCAA · Paper 2 · 1 mark
    The position (in cm) of a particle is given by $x=\cos(4t)$, where $t$ is time (in seconds). The velocity of the particle when $t=5$ is
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15. Q2 · 2022 QCAA · Paper 2 · 1 mark
    Identify the correct features of the function $f(x)=xe^x$.
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16. Q3 · 2022 QCAA · Paper 2 · 1 mark
    The derivative of the function $f(x)$ is given by $f'(x)=\sin(x^3)$ for the domain $-1.8<x<1.8$. The number of points of inflection that the graph of $f(x)$ has on this interval is
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17. Q4 · 2022 QCAA · Paper 2 · 1 mark
    The distribution for a sample proportion $\hat p$ has a mean of $0.15$ and a standard deviation of $0.0345$. The sample size is
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18. Q5 · 2022 QCAA · Paper 2 · 1 mark
    The continuous random variable $X$ has the probability density function $$f(x)=\begin{cases}\dfrac{\cos x}{2},&-\dfrac\pi2\le x\le\dfrac\pi2,\\0,&\text{otherwise.}\end{cases}$$ The standard deviation of $X$ is
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19. Q6 · 2022 QCAA · Paper 2 · 1 mark
    A stall at the school fete sells cups of lemonade. Assuming the amount of lemonade in a cup is normally distributed with a mean of 60 mL and a standard deviation of 3 mL, 80% of the cups contain more than
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20. Q7 · 2022 QCAA · Paper 2 · 1 mark
    A marble moves in one direction in a straight line with velocity $v=2\ln(t+1)$ (in metres per second) where $t$ is time (in seconds) since the marble passed through the origin. Determine the distance from the origin the marble has rolled after 4 seconds.
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21. Q8 · 2022 QCAA · Paper 2 · 1 mark
    Determine the equation of the asymptote of the function $f(x)=\log_9(x-3)-4$.
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22. Q9 · 2022 QCAA · Paper 2 · 1 mark
    Determine the length of side $AB$ in triangle $ABC$.
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23. Q10 · 2022 QCAA · Paper 2 · 1 mark
    The solution of $e^{2x-3}=42$ is
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24. Q11 · 2022 QCAA · Paper 2 · 7 marks
    A salesperson has a 20% probability of making a sale to each customer who enters the store. Each sale is independent of all other sales.
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