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WJEC Further Unit 1 Specimen Q1
7 marks Standard +0.8
Use mathematical induction to prove that \(4^n + 2\) is divisible by 6 for all positive integers \(n\). [7]
WJEC Further Unit 1 Specimen Q2
11 marks Standard +0.3
Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).
  1. Give your answer in Cartesian form [7]
  2. Give your answer in modulus-argument form. [4]
WJEC Further Unit 1 Specimen Q3
6 marks Challenging +1.2
Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series $$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$ Express your answer as a product of linear factors. [6]
WJEC Further Unit 1 Specimen Q4
7 marks Standard +0.8
The roots of the equation $$x^3 - 4x^2 + 14x - 20 = 0$$ are denoted by \(\alpha\), \(\beta\), \(\gamma\).
  1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = -12.$$ Explain why this result shows that exactly one of the roots of the above cubic equation is real. [3]
  2. Given that one of the roots is \(1 + 3i\), find the other two roots. Explain your method for each root. [4]
WJEC Further Unit 1 Specimen Q5
9 marks Standard +0.8
The complex number \(z\) is represented by the point \(P(x, y)\) in an Argand diagram and $$|z - 3| = 2|z + i|.$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre. [9]
WJEC Further Unit 1 Specimen Q6
9 marks Standard +0.8
The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \((x, y)\) is transformed to the point \((x + 1, y - 2)\), followed by an anticlockwise rotation through \(90°\) about the origin.
  1. Find the \(3 \times 3\) matrix representing \(T\). [6]
  2. Show that \(T\) has no fixed points. [3]
WJEC Further Unit 1 Specimen Q7
9 marks Standard +0.3
The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P(x, y)\) and \(Q(u,v)\) in Argand diagrams and $$w = z(1 + z)$$
  1. Show that $$v = y(1 + 2x)$$ and find an expression for \(u\) in terms of \(x\) and \(y\). [4]
  2. The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = au^2 + bu\), where \(a\) and \(b\) are constants whose values are to be determined. [5]
WJEC Further Unit 1 Specimen Q8
12 marks Standard +0.3
The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).
    1. Find the vector \(\overrightarrow{AB}\).
    2. Write down the vector equation of the line \(L\). [3]
  2. The plane \(\Pi\) has equation \(x + 3y - 2z = 5\).
    1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
    2. Find the acute angle between \(L\) and \(\Pi\). [9]
WJEC Further Unit 2 2018 June Q1
8 marks Challenging +1.8
The random variable \(X\) has the binomial distribution B(12, 0·3). The independent random variable \(Y\) has the Poisson distribution Po(4). Find
  1. \(E(XY)\), [2]
  2. Var\((XY)\). [6]
WJEC Further Unit 2 2018 June Q2
15 marks Standard +0.8
The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{x^3}{432}(8-x) & \text{for } 0 \leq x \leq 6, \\ 1 & \text{for } x > 6. \end{cases}$$
  1. Find \(P(X > 5)\). [2]
  2. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours. [2]
  3. Show that the upper quartile of the distribution lies between 4·5 and 4·6. [3]
  4. Find \(f(x)\), the probability density function for \(X\). [3]
  5. Find the mean lifetime of the batteries in hours. [4]
  6. The graph of \(f(x)\) is given below. \includegraphics{figure_1} Give a reason why the model may not be appropriate. [1]
WJEC Further Unit 2 2018 June Q3
11 marks Standard +0.3
A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1, the player wins £5. If the number is between 2 and 13 inclusive, the player wins £1. If the number is greater than 13, the player loses.
  1. Find the probability distribution of the player's profit. [5]
  2. Find the mean and standard deviation of the player's profit. [4]
  3. Given that 200 people play the game, calculate
    1. the expected number of players who win some money,
    2. the expected profit for the fete. [2]
WJEC Further Unit 2 2018 June Q4
9 marks Standard +0.3
On a Welsh television game show, contestants are asked to guess the weights of a random sample of seven cows. The game show judges want to investigate whether there is positive correlation between the actual weights and the estimated weights. The results are shown below for one contestant.
