Questions — WJEC (504 questions)

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WJEC Further Unit 4 Specimen Q6
7 marks Standard +0.3
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 3 & 2 \\ 3 & 2 & 5 \end{pmatrix}.$$
  1. Find
    1. the adjugate matrix of \(\mathbf{M}\),
    2. hence determine the inverse matrix \(\mathbf{M}^{-1}\). [5]
  2. Use your result to solve the simultaneous equations \begin{align} 2x + y + 3z &= 13
    x + 3y + 2z &= 13
    3x + 2y + 5z &= 22 \end{align} [2]
WJEC Further Unit 4 Specimen Q7
10 marks Standard +0.8
The function \(f\) is defined by $$f(x) = \frac{8x^2 + x + 5}{(2x + 1)(x^2 + 3)}.$$
  1. Express \(f(x)\) in partial fractions. [4]
  2. Hence evaluate $$\int_2^5 f(x)dx,$$ giving your answer correct to three decimal places. [6]
WJEC Further Unit 4 Specimen Q8
10 marks Challenging +1.2
The curve \(y = 1 + x^3\) is denoted by \(C\).
  1. A bowl is designed by rotating the arc of \(C\) joining the points \((0,1)\) and \((2,9)\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl. [5]
  2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\). [5]
WJEC Further Unit 4 Specimen Q9
14 marks Challenging +1.2
  1. Use mathematical induction to prove de Moivre's Theorem, namely that $$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta,$$ where \(n\) is a positive integer. [7]
    1. Use this result to show that $$\sin 5\theta = a \sin^5 \theta - b \sin^3 \theta + c \sin \theta,$$ where \(a\), \(b\) and \(c\) are positive integers to be found.
    2. Hence determine the value of \(\lim_{\theta \to 0} \frac{\sin 5\theta}{\sin \theta}\). [7]
WJEC Further Unit 4 Specimen Q10
11 marks Standard +0.8
Consider the differential equation $$\frac{dy}{dx} + 2y \tan x = \sin x, \quad 0 < x < \frac{\pi}{2}.$$
  1. Find an integrating factor for this differential equation. [4]
  2. Solve the differential equation given that \(y = 0\) when \(x = \frac{\pi}{4}\), giving your answer in the form \(y = f(x)\). [7]
WJEC Further Unit 4 Specimen Q11
17 marks Challenging +1.3
  1. Show that $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad \text{where } -1 < x < 1.$$ [4]
  2. Given that $$a \cosh x + b \sinh x \equiv r \cosh(x + \alpha), \quad \text{where } a > b > 0,$$ show that $$\alpha = \frac{1}{2} \ln \left(\frac{a+b}{a-b}\right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\). [7]
  3. Hence solve the equation $$5 \cosh x + 4 \sinh x = 10,$$ giving your answers correct to three significant figures. [6]
WJEC Further Unit 4 Specimen Q12
16 marks Challenging +1.2
The function \(f\) is given by $$f(x) = e^x \cos x.$$
  1. Show that \(f''(x) = -2e^x \sin x\). [2]
  2. Determine the Maclaurin series for \(f(x)\) as far as the \(x^4\) term. [6]
  3. Hence, by differentiating your series, determine the Maclaurin series for \(e^x \sin x\) as far as the \(x^3\) term. [4]
  4. The equation $$10e^x \sin x - 11x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places. [4]
WJEC Further Unit 5 2019 June Q1
8 marks Standard +0.3
A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below. 32.1 \quad 29.9 \quad 31.0 \quad 31.1 \quad 32.5 \quad 30.8 \quad 30.7 \quad 31.5
  1. Calculate a 95\% confidence interval for \(\mu\) based on this sample. [7]
  2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations. [1]
WJEC Further Unit 5 2019 June Q2
6 marks Standard +0.3
The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.
  1. Find the mean and the variance of \(X\). [2]
  2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]
WJEC Further Unit 5 2019 June Q3
9 marks Challenging +1.2
The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.
  1. What is the largest acceptable standard deviation? [3]
The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.
  1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [6]
WJEC Further Unit 5 2019 June Q4
11 marks Standard +0.8
Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder \(A\) was used by 12 players, randomly selected, and their mean weight gain was 900 g. Protein powder \(B\) was used by the other 10 players and their mean weight gain was 870 g. Let \(\mu_A\) and \(\mu_B\) be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder \(A\) and protein powder \(B\) respectively.
  1. Calculate a 98\% confidence interval for \(\mu_A - \mu_B\). [4]
  2. In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer. [2]
  3. Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder \(A\) over protein powder \(B\). [4]
  4. State one non-statistical assumption you have made in order to reach these conclusions. [1]
WJEC Further Unit 5 2019 June Q5
11 marks Standard +0.3
To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee must both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.
