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 9 of 26

1. Q1 · 2020 QCAA · Paper 1 · 1 mark
   The graphs of $f(x)=e^x$ and $g(x)=x^2-1$ are shown. The area of the shaded section bounded by these graphs between the lines $x=0$ and $x=1$ is
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2. Q2 · 2020 QCAA · Paper 1 · 1 mark
   Determine $$\int \frac{e^x+1}{e^x}\,dx.$$
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3. Q3 · 2020 QCAA · Paper 1 · 1 mark
   Determine $$2\int(4x+6)^3\,dx.$$
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4. Q4 · 2020 QCAA · Paper 1 · 1 mark
   Pulse rates of adult men are approximately normally distributed with a mean of 70 and a standard deviation of 8. Which of the following choices correctly describes how to determine the proportion of men that have a pulse rate greater than 78?
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5. Q5 · 2020 QCAA · Paper 1 · 1 mark
   The equation of the tangent to the curve $f(t)=te^t$ at $t=1$ is
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6. Q6 · 2020 QCAA · Paper 1 · 1 mark
   If the probability of success in a Bernoulli trial is 0.30, the variance is
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7. Q7 · 2020 QCAA · Paper 1 · 1 mark
   The life expectancy (in years) of an electronic component can be represented by the probability density function $$p(x)=\begin{cases}\frac{1}{x^2},&x\ge1,\\0,&\text{otherwise.}\end{cases}$$ The probability that the component lasts between 1 and 10 years is
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8. Q8 · 2020 QCAA · Paper 1 · 1 mark
   A test includes six multiple choice questions. Each question has four options for the answer. If the answers are guessed, the probability of getting at most two questions correct is represented by
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9. Q9 · 2020 QCAA · Paper 1 · 1 mark
   Determine $$\int \frac{x+1}{x^2+2x}\,dx.$$
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10. Q10 · 2020 QCAA · Paper 1 · 1 mark
    Two types of material (A and B) are being tested for their ability to withstand different temperatures. A random selection of both materials was subjected to extreme temperature changes and then classified according to their condition after they were removed from the testing facility. The results are shown in the table.

    | | A | B | Total | | --- | --- |…
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11. Q11 · 2020 QCAA · Paper 1 · 3 marks
    Determine the derivative of each of the following with respect to $x$.
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12. Q12 · 2020 QCAA · Paper 1 · 5 marks
    An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration $a$ (in $\mathrm{m\,s^{-2}}$) of the object is given by $$a(t)=\pi\cos(\pi t),\quad t\ge0,$$ where $t$ is time in seconds. The velocity at $t=1$ is $0.5\;\mathrm{m\,s^{-1}}$.
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13. Q13 · 2020 QCAA · Paper 1 · 7 marks
    A function is defined as $f(x)=x(\ln(x))^2$, $x>0$. The graph of the function is shown and has a local maximum at point A and a global minimum at point B. The derivative of the function is given by $f'(x)=2\ln(x)+(\ln(x))^2$, $x>0$.
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14. Q14 · 2020 QCAA · Paper 1 · 3 marks
    Determine the area of the triangle shown.
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15. Q15 · 2020 QCAA · Paper 1 · 4 marks
    Solve the following equations.
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16. Q16 · 2020 QCAA · Paper 1 · 4 marks
    Consider the graph of $f(x)$ shown. Identify the graph of the second derivative $f''(x)$ from the graphs in Diagram 1, Diagram 2 and Diagram 3. Justify your decisions using mathematical reasoning.
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17. Q17 · 2020 QCAA · Paper 1 · 6 marks
    The volume of water in a tank is represented by a function of the form $$V(t)=Ae^{kt},$$ where $V$ is in litres and $t$ is in minutes. Initially, the volume is 100 litres and it is decreasing by 50 litres per minute. Determine the time at which the volume is decreasing at the rate of $\frac{50}{7}$ litres per minute. Express your answer in the form $\ln(a)$.
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18. Q18 · 2020 QCAA · Paper 1 · 6 marks
    The function $f(x)$ has the form $f(x)=3\log_2(x+a)+b$. The function $g(x)$ has the form $g(x)=-\log_3(x+c)+5$. A section of the graphs of the two functions is shown. Determine the values of $a$, $b$ and $c$.
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19. Q19 · 2020 QCAA · Paper 1 · 6 marks
    A horizontal point of inflection is a point of inflection that is also a stationary point. Determine the value/s of $k$ for which the graph of $$f(x)=\frac{\ln(x)}{k}-\frac{kx}{x+1}$$ has only one horizontal point of inflection.
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20. Q20 · 2020 QCAA · Paper 1 · 6 marks
    At the end of the first stage of its growth cycle, a species of tree has a height of 5 metres and a trunk radius of 15 cm. In the second stage of its growth cycle, the tree stays at this height for the next 10 years. However, the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function $$r(t)=1.5+\sin\left(\frac{…
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21. Q1 · 2020 QCAA · Paper 2 · 1 mark
    The limit of $$\frac{12^h-1}{h}$$ as $h$ approaches 0 is closest to
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22. Q2 · 2020 QCAA · Paper 2 · 1 mark
    The pH of a substance is a measure of its acidity and is given by the formula $\mathrm{pH}=-\log_{10}[\mathrm H^+]$ where $[\mathrm H^+]$ is the concentration of hydrogen ions in moles per litre. If a solution has a pH equal to 0.2, the concentration of hydrogen ions in moles per litre is closest to
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23. Q3 · 2020 QCAA · Paper 2 · 1 mark
    Let $R$ be the region enclosed by the graph of $y=xe^x$, the $x$-axis, and the lines $x=-1$ and $x=1$. The area of $R$ is closest to
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24. Q4 · 2020 QCAA · Paper 2 · 1 mark
    Consider the function $f(x)=\log_p(x+q)$ where $p>1$ and $0<q<1$. Which of the following could be the graph of $f(x)$?
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