CowABCDEFG
Actual weight, kg61411057181001889770682
Estimated weight, kg70015008501400750900800
  1. Calculate Spearman's rank correlation coefficient for this data set. [5]
  2. Stating your hypotheses clearly, determine whether or not there is evidence at the 5% significance level of a positive association between the actual weights and the weights as estimated by this contestant. [3]
  3. One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement. [1]
WJEC Further Unit 2 2018 June Q5
12 marks Standard +0.3
A life insurance saleswoman investigates the number of policies she sells per day. The results for a random sample of 50 days are shown in the table below.
Number of policies sold0123456
Number of days229121591
She sees the same fixed number of clients each day. She would like to know whether the binomial distribution with parameters 6 and 0·6 is a suitable model for the number of policies she sells per day.
  1. State suitable hypotheses for a goodness of fit test. [1]
  2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.
    Number of policies sold0123456
    Observed229121591
    Expected0·2051·8436·912\(d\)\(e\)9·3312·333
    1. Calculate the values of \(d\) and \(e\).
    2. Carry out the test using a 10% level of significance and draw a conclusion in context. [10]
  3. What do the parameters 6 and 0·6 mean in this context? [1]
WJEC Further Unit 2 2018 June Q6
10 marks Moderate -0.3
A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.
Fewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualificationTotal
Less than £20 0001832202810108
£20 000 to £60 000509511215550462
More than £60 0003222935594
Total7114916121865664
By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.
  1. State the null and alternative hypotheses. [1]
  2. The table below shows the expected values. Calculate the value of \(k\). [2]
    Expected valuesFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20 000\(k\)24·2326·1935·4610·57
    £20 000 to £60 00049·40103·67112·02151·6845·23
    More than £60 00010·0521·0922·7930·869·20
  3. The following computer output is obtained. Calculate the values of \(m\) and \(n\). [2]
    Chi Squared ContributionsFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20 0003·604530799\(m\)1·461651·56860·03098
    £20 000 to £60 0000·0072727350·725354E-060·072640·50396
    More than £60 0004·9466198630·038971·690810·55498\(n\)
    X-squared = 19·61301, df = 8, p-value = 0·0119
    1. Without carrying out any further calculations, explain how X-squared = 19·61301 (the \(\chi^2\) test statistic) was calculated. [2]
    2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c). [2]
  5. The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement. [1]
WJEC Further Unit 2 2018 June Q7
7 marks Moderate -0.3
A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction. The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \includegraphics{figure_2}
  1. Calculate the equation of the least squares regression line of Job satisfaction (\(y\)) on Interesting work (\(x\)), given the following summary statistics. [5] \(\sum x = 46 \cdot 2\), \quad \(\sum y = 898\), \quad \(S_{xx} = 3 \cdot 48\) \(S_{xy} = 49 \cdot 45\), \quad \(S_{yy} = 1437 \cdot 714\), \quad \(n = 14\)
  2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90. [2]
WJEC Further Unit 2 2023 June Q1
7 marks Moderate -0.8
The random variable \(X\) has mean 17 and variance 64. The independent random variable \(Y\) has mean 10 and variance 16. Find the value of
  1. E\((4Y - 2X + 1)\), [2]
  2. Var\((4Y - 5X + 3)\), [2]
  3. E\((X^2 Y)\). [3]
WJEC Further Unit 2 2023 June Q2
8 marks Moderate -0.3
For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \((19, 20)\).
  1. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\). [6]
  2. State two reasons why your prediction in part (a) may not be reliable. [2]
WJEC Further Unit 2 2023 June Q3
11 marks Standard +0.3
It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
  1. Find the probability that the lifetime of a randomly selected hair dryer is between 1·8 and 2·5 years. [4]
  2. Given that 20% of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\). [3]
  3. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers. [3]
  4. State one assumption that you have made in part (c). [1]
WJEC Further Unit 2 2023 June Q4
12 marks Standard +0.3
A continuous random variable \(X\) has cumulative distribution function \(F\) given by $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{1}{4}x & \text{for } 0 \leqslant x \leqslant 2, \\ \frac{1}{480}x^4 + \frac{7}{15} & \text{for } 2 < x \leqslant b, \\ 1 & \text{for } x > b. \end{cases}$$
  1. Show that \(b = 4\). [2]
  2. Find P\((X \leqslant 2 \cdot 5)\). [2]
  3. Write down the value of the lower quartile of \(X\). [1]
  4. Find the value of the upper quartile of \(X\). [3]
  5. Find, correct to three significant figures, the value of \(k\) that satisfies the equation P\((X > 3 \cdot 5) = \text{P}(X < k)\). [4]
WJEC Further Unit 2 2023 June Q5
12 marks Standard +0.3
  1. Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient. [2]
  2. A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
    TractorFarmer's rankPTO (horsepower)Price (£1000s)
    A177·580
    B687·945
    C553·047
    D441·053
    E2112·060
    F390·061
    Spearman's rank correlation coefficient between the farmer's ranks and the price is 0·9429.
