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 26 of 26

1. Q369 · Original practice · 3 marks
   It is known that $\int_1^5 f(x)\,dx=12$ and $\int_1^3 f(x)\,dx=7$.
   Further integration
2. Q370 · Original practice · 3 marks
   A positive function has the following recorded ordinates.

   $$\begin{array}{c|rrrr}x&0&1&2&3\\\hline f(x)&2&k&5&8\end{array}$$

   A three-strip trapezoidal estimate of its area on $[0,3]$ is $15$.
   Further integration
3. Q371 · Original practice · 6 marks
   The density of a continuous random variable is

   $$f(x)=\begin{cases}x/4,&0\leq x\leq2,\\(4-x)/4,&2<x\leq4,\\0,&\text{otherwise}.\end{cases}$$
   Continuous random variables and the normal distribution
4. Q372 · Original practice · 4 marks
   The histogram models a viewing time $X$ minutes. Its vertical axis is relative frequency density, and values within each class interval are modelled as uniform. The three class intervals are $[0,1)$, $[1,3)$ and $[3,6]$.
   Continuous random variables and the normal distribution
5. Q373 · Original practice · 5 marks
   Test scores are normally distributed. The 20th percentile is $82$ and the 80th percentile is $98$.
   Continuous random variables and the normal distribution
6. Q374 · Original practice · 5 marks
   An outlet sells two types of rechargeable battery. Type A accounts for $70\%$ of batteries sold and has operating time $N(8,1^2)$ hours. Type B accounts for $30\%$ and has operating time $N(10,2^2)$ hours. A randomly selected battery operates for more than $10$ hours.
   Continuous random variables and the normal distribution
7. Q375 · Original practice · 5 marks
   A computer simulates $100$ independent samples, each containing $10$ Bernoulli trials with success probability $p=0.3$. The graph shows the observed frequencies of the sample proportions. No other sample proportions occurred in this simulation.
   Sampling and proportions
8. Q376 · Original practice · 5 marks
   A population proportion is greater than $0.5$. For independent random samples of size $100$, the standard deviation of the sample proportion is $0.04$.
   Sampling and proportions
9. Q377 · Original practice · 5 marks
   Two independent random samples from the same population have no respondents in common. In sample A, $72$ of $120$ respondents support a proposal. In sample B, $90$ of $180$ support it. Use $z=1.96$.
   Interval estimates for proportions
10. Q378 · Original practice · 6 marks
    For $0\leq t\leq12$, a reservoir level is modelled by $H(t)=7+3\sin(\pi t/6)$ metres, where $t$ is measured in hours. An alarm sounds only when the level is above $9$ m and rising at more than $1$ m/hour. An engineer claims that the alarm will sound for at least $20$ minutes during this period. Evaluate the reasonableness of the claim.
    Further applications of differentiation