  1. Explain why the Wilcoxon signed rank test is appropriate. [2]
The scores awarded are shown below.
StudentABCDEFGH
Experienced Examiner1081099295145148134120
Trainee1141169593137144133110
    1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a 5\% significance level.
    2. What conclusion should be reached about the suitability of the trainee to qualify? [9]
WJEC Further Unit 5 2019 June Q6
10 marks Standard +0.3
A manufacturer of batteries for electric cars claims that an hour of charge can power a certain model of car to travel for an average of 123 miles. An electric car company and a consumer, Hopcyn, both wish to test the validity of the manufacturer's claim.
  1. Explain why Hopcyn may want to use a one-sided test and why the car company may want to use a two-sided test. [2]
To test the validity of this claim, Hopcyn collects data from a random sample of 90 drivers of this model of car to see how far they travelled, \(X\) miles, on an hour of charge. He produced the following summary statistics. $$\sum x = 11007 \quad \sum x^2 = 1361913$$
    1. Assuming Hopcyn uses a one-sided test, state the hypotheses.
    2. Test at the 5\% significance level whether the manufacturer's claim is correct. [8]
WJEC Further Unit 5 2019 June Q7
7 marks Standard +0.3
Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers. Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28 New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15 Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]
WJEC Further Unit 5 2019 June Q8
18 marks Challenging +1.2
The random variable \(X\) has probability density function $$f(x) = 1 + \frac{3\lambda x}{2} \quad \text{for } -\frac{1}{2} \leqslant x \leqslant \frac{1}{2},$$ $$f(x) = 0 \quad \text{otherwise,}$$ where \(\lambda\) is an unknown parameter such that \(-1 \leqslant \lambda \leqslant 1\).
    1. Find E\((X)\) in terms of \(\lambda\).
    2. Show that \(\text{Var}(X) = \frac{16 - 3\lambda^2}{192}\). [6]
  2. Show that P\((X > 0) = \frac{8 + 3\lambda}{16}\). [2]
In order to estimate \(\lambda\), \(n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\overline{X}\).
    1. Identify the distribution of \(Y\).
    2. Show that \(T_1\) is an unbiased estimator for \(\lambda\), where $$T_1 = \frac{16Y}{3n} - \frac{8}{3}.$$ [4]
    1. Show that \(\text{Var}(T_1) = \frac{64 - 9\lambda^2}{9n}\).
    2. Given that \(T_2\) is also an unbiased estimator for \(\lambda\), where $$T_2 = 8\overline{X},$$ find an expression for Var\((T_2)\) in terms of \(\lambda\) and \(n\).
    3. Hence, giving a reason, determine which is the better estimator, \(T_1\) or \(T_2\). [6]
WJEC Further Unit 5 2024 June Q1
9 marks Moderate -0.3
During practice sessions, a basketball coach makes his players run several 'line drills'.
  1. He records the times taken, in seconds, by one of his players to run the first 'line drill' on a random sample of 8 practice sessions. The results are shown below. 29.4 \quad 31.1 \quad 28.9 \quad 30.0 \quad 29.9 \quad 30.4 \quad 29.7 \quad 30.2 Assuming that these data come from a normal distribution with mean \(\mu\) and variance 0.6, calculate a 95\% confidence interval for \(\mu\). [5]
  2. State the two ways in which the method used to calculate the confidence interval in part (a) would change if the variance were unknown. [2]
  3. During a practice session, a player recorded a mean time of 35.6 seconds for 'line drills'.
    1. Give a reason why this player may not be the same as the player in part (a).
    2. Give a reason why this player could be the same as the player in part (a). [2]
WJEC Further Unit 5 2024 June Q2
9 marks Standard +0.8
In country A, the median daily caffeine intake per student who drinks coffee is 120 mg. A university professor who oversees a foreign exchange programme believes that students visiting from country B drink more coffee and therefore have a greater daily caffeine intake from coffee. On a randomly chosen day, the caffeine intake, in mg, from coffee consumption by each of 15 randomly selected students from country B is given below. 136 \quad 149 \quad 202 \quad 0 \quad 110 \quad 0 \quad 100 \quad 180 0 \quad 187 \quad 0 \quad 0 \quad 138 \quad 197 \quad 115 The professor suspects that the students with zero caffeine intake do not drink coffee, and decides to ignore those students and instead focus on the coffee-drinking students.
  1. Conduct an appropriate Wilcoxon test at a significance level as close to 5\% as possible. State your conclusion in context. [8]
  2. State one limitation of this investigation. [1]
WJEC Further Unit 5 2024 June Q3
7 marks Moderate -0.8
Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.
