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 18 of 26

1. Q177 · Original practice · 1 mark
   The area under the line $y=4-x$ and above the $x$-axis for $0\le x\le4$ is
   Further integration
2. Q178 · Original practice · 2 marks
   The values $f(0)=2$, $f(1)=3$, $f(2)=5$, $f(3)=4$ and $f(4)=2$ are given. Use the trapezoidal rule with width $1$ to estimate $\displaystyle\int_0^4 f(x)\,dx$.
   Further integration
3. Q179 · Original practice · 4 marks
   The graphs of $y=2x$ and $y=x^2$ are shown. Determine the area of the bounded region between the curves.
   Further integration
4. Q180 · Original practice · 5 marks
   A tank contains $80$ L at $t=0$. For $0\le t\le6$ minutes, its volume changes at the rate $\dfrac{dV}{dt}=12-\dfrac12t^2$ litres per minute. Determine the time at which the volume is greatest, the greatest volume, and the volume at $t=6$.
   Further integration
5. Q181 · Original practice · 1 mark
   Two sides of a triangle have lengths $5$ cm and $7$ cm with included angle $60^\circ$. The third side has length
   Trigonometry
6. Q182 · Original practice · 1 mark
   In triangle $ABC$, $A=40^\circ$, $B=65^\circ$ and side $a=9$ cm. Side $b$ is closest to
   Trigonometry
7. Q183 · Original practice · 2 marks
   Two sides of a triangular reserve are $14$ km and $9$ km long and include an angle of $35^\circ$. Determine the area of the reserve to one decimal place.
   Trigonometry
8. Q184 · Original practice · 4 marks
   An aircraft flies $30$ km on a bearing of $025^\circ$, then $40$ km on a bearing of $110^\circ$. Determine its distance and bearing from its starting point.
   Trigonometry
9. Q185 · Original practice · 5 marks
   Points $A$ and $B$ are $150$ m apart, with $B$ due north of $A$. A point $C$ is observed on a bearing of $070^\circ$ from $A$ and a bearing of $120^\circ$ from $B$, as shown. Determine $AC$ and the perpendicular eastward distance of $C$ from the line $AB$.
   Trigonometry
10. Q186 · Original practice · 1 mark
    A continuous random variable is uniformly distributed on $1\le x\le5$. Its probability density is
    Continuous random variables and the normal distribution
11. Q187 · Original practice · 1 mark
    A normally distributed variable has mean $70$ and standard deviation $4$. The $z$-score corresponding to $x=62$ is
    Continuous random variables and the normal distribution
12. Q188 · Original practice · 2 marks
    Let $X\sim N(40,5^2)$. The shaded region in the graph represents $P(X>45)$. Determine this probability to four decimal places.
    Continuous random variables and the normal distribution
13. Q189 · Original practice · 5 marks
    The cumulative distribution function of a continuous random variable $X$ is $F(x)=0$ for $x<0$, $F(x)=x^2/9$ for $0\le x\le3$, and $F(x)=1$ for $x>3$. Determine the probability density function of $X$, $E(X)$ and $P(X>2)$.
    Continuous random variables and the normal distribution
14. Q190 · Original practice · 5 marks
    A normal random variable $X$ satisfies $P(X<68)=0.05$ and $P(X>82)=0.20$. Determine the mean and standard deviation of $X$ to three significant figures.
    Continuous random variables and the normal distribution
15. Q191 · Original practice · 1 mark
    Which method is most likely to give a random sample of customers from a database of $5000$ customers?
    Sampling and proportions
16. Q192 · Original practice · 1 mark
    For random samples of size $300$ from a population with proportion $p=0.27$, the variance of $\hat p$ is
    Sampling and proportions
17. Q193 · Original practice · 3 marks
    A population proportion is $p=0.35$. For random samples of size $100$, the sampling distribution of $\hat p$ is approximated by the normal curve shown. Determine its mean and standard deviation, and estimate $P(0.30<\hat p<0.40)$.
    Sampling and proportions
18. Q194 · Original practice · 3 marks
    For random samples from a population with proportion $p=0.36$, the standard deviation of $\hat p$ is $0.020$. Determine the sample size $n$.
    Sampling and proportions
19. Q195 · Original practice · 5 marks
    A population has proportion $p=0.15$. Determine the minimum sample size $n$ such that, using the normal approximation, $P(\hat p>0.20)<0.025$.
    Sampling and proportions
20. Q196 · Original practice · 1 mark
    An approximate confidence interval for a population proportion is $(0.33,0.47)$. The sample proportion used to construct the interval is
    Interval estimates for proportions
21. Q197 · Original practice · 1 mark
    For the same sample size and sample proportion, changing from a 95% confidence interval to a 99% confidence interval will generally
    Interval estimates for proportions
22. Q198 · Original practice · 3 marks
    A seed trial randomly selects $320$ seeds from a large batch, and $208$ germinate. Using $z=1.645$, construct an approximate 90% confidence interval for the germination proportion of the batch.
    Interval estimates for proportions
23. Q199 · Original practice · 4 marks
    Five confidence intervals, A–E, are shown together with the true population proportion $p=0.60$.

    (a) Identify the intervals that do not contain $p$.

    (b) Explain the meaning of a 95% confidence level in the context of repeated random sampling.
    Interval estimates for proportions
24. Q200 · Original practice · 5 marks
    A survey of $400$ people gives $\hat p=0.58$. Its approximate 95% confidence interval uses $z=1.96$. A new survey is planned using the same planning value $p=0.58$, but the new interval will have 99% confidence using $z=2.576$. Determine the minimum sample size required so that the new 99% interval has margin of error no greater than the current 95% interval…
    Interval estimates for proportions