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 22 of 26

1. Q273 · Original practice · 5 marks
   Let $f(x)=\dfrac{x}{x^2+4}$.
   Differentiation of trigonometric functions and differentiation rules
2. Q274 · Original practice · 3 marks
   For $f(x)=3\sin(2x)+x$,
   Differentiation of trigonometric functions and differentiation rules
3. Q275 · Original practice · 5 marks
   The height of a decorative wave is modelled by $h(x)=2\sin^2x$ for $0\leq x\leq\pi$.
   Differentiation of trigonometric functions and differentiation rules
4. Q276 · Original practice · 6 marks
   Let $f(x)=x^3-3x$.
   Further applications of differentiation
5. Q277 · Original practice · 5 marks
   A rectangular herb garden is built against a straight wall. A total of $36$ m of fencing is used for its other three sides. Let $x$ metres be the side length perpendicular to the wall.
   Further applications of differentiation
6. Q278 · Original practice · 6 marks
   Squares of side $x$ cm are cut from each corner of a $30$ cm by $18$ cm sheet. The sides are folded up to make an open box.
   Further applications of differentiation
7. Q279 · Original practice · 5 marks
   A cup of tea has temperature $T(t)=22+68e^{-0.12t}$ degrees Celsius, $t$ minutes after pouring.
   Differentiation of exponential and logarithmic functions
8. Q280 · Original practice · 4 marks
   Let $f(x)=e^x\cos x$.
   Differentiation of trigonometric functions and differentiation rules
9. Q281 · Original practice · 4 marks
   A function satisfies $f^{\prime}(x)=6x^2+4e^{2x}-\dfrac3x$ for $x>0$ and $f(1)=2e^2$.
   Introduction to integration
10. Q282 · Original practice · 6 marks
    A delivery robot moves along a straight corridor with acceleration $a(t)=6t-4\ \mathrm{m/s^2}$. At $t=0$, its velocity is $1\ \mathrm{m/s}$ and its position is $2$ m from the reference point.
    Introduction to integration
11. Q283 · Original practice · 6 marks
    A small cart has velocity $v(t)=e^t-2\ \mathrm{m/s}$ for $0\leq t\leq\ln4$.
    Further integration
12. Q284 · Original practice · 4 marks
    The growth rate of a vine is $h^{\prime}(t)=\dfrac{12}{t+2}$ cm per week, for $t\geq0$. Its initial height is $15$ cm.
    Introduction to integration
13. Q285 · Original practice · 5 marks
    The number $X$ of bonus stamps in a pack has the distribution $P(X=0)=k$, $P(X=1)=2k$, $P(X=2)=k$ and $P(X=4)=2k$.
    Discrete random variables
14. Q286 · Original practice · 5 marks
    A ride operator finds that a randomly selected visitor chooses the spinning ride with probability $0.7$. Choices by the next $12$ visitors are assumed independent. Let $X$ be the number who choose it.
    Discrete random variables
15. Q287 · Original practice · 5 marks
    A spinner pays a prize $W$ dollars. The prizes $0$, $4$ and $10$ occur with probabilities $0.5$, $0.3$ and $0.2$, respectively. A player pays a fee $d$ dollars and receives net gain $G=W-d$.
    Discrete random variables
16. Q288 · Original practice · 4 marks
    A puzzle room offers two rounds. In each round, a team attempts three locks. Each lock opens with probability $0.4$, independently of all other attempts. A round is completed only if all three locks open.
    Discrete random variables
17. Q289 · Original practice · 4 marks
    Evaluate the following integrals exactly.
    Further integration
18. Q290 · Original practice · 5 marks
    The graph of $y=x^2-1$ is shown for $-2\leq x\leq2$.
    Further integration
19. Q291 · Original practice · 5 marks
    The curves $y=x^2$ and $y=x+2$ enclose a finite region.
    Further integration
20. Q292 · Original practice · 5 marks
    A light sculpture has vertical profile $y=2\sin x-1$ for $0\leq x\leq\pi$.
    Further integration
21. Q293 · Original practice · 4 marks
    A mural has a curved upper edge above a horizontal baseline. Its measured heights $h$ metres at distances $x$ metres are shown: $\begin{array}{c|ccccc}x&0&1&2&3&4\\\hline h&1.2&2.1&2.8&2.0&1.3\end{array}$.
    Further integration
22. Q294 · Original practice · 5 marks
    The marginal production cost of $x$ souvenir pins is modelled by $C^{\prime}(x)=0.04x+2+\dfrac{20}{x+10}$ dollars per pin, for $x\geq0$. The fixed cost is $C(0)=50$ dollars.
    Further integration
23. Q295 · Original practice · 6 marks
    Water enters a tank at rate $I(t)=8+4\sin(\pi t/6)$ litres per minute, while water leaves at $6$ litres per minute. Initially the tank contains $20$ litres. The model applies for $0\leq t\leq12$.
    Further integration
24. Q296 · Original practice · 4 marks
    Let $F(x)=\displaystyle\int_0^{x^2}(1+\sin t)\,dt$.
    Further integration