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 19 of 26

1. Q201 · Original practice · 1 mark
   For $f(x)=x^{2} e^{x}, f'(x)$ is
   Differentiation of exponential and logarithmic functions
2. Q202 · Original practice · 1 mark
   The domain of $h(x)=\ln (5-3x)$ is
   Functions and relations
3. Q203 · Original practice · 1 mark
   The exact area under $y=3x^{2}$ from $x=0$ to $x=2$ is
   Introduction to integration
4. Q204 · Original practice · 1 mark
   The solutions of $\cos (\theta )=-\frac{\sqrt{3}}{2}$ for $0\le \theta \le 2\pi$ are
   Trigonometry
5. Q205 · Original practice · 1 mark
   The graph $y=3f(2x-4)-1$ is obtained from $y=f(x)$ by
   Functions and relations
6. Q206 · Original practice · 1 mark
   If $P(A)=0.6, P(B)=0.5$ and $P(A\cap B)=0.3$, then A and B are
   Probability
7. Q207 · Original practice · 1 mark
   If $3e^{0.2t}=12$, then $t$ equals
   Exponential functions
8. Q208 · Original practice · 1 mark
   For $X\sim B(6,0.25), P(X=0)$ equals
   Discrete random variables
9. Q209 · Original practice · 1 mark
   If $X\sim N(40,5^{2})$, the $z$-score for $x=50$ is
   Continuous random variables and the normal distribution
10. Q210 · Original practice · 1 mark
    For a sample proportion $p$-hat$=0.40$ with $n=100$, an approximate standard error is
    Sampling and proportions
11. Q211 · Original practice · 4 marks
    Let $f(x)=\sqrt{3x+6}-2$.
    Functions and relations
12. Q212 · Original practice · 5 marks
    Let $f(x)=x^{3}-6x^{2}+9x+4$.
    Further applications of differentiation
13. Q213 · Original practice · 5 marks
    The curves $y=6-0.5x^{2}$ and $y=x+2$ enclose a finite region.
    Further integration
14. Q214 · Original practice · 5 marks
    A screening test has sensitivity $0.92$ and specificity $0.95$. In the tested population, $4\%$ have the condition.
    Conditional probability
15. Q215 · Original practice · 4 marks
    A continuous random variable $X$ has density $f(x)=k x(4-x)$ for $0\le x\le 4$ and $0$ otherwise.
    Continuous random variables and the normal distribution
16. Q216 · Original practice · 5 marks
    The mass $X$ of $a$ manufactured component is modelled by $N(250,4^{2})$ grams.
    Continuous random variables and the normal distribution
17. Q217 · Original practice · 6 marks
    A rectangular poster has area $600\,\mathrm{cm}^{2}$. It has $2\,\mathrm{cm}$ margins on the left and right and $3\,\mathrm{cm}$ margins at the top and bottom. Let the poster width be $w cm$.
    Further applications of differentiation
18. Q218 · Original practice · 5 marks
    For $0\le x\le 2\pi$, solve $2\sin ^{2} x-3\sin x+1=0$.
    Trigonometry
19. Q219 · Original practice · 6 marks
    A random sample of $225$ voters gives $126$ who support a proposal. Use $z=1.96$ for an approximate $95\%$ confidence interval for the population proportion.
    Interval estimates for proportions
20. Q220 · Original practice · 1 mark
    The solution of $7e^{0.3t}=21$ is
    Exponential functions
21. Q221 · Original practice · 1 mark
    If $X\sim N(80,10^{2}), P(X<60)$ is approximately
    Continuous random variables and the normal distribution
22. Q222 · Original practice · 1 mark
    $\frac{d}{\mathrm{d}x} \ln (3x^{2}+1)$ equals
    Differentiation of exponential and logarithmic functions
23. Q223 · Original practice · 1 mark
    The period of $4\sin (\frac{\pi t}{6})+2$ is
    Trigonometry
24. Q224 · Original practice · 1 mark
    If $X\sim B(10,0.3), P(X=2)$ is
    Discrete random variables