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 12 of 26

1. Q33 · Original practice · 2 marks
   A normally distributed variable $X$ has mean 60 and standard deviation 8. Determine $P(X\le68)$ to four decimal places.
   Continuous random variables and the normal distribution
2. Q34 · Original practice · 3 marks
   A population has proportion $p=0.60$. For random samples of size 250, determine the mean and standard deviation of the sampling distribution of $\hat p$.
   Sampling and proportions
3. Q35 · Original practice · 3 marks
   A random sample of 400 gives $\hat p=0.48$. Using $z=1.96$, determine an approximate 95% confidence interval for the population proportion.
   Interval estimates for proportions
4. Q36 · Original practice · 5 marks
   Consider $f(x)=x^2e^{-x}$. Determine all stationary points for $x\ge0$ and classify each as a local maximum or local minimum.
   Differentiation of exponential and logarithmic functions
5. Q37 · Original practice · 5 marks
   For $h(x)=\sin x\cos x$ on $0\le x\le\pi$, determine the stationary points and classify them.
   Differentiation of trigonometric functions and differentiation rules
6. Q38 · Original practice · 5 marks
   A 30 cm by 20 cm rectangular sheet has squares of side $x$ cm cut from each corner. The sides are folded up to form an open box. Determine the value of $x$ that maximises the volume and state the maximum volume to the nearest cubic centimetre.
   Further applications of differentiation
7. Q39 · Original practice · 4 marks
   A particle moves along a straight line with velocity $v(t)=t^2-4t+3$ for $0\le t\le4$. Determine the total distance travelled.
   Introduction to integration
8. Q40 · Original practice · 4 marks
   A game pays $\char36 0$, $\char36 50$ or $\char36 100$ with probabilities $0.5$, $0.3$ and $0.2$ respectively. Determine the fair entry price and the variance of the payout.
   Discrete random variables
9. Q41 · Original practice · 3 marks
   For $0\le x\le1$, the graph $y=kx(1-x)$ lies above the $x$-axis. The area between the graph and the $x$-axis is 2 square units. Determine $k$.
   Further integration
10. Q42 · Original practice · 5 marks
    In triangle $ABC$, $AB=120$ m, $AC=80$ m and $\angle A=70^\circ$. Determine $BC$ and $\angle B$ to one decimal place.
    Trigonometry
11. Q43 · Original practice · 5 marks
    A normally distributed random variable $X$ satisfies $P(X<42)=0.10$ and $P(X<58)=0.90$. Determine the mean and standard deviation of $X$ to three significant figures.
    Continuous random variables and the normal distribution
12. Q44 · Original practice · 5 marks
    A manufacturer claims that the proportion of defective components is $p=0.08$. In a random sample of 300 components, 34 are defective. Assuming the claim is correct, use a normal approximation to determine the probability of obtaining a sample proportion at least as large as the observed value. Comment on whether the result would be unusual at the 5% level.
    Sampling and proportions
13. Q45 · Original practice · 4 marks
    A pilot study estimates a population proportion as $0.37$. Determine the minimum sample size required so that an approximate 95% confidence interval has margin of error at most $0.03$, using the pilot estimate in the calculation.
    Interval estimates for proportions
14. Q46 · Original practice · 4 marks
    The curves $y=x^2$ and $y=kx$, where $k>0$, enclose a bounded region of area $\dfrac92$. Determine $k$.
    Further integration
15. Q47 · Original practice · 5 marks
    Surveyors measure a baseline $AB=75$ m. A point $C$ is observed such that $\angle CAB=52^\circ$ and $\angle CBA=68^\circ$. Determine the perpendicular distance from $C$ to $AB$, to the nearest metre.
    Trigonometry
16. Q48 · Original practice · 5 marks
    A continuous random variable $X$ has the triangular density shown, given by $f(x)=x$ for $0\le x\le1$ and $f(x)=2-x$ for $1<x\le2$. Determine $P(X>1.5\mid X>0.5)$.
    Continuous random variables and the normal distribution
17. Q49 · Original practice · 4 marks
    A poll of 600 randomly selected people records a sample proportion of $0.54$ supporting a proposal. Assuming the true population proportion is $0.50$, use a normal approximation to determine the probability of obtaining a sample proportion of at least $0.54$.
    Sampling and proportions
18. Q50 · Original practice · 6 marks
    Two independent surveys estimate population proportions. Survey A has $n=400$ and $\hat p=0.52$. Survey B has $n=900$ and $\hat p=0.48$. Construct approximate 95% confidence intervals for both proportions using $z=1.96$. State whether the intervals overlap, and explain whether overlap alone proves that the two population proportions are equal.
    Interval estimates for proportions
19. Q51 · Original practice · 1 mark
    If $f(x)=e^{3x-x^2}$, then $f'(x)$ is
    Differentiation of exponential and logarithmic functions
20. Q52 · Original practice · 1 mark
    If $y=\sin(2x)+\cos x$, then $\dfrac{dy}{dx}$ is
    Differentiation of trigonometric functions and differentiation rules
21. Q53 · Original practice · 1 mark
    The graph of $f(x)=x^3-3x$ is shown. The $x$-coordinates of its stationary points are
    Further applications of differentiation
22. Q54 · Original practice · 1 mark
    An antiderivative of $4x^3+2$ is
    Introduction to integration
23. Q55 · Original practice · 1 mark
    A discrete random variable $X$ has $P(X=0)=0.25$, $P(X=1)=0.50$ and $P(X=2)=0.25$. Determine $\operatorname{Var}(X)$.
    Discrete random variables
24. Q56 · Original practice · 1 mark
    Evaluate $\displaystyle\int_0^1 e^{2x}\,dx$.
    Further integration