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 24 of 26

1. Q321 · Original practice · 4 marks
   From $225$ independent responses, $144$ people choose option A. A student writes the $95\%$ interval as $0.64\pm1.96\sqrt{0.64(0.36)}/225$.
   Interval estimates for proportions
2. Q322 · Original practice · 4 marks
   A polling team has no prior estimate of the population proportion. It wants a $95\%$ approximate confidence interval with planned margin of error at most $0.03$. Use $z=1.96$.
   Interval estimates for proportions
3. Q323 · Original practice · 8 marks
   Visitors arrive at a night market at rate $R(t)=120(e^{-t/2}-e^{-t})$ people per hour for $t\geq0$. Let $N(t)$ be the modelled number who have arrived by time $t$, with $N(0)=0$.
   Further integration
4. Q324 · Original practice · 8 marks
   A small stage drone repeats a vertical routine $h(t)=3+2\sin(\pi t/6)$ metres, where $t$ is measured in seconds, for $0\leq t\leq12$. A photograph is taken at a random time $T$ uniformly distributed on $[0,12]$.
   Continuous random variables and the normal distribution
5. Q325 · Original practice · 7 marks
   A random variable $X$ has density $f(x)=\dfrac{k}{x+1}$ for $0\leq x\leq e-1$, and zero otherwise.
   Continuous random variables and the normal distribution
6. Q326 · Original practice · 6 marks
   Two rigid arms of lengths $10$ m and $14$ m form two sides of a triangular projection. Their included angle is $\theta(t)=\pi t/12$ radians, where $0<t\leq6$ seconds.
   Trigonometry
7. Q327 · Original practice · 6 marks
   The setup time for one independently prepared escape-room station is normally distributed with mean $18$ minutes and standard deviation $3$ minutes. A station is ready on time if setup takes no more than $21$ minutes. Ten stations are prepared independently.
   Discrete random variables
8. Q328 · Original practice · 7 marks
   A hinged triangular shade has sides of lengths $4$ m and $6$ m meeting at a random angle $X$ radians. Its angle density is $f(x)=\dfrac12\sin x$ for $0\leq x\leq\pi$, and zero otherwise.
   Continuous random variables and the normal distribution
9. Q329 · Original practice · 7 marks
   A visitor's arrival time $X$ hours has density $f(x)=\dfrac34x(2-x)$ for $0\leq x\leq2$, and zero otherwise. An organiser chooses a half-hour prize window $[a,a+1/2]$, where $0\leq a\leq3/2$.
   Continuous random variables and the normal distribution
10. Q330 · Original practice · 6 marks
    A survey costs $$400$to organise plus$$5$per completed response. A team wants a$95%$approximate confidence interval with planned margin of error at most$0.04$, for any population proportion. Use$z=1.96$.
    Interval estimates for proportions
11. Q331 · Original practice · 8 marks
    A camera slider has velocity $v(t)=2\cos t-e^{-t}$ metres per second for $0\leq t\leq\pi$. Its initial position is $s(0)=0$.
    Further integration
12. Q332 · Original practice · 8 marks
    A rectangular logo has its lower edge on the $x$-axis, left edge on the $y$-axis and upper-right corner on $y=e^{-x}$. Its width is $b>0$ units.
    Further applications of differentiation
13. Q333 · Original practice · 7 marks
    A curved garden border has height $f(x)=5+\cos(\pi x/4)$ metres for $0\leq x\leq4$.
    Further integration
14. Q334 · Original practice · 6 marks
    A cutting machine produces lengths $X$ mm with normal distribution of adjustable mean $\mu$ and fixed standard deviation $3$ mm. A piece is accepted if $100\leq X\leq110$. Let $P(\mu)$ be its acceptance probability. You are given

    $$P^{\prime}(\mu)=\frac1{3\sqrt{2\pi}}\left[e^{-(100-\mu)^2/18}-e^{-(110-\mu)^2/18}\right].$$
    Continuous random variables and the normal distribution
15. Q335 · Original practice · 6 marks
    Two sides of a triangular fabric panel have total length $20$ m and meet at $60^\circ$. Their lengths are $x$ and $20-x$ metres.
    Trigonometry
16. Q336 · Original practice · 8 marks
    A waiting-time model has density $f_a(x)=\dfrac{ae^{-ax}}{1-e^{-2a}}$ for $0\leq x\leq2$, and zero otherwise, where $a>0$. Measurements indicate that $P(X\leq1)=3/4$.
    Continuous random variables and the normal distribution
17. Q337 · Original practice · 8 marks
    A robot's position is $s(t)=t^3-6t^2+9t$ metres for $0\leq t\leq4$ seconds. An inspection occurs at a random time $T$ uniformly distributed on $[0,4]$.
    Further integration
18. Q338 · Original practice · 10 marks
    A festival entrance has upper profile $y=4\sin(\pi x/12)$ metres for $0\leq x\leq12$. A rectangular screen stands on the baseline and is centred at $x=6$. Its top corners touch the profile, and its height is $h$ metres, with $0<h<4$.
    Further applications of differentiation
19. Q339 · Original practice · 1 mark
    Determine $\displaystyle\lim_{h\to0}\frac{e^{2h}-1}{h}$.
    Differentiation of exponential and logarithmic functions
20. Q340 · Original practice · 1 mark
    The line $L$ is perpendicular to the tangent to $y=\ln(3x+1)$ at $x=1$. Determine the gradient of $L$.
    Differentiation of exponential and logarithmic functions
21. Q341 · Original practice · 1 mark
    The graph of $f^{\prime}(x)=(x-1)^2(x-3)$ is shown for $0<x<4$. Which statement describes the local extrema of $f$ on this interval?
    Further applications of differentiation
22. Q342 · Original practice · 1 mark
    Which experiment is suitable for an exact binomial model for the number of successes?
    Discrete random variables
23. Q343 · Original practice · 1 mark
    $X\sim\operatorname{Bin}(50,0.6)$. Determine, to four decimal places, the probability that $X$ is within one standard deviation of its mean.
    Discrete random variables
24. Q344 · Original practice · 1 mark
    A student compares their results with three normally distributed class results. The table gives the class mean, standard deviation and student result.

    $$\begin{array}{c|ccc}\text{Subject}&\text{Mean}&\text{Standard deviation}&\text{Result}\\\hline A&60&12&78\\B&70&8&80\\C&55&15&76\end{array}$$

    In which subject is the student at the highest percentile?
    Continuous random variables and the normal distribution