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 14 of 26

1. Q81 · Original practice · 2 marks
   Evaluate $\displaystyle\int_1^3\frac{2x}{x^2+1}\,dx$.
   Further integration
2. Q82 · Original practice · 3 marks
   From point $P$, landmark $A$ is $13$ km away and landmark $B$ is $9$ km away. The angle $APB$ is $58^\circ$, as shown. Determine $AB$ to the nearest $0.1$ km.
   Trigonometry
3. Q83 · Original practice · 3 marks
   $X\sim N(72,6^2)$. Determine $P(65\le X\le80)$ to four decimal places.
   Continuous random variables and the normal distribution
4. Q84 · Original practice · 3 marks
   A population has proportion $p=0.35$. For random samples of size $500$, use a normal approximation to determine $P(\hat p>0.40)$ to four decimal places.
   Sampling and proportions
5. Q85 · Original practice · 3 marks
   In a random sample of $250$ people, $68$ answer yes. Using $z=1.645$, construct an approximate 90% confidence interval for the population proportion.
   Interval estimates for proportions
6. Q86 · Original practice · 4 marks
   For $f(x)=\dfrac{\ln x}{x}$, $x>0$, determine the stationary point and classify it.
   Differentiation of exponential and logarithmic functions
7. Q87 · Original practice · 5 marks
   For $f(x)=\sin x+\dfrac12\cos(2x)$ on $0\le x\le\pi$, determine all interior stationary points.
   Differentiation of trigonometric functions and differentiation rules
8. Q88 · Original practice · 5 marks
   A closed cylindrical can has volume $500\pi\text{ cm}^3$. Let its radius be $r$ cm and height be $h$ cm. Determine the radius and height that minimise its total surface area.
   Further applications of differentiation
9. Q89 · Original practice · 5 marks
   A particle has acceleration $a(t)=6t-12$, velocity $v(0)=9$ and position $s(0)=4$. Determine the total distance travelled for $0\le t\le4$.
   Introduction to integration
10. Q90 · Original practice · 5 marks
    A discrete random variable $X$ has $P(X=x)=kx$ for $x=1,2,3,4$. Determine $k$, $E(X)$ and $\operatorname{Var}(X)$.
    Discrete random variables
11. Q91 · Original practice · 4 marks
    Determine the area enclosed by the curves $y=4-x^2$ and $y=x+2$.
    Further integration
12. Q92 · Original practice · 5 marks
    A boat travels $18$ km on a bearing of $035^\circ$, then $25$ km on a bearing of $120^\circ$, as shown. Determine its distance and bearing from the starting point.
    Trigonometry
13. Q93 · Original practice · 5 marks
    The lifetime $X$ of a component is normally distributed with mean $120$ hours and standard deviation $18$ hours. A component is classified as long-life if it is in the top 5% of lifetimes. Determine the long-life cutoff, and then determine $P(X>150\mid X\text{ is long-life})$.
    Continuous random variables and the normal distribution
14. Q94 · Original practice · 5 marks
    A population proportion is $p=0.12$. For random samples of size $250$, use a normal approximation to determine (a) $P(0.08<\hat p<0.15)$ and (b) the value $c$ such that $P(\hat p<c)=0.95$.
    Sampling and proportions
15. Q95 · Original practice · 5 marks
    A sample has $\hat p=0.42$ and $n=400$. A symmetric confidence interval is constructed with margin of error $0.048$. Determine the corresponding confidence level, to the nearest percent.
    Interval estimates for proportions
16. Q96 · Original practice · 5 marks
    The curves $y=k-x^2$ and $y=x$, where $k>0$, enclose a region of area $\dfrac92$ square units, as shown. Determine $k$.
    Further integration
17. Q97 · Original practice · 6 marks
    Points $A$ and $B$ are $80$ m apart on an east–west line, with $B$ due east of $A$. A tower $T$ is observed from $A$ on a bearing of $035^\circ$ and from $B$ on a bearing of $315^\circ$. Determine $AT$ and the perpendicular distance from $T$ to line $AB$, to the nearest metre.
    Trigonometry
18. Q98 · Original practice · 5 marks
    A continuous random variable has the triangular density shown, with $f(x)=cx$ for $0\le x\le2$ and $f(x)=c(4-x)$ for $2<x\le4$. Determine $c$ and $P(X<1.5\mid X<3)$.
    Continuous random variables and the normal distribution
19. Q99 · Original practice · 5 marks
    For random samples of size $400$ from a population with unknown proportion $p$, suppose $P(\hat p<0.45)=0.10$ under a normal approximation. Determine $p$ to three decimal places.
    Sampling and proportions
20. Q100 · Original practice · 7 marks
    In a survey of $800$ people, $456$ support a proposal. (a) Using $z=2.576$, construct an approximate 99% confidence interval for the population proportion. (b) A new survey will use a 95% confidence interval and the same planning estimate of the proportion. Determine the minimum sample size needed so that its margin of error is at most half the margin from p…
    Interval estimates for proportions
21. Q101 · Original practice · 1 mark
    For $f(x)=7e^{-3x}+4$, determine $f'(0)$.
    Differentiation of exponential and logarithmic functions
22. Q102 · Original practice · 1 mark
    Solve $\ln(x-2)=1$ for $x>2$.
    Differentiation of exponential and logarithmic functions
23. Q103 · Original practice · 1 mark
    For $f(x)=e^x\ln x$, $x>0$, determine the gradient of the tangent at $x=1$.
    Differentiation of exponential and logarithmic functions
24. Q104 · Original practice · 3 marks
    A population is modelled by $P(t)=1200e^{-0.08t}$, where $t$ is measured in years. Determine $P'(5)$ and interpret its meaning in context.
    Differentiation of exponential and logarithmic functions