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 25 of 26

1. Q345 · Original practice · 1 mark
   The shaded rectangles use left endpoints and four equal strips to estimate $\int_0^4 e^{0.2x}\,dx$. Determine the estimate to three decimal places.
   Further integration
2. Q346 · Original practice · 1 mark
   Use the trapezoidal rule with six equal strips to estimate $\displaystyle\int_0^3\frac{1}{x+1}\,dx$. Give the estimate to three decimal places.
   Further integration
3. Q347 · Original practice · 1 mark
   A continuous random variable has cumulative distribution function $F(x)=x^3/8$ for $0\leq x\leq2$. Determine its 75th percentile to three decimal places.
   Continuous random variables and the normal distribution
4. Q348 · Original practice · 1 mark
   The distribution of a sample proportion has mean $0.35$ and standard deviation $0.025$. Determine the sample size.
   Sampling and proportions
5. Q349 · Original practice · 1 mark
   In a random sample of $625$ residents, $400$ support a community project. Use $z=1.96$ to determine a 95% approximate confidence interval for the population proportion. Round endpoints to three decimal places.
   Interval estimates for proportions
6. Q350 · Original practice · 1 mark
   The graphs of $y=e^{-x}$ and $y=x$ intersect for $x>0$. Determine the x-coordinate of their intersection to three decimal places.
   Differentiation of exponential and logarithmic functions
7. Q351 · Original practice · 5 marks
   The graph of $f(x)=a\ln(x+2)+b$ has vertical asymptote $x=-2$ and passes through $(0,1)$ and $(2,3)$, as shown.
   Differentiation of exponential and logarithmic functions
8. Q352 · Original practice · 5 marks
   Let $f(x)=e^{2x}\sin(3x)$.
   Differentiation of trigonometric functions and differentiation rules
9. Q353 · Original practice · 5 marks
   Consider $g(x)=e^x/(x+2)$ for $x>-2$.
   Differentiation of trigonometric functions and differentiation rules
10. Q354 · Original practice · 3 marks
    A student claims that the graph of $f(x)=x^4/12$ has a point of inflection at $x=0$ because $f^{\prime\prime}(0)=0$. Evaluate the reasonableness of the claim.
    Further applications of differentiation
11. Q355 · Original practice · 6 marks
    The graph of $f^{\prime}(x)=x^2-4$ is shown on the left. It is known that $f(0)=1$. Use the grid on the right for your sketch.
    Further applications of differentiation
12. Q356 · Original practice · 4 marks
    The derivative of a function is $f^{\prime}(x)=e^x(x-2)$. Determine the interval on which $f$ is both decreasing and concave up. Justify your answer.
    Further applications of differentiation
13. Q357 · Original practice · 6 marks
    A closed cylindrical container must hold $32\pi\ \mathrm{cm}^3$. Its radius is $r$ cm and height is $h$ cm. Material for its two circular ends costs $3$ cents per $\mathrm{cm}^2$, while material for its curved side costs $1$ cent per $\mathrm{cm}^2$. Ignore seams and waste.
    Further applications of differentiation
14. Q358 · Original practice · 5 marks
    A water level varies sinusoidally. The minimum level is $4$ m, the maximum is $12$ m and the period is $10$ hours. A minimum occurs at $t=0$.
    Further applications of differentiation
15. Q359 · Original practice · 4 marks
    $f^{\prime}(x)=3\cos(2x)-2e^{-x}$ and $f(\pi)=4$. Determine $f(x)$ exactly.
    Introduction to integration
16. Q360 · Original practice · 5 marks
    A particle has acceleration $a(t)=2\cos t$ m/s$^2$ for $t\geq0$. Its velocity at $t=0$ is $1$ m/s and its position at $t=\pi/2$ is $3$ m.
    Introduction to integration
17. Q361 · Original practice · 5 marks
    Two particles move on a straight line with velocities $v_A(t)=t^2-2t+2$ and $v_B(t)=2t+2$ m/s for $t\geq0$. They have equal velocities at $t=0$.
    Introduction to integration
18. Q362 · Original practice · 5 marks
    A fair four-sided die gives $X\in\{1,2,3,4\}$, each with probability $1/4$. A game pays $Y=4X-7$ dollars.
    Discrete random variables
19. Q363 · Original practice · 4 marks
    A kiosk records the number $X$ of additional items bought in $60$ transactions.

    $$\begin{array}{c|rrrr}x&0&1&2&3\\\hline\text{Frequency}&18&24&12&6\end{array}$$

    Use relative frequencies as estimates of the probabilities.
    Discrete random variables
20. Q364 · Original practice · 4 marks
    Two independent-trial systems use success indicators $A$ and $B$. For a single trial, $P(A=1)=0.3$ and $P(B=1)=0.6$.
    Discrete random variables
21. Q365 · Original practice · 4 marks
    $X\sim\operatorname{Bin}(n,p)$ has mean $6$ and variance $9/2$.
    Discrete random variables
22. Q366 · Original practice · 4 marks
    A component passes a specialised stress test with probability $0.15$. Tests of different components are independent. Determine the minimum number of components that must be tested for the probability of at least one pass to be at least $0.90$.
    Discrete random variables
23. Q367 · Original practice · 6 marks
    A supplier compares two independent-trial inspection plans. Plan A approves a batch if at least four of five sampled items pass; each item passes with probability $0.7$. Plan B approves a batch only if all four sampled items pass; each item passes with probability $0.8$. A manager claims that the approval probability for Plan A exceeds that for Plan B by at…
    Discrete random variables
24. Q368 · Original practice · 5 marks
    The curve is $y=x^2+1$ for $0\leq x\leq2$.
    Further integration