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 20 of 26

1. Q225 · Original practice · 1 mark
   For $p$-hat$=0.64, n=400$ and $z=1.96$, the approximate margin of error is closest to
   Interval estimates for proportions
2. Q226 · Original practice · 1 mark
   $\int_0^{2} e^{2x} \mathrm{d}x$ equals
   Further integration
3. Q227 · Original practice · 1 mark
   If $f(x)=2x-5$, then $f^{-1}(x)$ is
   Functions and relations
4. Q228 · Original practice · 1 mark
   If $f(x)=2x$ for $0\le x\le 1$, then $P(X<0.5)$ is
   Continuous random variables and the normal distribution
5. Q229 · Original practice · 1 mark
   For $f(x)=x^{3}-3x$, a stationary point occurs at
   Further applications of differentiation
6. Q230 · Original practice · 4 marks
   A medication concentration is modelled by $C(t)=18e^{-0.22t} mg L^{-1}$ for $t\ge 0$.
   Exponential functions
7. Q231 · Original practice · 5 marks
   Let $X\sim B(20,0.35)$.
   Discrete random variables
8. Q232 · Original practice · 4 marks
   A function is $f(x)=\ln (x-1)+2$.
   Functions and relations
9. Q233 · Original practice · 5 marks
   The rate of water flow into a tank is $R(t)=5+2\sin (\frac{\pi t}{6}) L min^{-1}$ for $0\le t\le 12$.
   Further integration
10. Q234 · Original practice · 5 marks
    Scores are approximately $N(68,12^{2})$. A scholarship is awarded to the top $8\%$ of students.
    Continuous random variables and the normal distribution
11. Q235 · Original practice · 5 marks
    A lifetime $X$ (hours) has density $f(x)=c e^{-0.5x}$ for $x\ge 0$.
    Continuous random variables and the normal distribution
12. Q236 · Original practice · 5 marks
    In a sample of $500$ customers, $315$ prefer option A. Use $z=1.96$.
    Interval estimates for proportions
13. Q237 · Original practice · 6 marks
    A closed cylindrical can must hold $750\,\mathrm{cm}^{3}$. Let $r$ be its radius.
    Further applications of differentiation
14. Q238 · Original practice · 6 marks
    Two independent random samples have means $52.4$ and $49.8$, standard deviations $6.0$ and $5.5$, and sample sizes $100$ and $120$ respectively.
    Statistical inference
15. Q239 · Original practice · 1 mark
    A lantern display has brightness $B(t)=120e^{-0.3t}$, where $t$ is measured in minutes. Which expression gives $B^{\prime}(t)$?
    Differentiation of exponential and logarithmic functions
16. Q240 · Original practice · 1 mark
    The graph represents $y=\ln(x-2)+1$. Which pair states its domain and vertical asymptote?
    Differentiation of exponential and logarithmic functions
17. Q241 · Original practice · 1 mark
    For $f(x)=x^2\sin x$, the derivative is
    Differentiation of trigonometric functions and differentiation rules
18. Q242 · Original practice · 1 mark
    If $y=\cos(3x-1)$, then $\dfrac{dy}{dx}$ is
    Differentiation of trigonometric functions and differentiation rules
19. Q243 · Original practice · 1 mark
    The graph shows $f^{\prime}(x)=(x+2)(x-1)$. At which value of $x$ does $f$ have a local maximum?
    Further applications of differentiation
20. Q244 · Original practice · 1 mark
    For $f(x)=x^3-6x^2+9x+4$, the point of inflection is
    Further applications of differentiation
21. Q245 · Original practice · 1 mark
    Which is an antiderivative of $6\cos(2x)$?
    Introduction to integration
22. Q246 · Original practice · 1 mark
    A moving tile has velocity $v(t)=4t-6$ and position $s(0)=5$. Which expression gives its position?
    Introduction to integration
23. Q247 · Original practice · 1 mark
    An arcade bonus $X$ has the probability distribution shown. What is $E(X)$?
    Discrete random variables
24. Q248 · Original practice · 1 mark
    Each of five independent mystery envelopes contains a gold card with probability $0.2$. The probability that exactly one envelope contains a gold card is
    Discrete random variables