    1. Test at the 5% significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm. [4]
    2. Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO. [4]
    3. How should the tractor salesman interpret the results in (i) and (ii)? [2]
WJEC Further Unit 2 2023 June Q6
20 marks Standard +0.8
A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that 10% of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution B\((20, 0 \cdot 1)\). The numbers of boats that were not taken for 110 randomly selected days are given below.
Number of boats not taken012345 or more
Frequency1035292583
  1. State suitable hypotheses to carry out a goodness of fit test. [1]
  2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.
    Number of boats not taken012345 or more
    Observed1035292583
    Expected\(f\)29·72\(g\)20·919·884·75
    1. Calculate the values of \(f\) and \(g\).
    2. By completing the test, give the conclusion the manager should reach. [10]
The cost of hiring a boat is £15. Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give £20 (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
    1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
    2. Hence calculate the expected daily net income for the company following the manager's decision. [8]
  2. Is the manager justified in her decision? Give a reason for your answer. [1]
WJEC Further Unit 2 Specimen Q1
7 marks Challenging +1.8
The random variable \(X\) has mean14 and standard deviation 5. The independent random variable \(Y\) has mean 12 and standard deviation 3. The random variable \(W\) is given by \(W = XY\). Find the value of
  1. E(W), [1]
  2. Var(W). [6]
WJEC Further Unit 2 Specimen Q2
13 marks Standard +0.3
The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function $$f(t) = \frac{1}{2500}t(100-t^2) \quad \text{for } 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise.}$$
  1. Determine the mean queueing time. [3]
    1. Find the cumulative distribution function, \(F(t)\), of \(T\).
    2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
    3. Find the median queueing time. [10]
WJEC Further Unit 2 Specimen Q3
9 marks Standard +0.3
A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.
StudentABCDEFGH
History mark7359834957826760
Geography mark5551585944664967
  1. Calculate Spearman's rank correlation coefficient for this data set. [6]
  2. Hence determine whether or not, at the 5% significance level, there is evidence of a positive association between marks in History and marks in Geography. [2]
  3. Explain why it might not have been appropriate to use Pearson's product moment correlation coefficient to test association using this data set. [1]
WJEC Further Unit 2 Specimen Q4
9 marks Moderate -0.8
A year 12 student wishes to study at a Welsh university. For a randomly chosen year between 2000 and 2017 she collected data for seven universities in Wales from the Complete University Guide website. The data are for the variables: • 'Entry standards' – the average UCAS tariff score of new undergraduate students; • 'Student satisfaction' – a measure of student views of the teaching quality at the university taken from the National Student Survey (maximum 5); • 'Graduate prospects' – a measure of the employability of a university's first degree graduates (maximum 100); • 'Research quality' – a measure of the quality of the research undertaken in the university (maximum 4).
  1. Pearson's product-moment correlation coefficients, for each pairing of the four variables, are shown in the table below. Discuss the correlation between graduate prospects and the other three variables. [2]
    VariableEntry standardsStudent satisfactionGraduate prospectsResearch quality
    Entry standards1
    Student satisfaction-0.0301
    Graduate prospects0.7720.2361
    Research quality0.8660.0660.8271
  2. Calculate the equation of the least squares regression line to predict 'Entry standards'(y) from 'Research quality'(x), given the summary statistics: $$\sum x = 22.24, \sum y = 2522, S_{xx} = 1.0542, S_{xy} = 20193.5, S_{yy} = 122.72.$$ [5]
  3. The data for one of the Welsh universities are missing. This university has a research quality of 3.00. Use your equation to predict the entry standard for this university. [2]