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 23 of 26

1. Q297 · Original practice · 6 marks
   In triangle $ABC$, $\angle A=35^\circ$, $BC=8$ cm and $AC=12$ cm. Two different triangles are possible.
   Trigonometry
2. Q298 · Original practice · 6 marks
   Two kayakers leave jetty $A$. Kayaker $B$ travels $16$ km on a bearing of $065^\circ$ and kayaker $C$ travels $22$ km on a bearing of $135^\circ$.
   Trigonometry
3. Q299 · Original practice · 4 marks
   Two observers stand on level ground along the same straight line from the base of a vertical lookout. Observer $P$ is $30$ m farther from the base than observer $Q$. Their angles of elevation to the top are $30^\circ$ and $60^\circ$, respectively.
   Trigonometry
4. Q300 · Original practice · 4 marks
   A rectangular rooftop is $8$ m by $6$ m. A vertical $12$ m mast stands at corner $B$, diagonally opposite corner $A$. The top of the mast is $T$.
   Trigonometry
5. Q301 · Original practice · 4 marks
   A triangular sail has side lengths $13$ m, $14$ m and $15$ m. Let $\theta$ be the angle between the $13$ m and $14$ m sides.
   Trigonometry
6. Q302 · Original practice · 4 marks
   Three beacons form triangle $ABC$. The baseline $AB$ is $12$ km, $\angle A=40^\circ$ and $\angle B=65^\circ$.
   Trigonometry
7. Q303 · Original practice · 3 marks
   A circular window has radius $6$ cm. A chord subtends an angle of $\dfrac{2\pi}{3}$ radians at its centre. The smaller segment is shaded.
   Trigonometry
8. Q304 · Original practice · 6 marks
   A random variable $X$ has density $f(x)=kx^2$ for $0\leq x\leq1$, and zero otherwise.
   Continuous random variables and the normal distribution
9. Q305 · Original practice · 5 marks
   The time $X$ hours until a randomly chosen visitor leaves an exhibit has density $f(x)=\dfrac{2-x}{2}$ for $0\leq x\leq2$, and zero otherwise.
   Continuous random variables and the normal distribution
10. Q306 · Original practice · 4 marks
    A loading time $X$ minutes has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=x^2/9$ for $0\leq x\leq3$, and $F(x)=1$ for $x>3$.
    Continuous random variables and the normal distribution
11. Q307 · Original practice · 4 marks
    Battery life $X$ hours is normally distributed with mean $48$ hours and standard deviation $6$ hours.
    Continuous random variables and the normal distribution
12. Q308 · Original practice · 4 marks
    The fill volume $X$ mL of a reusable bottle is normally distributed with mean $500$ mL. Exactly $10\%$ of bottles contain less than $495$ mL.
    Continuous random variables and the normal distribution
13. Q309 · Original practice · 3 marks
    Diameters $X$ mm of machine-made tokens are normally distributed with mean $32$ mm and standard deviation $3$ mm. Tokens are accepted if $a\leq X\leq b$. The acceptance interval is symmetric about the mean and contains $95\%$ of tokens.
    Continuous random variables and the normal distribution
14. Q310 · Original practice · 4 marks
    A randomly selected origami session lasts $X$ hours with density $f(x)=\dfrac32x(2-x)$ for $0\leq x\leq2$, and zero otherwise.
    Continuous random variables and the normal distribution
15. Q311 · Original practice · 4 marks
    In a very large population of commuters, $30\%$ use a bicycle at least once a week. A random sample of $200$ independent commuters is selected. Let $\widehat p$ be the sample proportion who do so.
    Sampling and proportions
16. Q312 · Original practice · 3 marks
    A council wants to estimate the proportion of residents who support a new skate park. It surveys every fifth person entering the existing skate park between $4$ pm and $6$ pm on Saturday.
    Sampling and proportions
17. Q313 · Original practice · 4 marks
    Two random-sampling procedures estimate the same population proportion $p=0.6$. Procedure A uses $n=100$ and procedure B uses $n=400$.
    Sampling and proportions
18. Q314 · Original practice · 4 marks
    A random sample of $400$ café customers includes $250$ who prefer reusable cups.
    Interval estimates for proportions
19. Q315 · Original practice · 4 marks
    A pilot survey suggests a population proportion of $0.4$. A larger survey will use a $95\%$ approximate confidence interval with planned margin of error at most $0.025$. Use $z=1.96$ and the pilot value for planning.
    Interval estimates for proportions
20. Q316 · Original practice · 4 marks
    In a random sample of $500$ residents, $280$ support a pedestrian bridge. A councillor claims that more than half of all residents support it. Use $z=1.96$.
    Interval estimates for proportions
21. Q317 · Original practice · 3 marks
    The diagram shows ten $95\%$ approximate confidence intervals obtained from independent random samples. The dashed line is the true population proportion $p=0.4$.
    Interval estimates for proportions
22. Q318 · Original practice · 4 marks
    Two survey plans use the same assumed population proportion. Plan A uses $400$ responses and a $95\%$ confidence level. Plan B uses a $99\%$ confidence level and aims for a planned margin of error no larger than that of Plan A. Use $z_{95}=1.96$ and $z_{99}=2.576$.
    Interval estimates for proportions
23. Q319 · Original practice · 5 marks
    Each of $240$ independently selected voters supports a proposal with probability $0.45$. Let $\widehat p$ be the sample proportion supporting it.
    Sampling and proportions
24. Q320 · Original practice · 3 marks
    A $95\%$ approximate confidence interval has unrounded endpoints $0.45$ and $0.55$. The construction used $\widehat p=0.5$ and $z=2$ as a convenient approximation.
    Interval estimates for proportions