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 16 of 26

1. Q129 · Original practice · 4 marks
   The line $y=3x$ and the parabola $y=x^2+2$ enclose a bounded region. Determine its area.
   Further integration
2. Q130 · Original practice · 6 marks
   A tank initially contains $V_0$ litres of water. For $0\le t\le10$ minutes, water enters at a rate $4e^{-0.2t}$ L/min while a pump removes water at a constant rate of $2$ L/min. Determine the minimum value of $V_0$ that guarantees the tank does not become empty during the 10-minute interval.
   Further integration
3. Q131 · Original practice · 1 mark
   Two sides of a triangle have lengths $7$ cm and $10$ cm and include an angle of $60^\circ$. The third side is closest to
   Trigonometry
4. Q132 · Original practice · 1 mark
   In triangle $ABC$, $A=30^\circ$, $a=5$ and $b=8$. How many distinct triangles satisfy these measurements?
   Trigonometry
5. Q133 · Original practice · 3 marks
   A boat travels $12$ km from $A$ to $B$ on a bearing of $060^\circ$, then $8$ km from $B$ to $C$ on a bearing of $150^\circ$. Determine the straight-line distance $AC$.
   Trigonometry
6. Q134 · Original practice · 4 marks
   Two observation points $A$ and $B$ lie on level ground on the same straight line with the base of a vertical tower, with $B$ $40$ m closer to the tower than $A$. The angles of elevation to the top are $28^\circ$ from $A$ and $46^\circ$ from $B$. Determine the height of the tower to the nearest metre.
   Trigonometry
7. Q135 · Original practice · 6 marks
   Ground stations $A$ and $B$ are $100$ m apart on an east–west line, with $B$ due east of $A$. A drone $D$ is observed from $A$ on a bearing of $035^\circ$ and from $B$ on a bearing of $300^\circ$. The angle of elevation of the drone from $A$ is $38^\circ$. Determine the height of the drone above the ground, to the nearest metre.
   Trigonometry
8. Q136 · Original practice · 1 mark
   A measurement $X$ is normally distributed with mean $68$ and standard deviation $3$. The value $X=74$ has $z$-score
   Continuous random variables and the normal distribution
9. Q137 · Original practice · 1 mark
   A continuous random variable has density $f(x)=k$ for $2\le x\le7$ and $f(x)=0$ otherwise. Determine $k$.
   Continuous random variables and the normal distribution
10. Q138 · Original practice · 2 marks
    Let $X\sim N(50,8^2)$. Determine the 90th percentile of $X$, to one decimal place.
    Continuous random variables and the normal distribution
11. Q139 · Original practice · 5 marks
    A continuous random variable has density $f(x)=kx$ for $0\le x\le3$ and $f(x)=0$ otherwise. Determine $k$, $E(X)$ and the standard deviation of $X$.
    Continuous random variables and the normal distribution
12. Q140 · Original practice · 5 marks
    Let $X\sim N(50,4^2)$ and $Y\sim N(60,6^2)$. Determine the value $k$ for which $P(X>k)=P(Y<k)$. Hence determine this common probability.
    Continuous random variables and the normal distribution
13. Q141 · Original practice · 1 mark
    Which procedure is most likely to produce a random sample of $80$ students from a school of $1200$ students?
    Sampling and proportions
14. Q142 · Original practice · 1 mark
    For a population proportion $p=0.64$ and random samples of size $400$, which distribution best approximates the sampling distribution of $\hat p$?
    Sampling and proportions
15. Q143 · Original practice · 3 marks
    A population has proportion $p=0.40$. For random samples of size $400$, use a normal approximation to determine $P(0.36<\hat p<0.44)$.
    Sampling and proportions
16. Q144 · Original practice · 3 marks
    For samples of size $400$, the sampling distribution of $\hat p$ has standard deviation $0.020$. Given that the population proportion satisfies $p>0.5$, determine $p$.
    Sampling and proportions
17. Q145 · Original practice · 5 marks
    A population proportion is $p=0.25$. Determine the minimum sample size $n$ such that, using the normal approximation, $P(|\hat p-p|<0.03)\ge0.95$.
    Sampling and proportions
18. Q146 · Original practice · 1 mark
    An approximate confidence interval for a population proportion is $(0.41,0.49)$. The margin of error is
    Interval estimates for proportions
19. Q147 · Original practice · 1 mark
    For the same confidence level and approximately the same sample proportion, increasing a sample size from $n$ to $4n$ changes the margin of error by approximately a factor of
    Interval estimates for proportions
20. Q148 · Original practice · 3 marks
    In a random sample of $300$ people, $168$ support a proposal. Using $z=1.96$, construct an approximate 95% confidence interval for the population proportion.
    Interval estimates for proportions
21. Q149 · Original practice · 4 marks
    A study uses $\hat p=0.40$ when planning a 90% confidence interval for a population proportion. Using $z=1.645$, determine the minimum sample size required for a margin of error no greater than $0.035$.
    Interval estimates for proportions
22. Q150 · Original practice · 5 marks
    A pilot survey of $100$ people gives $\hat p=0.24$. The same estimate is used to plan a final 99% confidence interval with margin of error at most $0.03$. Using $z=2.576$, determine the minimum final sample size. If the pilot responses may be included in the final sample, determine how many additional responses are required.
    Interval estimates for proportions
23. Q151 · Original practice · 1 mark
    For $f(x)=5e^{2x}-3$, determine $f'(0)$.
    Differentiation of exponential and logarithmic functions
24. Q152 · Original practice · 1 mark
    Solve $e^{2x}=7$.
    Differentiation of exponential and logarithmic functions