  1. Calculate an approximate 90\% confidence interval for \(p\), the probability that a randomly selected customer buys a steak pie. [6]
  2. Suppose that Tony carries out 50 such surveys and calculates 90\% confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of \(p\). [1]
WJEC Further Unit 5 2024 June Q4
11 marks Standard +0.3
The sports performance director at a university wishes to investigate whether there is a difference in the means of the specific gravities of blood of cyclists and runners. She models the distribution of specific gravity for cyclists as \(N\left(\mu_c, 8^2\right)\) and for runners as \(N\left(\mu_r, 10^2\right)\).
  1. State suitable hypotheses for this investigation. [1]
The mean specific gravity of blood of a random sample of 40 cyclists from the university was 1063. The mean specific gravity of blood of a random sample of 40 runners from the same university was 1060.
  1. Calculate and interpret the \(p\)-value for the data. [6]
  2. Suppose now that both samples were of size \(n\), instead of 40. Find the least value of \(n\) that would ensure that an observed difference of 3 in the mean specific gravities would be significant at the 1\% level. [4]
WJEC Further Unit 5 2024 June Q5
19 marks Challenging +1.8
The probability density function of the continuous random variable \(X\) is given by $$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$ $$f(x) = 0 \quad \text{otherwise.}$$ \(\overline{X}\) is the mean of a random sample of \(n\) observations of \(X\).
    1. Show that \(U = \frac{4\overline{X}}{3}\) is an unbiased estimator for \(\alpha\). [5]
    2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\)? [9]
  2. \(\overline{X}_1\) and \(\overline{X}_2\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2\) is also an unbiased estimator for \(\alpha\).
    1. Show that \(\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}\). [4]
    2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator. [1]
WJEC Further Unit 5 2024 June Q6
6 marks Standard +0.8
Alana is a PhD student researching language acquisition. She gives one group of randomly selected participants, Group A, 4 minutes to memorise 40 words that are similar in meaning. She gives a different, randomly selected group of participants, Group B, 4 minutes to memorise 40 words that are different in meaning. Alana believes that the students in Group B will do better than the students in Group A. The following results are the number of words recalled on testing the students from the two groups.
Group A32824161020221823212614
Group B302911253836281217
Conduct a Mann-Whitney U test at a significance level as close as possible to 5\% to test Alana's belief. [6]
WJEC Further Unit 5 2024 June Q7
19 marks Standard +0.8
A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.
  1. The mass \(W\), in kg, of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of \(\mu\). [3]
The mass \(X\), in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.
  1. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]
The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5. The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.
  1. Calculate the probability that the farmer's wife will move
    1. the einkorn,
    2. the corn. [5]
  2. The mass \(E\), in kg, of emmer stored in each individual container is normally distributed with mean 10.5 and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208.
    1. Find the value of \(\sigma\) that the farmer's son used.
    2. Explain why the value of \(\sigma\) that he used is unreasonable. [8]
WJEC Further Unit 5 Specimen Q1
13 marks Standard +0.8
Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm{N}(32, 4^2)\). You may assume that the times taken to complete the crossword on successive days are independent.
    1. Find the upper quartile of \(X\) and explain its meaning in context.
    2. Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
  2. Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm{N}(18, 2^2)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword. [6]
WJEC Further Unit 5 Specimen Q2
11 marks Standard +0.3
A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows. $$68.1 \quad 70.4 \quad 68.6 \quad 67.7 \quad 71.3 \quad 67.6 \quad 68.9 \quad 70.2 \quad 68.4 \quad 69.8$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).
  1. Determine a 95% confidence interval for \(\mu\). [9]
  2. The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95?' Explain why the answer to this question is no and give a correct interpretation. [2]
WJEC Further Unit 5 Specimen Q3
9 marks Challenging +1.2
A motoring organisation wishes to determine whether or not the petrol consumption of two different car models A and B are the same. A trial is therefore carried out in which 6 cars of each model are given 10 litres of petrol and driven at a predetermined speed around a track until the petrol is used up. The distances travelled, in miles, are shown below Model A: \(86.3 \quad 84.2 \quad 85.8 \quad 83.1 \quad 84.7 \quad 85.3\) Model B: \(84.9 \quad 85.9 \quad 84.8 \quad 86.5 \quad 85.2 \quad 85.5\) It is proposed to use a test with significance level 5% based on the Mann-Whitney statistic \(U\).
  1. State suitable hypotheses. [2]
  2. Find the critical region for the test. [3]
  3. Determine the value of \(U\) for the above data and state your conclusion in context. You must justify your answer. [4]