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 21 of 26

1. Q249 · Original practice · 1 mark
   A Bernoulli random variable has success probability $0.35$. Its variance is
   Discrete random variables
2. Q250 · Original practice · 1 mark
   The shaded region lies between $y=3-x$ and the $x$-axis for $0\leq x\leq3$. Its area is
   Further integration
3. Q251 · Original practice · 1 mark
   The graph shows a velocity function. The displacement over $0\leq t\leq4$ is $\int_0^4(t-2)\,dt$. What is the total distance travelled?
   Further integration
4. Q252 · Original practice · 1 mark
   Values of $f$ are $f(0)=2$, $f(1)=5$ and $f(2)=4$. The trapezoidal estimate of $\int_0^2 f(x)\,dx$ with two equal strips is
   Further integration
5. Q253 · Original practice · 1 mark
   In triangle $ABC$, $AB=7$ cm, $AC=5$ cm and $\angle BAC=60^\circ$. The length $BC$ is
   Trigonometry
6. Q254 · Original practice · 1 mark
   Two straight paths of lengths $8$ m and $10$ m meet at an angle of $30^\circ$. The area of the triangle they form is
   Trigonometry
7. Q255 · Original practice · 1 mark
   A drone flies from $A$ to $B$ on a bearing of $065^\circ$. The bearing of $A$ from $B$ is
   Trigonometry
8. Q256 · Original practice · 1 mark
   A random variable has probability density $f(x)=kx$ for $0\leq x\leq2$, and zero otherwise. What is $k$?
   Continuous random variables and the normal distribution
9. Q257 · Original practice · 1 mark
   For a continuous random variable with a probability density, $P(X=3)$ equals
   Continuous random variables and the normal distribution
10. Q258 · Original practice · 1 mark
    Cookie masses are normally distributed with mean $52$ g and standard deviation $4$ g. A cookie of mass $58$ g has standardised value
    Continuous random variables and the normal distribution
11. Q259 · Original practice · 1 mark
    The graph shows a normal density. The shaded area is approximately
    Continuous random variables and the normal distribution
12. Q260 · Original practice · 1 mark
    In a random sample of $240$ festival visitors, $156$ arrived by public transport. The sample proportion is
    Sampling and proportions
13. Q261 · Original practice · 1 mark
    For independent random samples of size $100$ from a population with proportion $p=0.4$, the standard deviation of the sample proportion is
    Sampling and proportions
14. Q262 · Original practice · 1 mark
    A school wants to estimate the proportion of all students who cycle to school. Which procedure is most likely to produce a representative sample?
    Sampling and proportions
15. Q263 · Original practice · 1 mark
    A $95\%$ approximate confidence interval for a population proportion is $(0.41,0.53)$. Its margin of error is
    Interval estimates for proportions
16. Q264 · Original practice · 1 mark
    A researcher reports a $95\%$ confidence interval for the proportion of households with a compost bin. Which interpretation is appropriate?
    Interval estimates for proportions
17. Q265 · Original practice · 1 mark
    For a fixed sample, increasing the confidence level from $90\%$ to $99\%$ produces an approximate confidence interval that is
    Interval estimates for proportions
18. Q266 · Original practice · 1 mark
    Using the same confidence level and assumed proportion, a sample of size $n$ gives a planned margin of error $0.04$. To reduce this to $0.02$, the required sample size is
    Interval estimates for proportions
19. Q267 · Original practice · 1 mark
    For $f(x)=\dfrac{\ln x}{x}$, where $x>0$, $f^{\prime}(x)$ is
    Differentiation of trigonometric functions and differentiation rules
20. Q268 · Original practice · 1 mark
    A player takes four independent shots, each successful with probability $0.6$. A badge is awarded for at least three successes. The probability of a badge is
    Discrete random variables
21. Q269 · Original practice · 3 marks
    Consider the equation $\ln(x-1)+\ln(x+1)=\ln8$.
    Differentiation of exponential and logarithmic functions
22. Q270 · Original practice · 4 marks
    The scent concentration from a diffuser is modelled by $C(t)=Ae^{-kt}$, where $C$ is measured in arbitrary units and $t$ in hours. Measurements give $C(0)=80$ and $C(3)=20$.
    Differentiation of exponential and logarithmic functions
23. Q271 · Original practice · 4 marks
    The curve $y=\ln(2x+1)$ is defined for $x>-\dfrac12$.
    Differentiation of exponential and logarithmic functions
24. Q272 · Original practice · 5 marks
    A pop-up shop models its hourly sales rate by $R(t)=60te^{-t}$ for $0\leq t\leq5$, where $t$ is hours after opening and $R$ is items per hour.
    Further applications of differentiation