Questions — WJEC (325 questions)

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WJEC Further Unit 1 2024 June Q8
  1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where
$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13
- 13 & 84 \end{array} \right]$$
  1. Determine the value of \(k\).
    Find the invariant points of \(T\).
WJEC Further Unit 1 2024 June Q9
9. Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), are defined by $$\begin{aligned} & \Pi _ { 1 } : 4 x - 3 y + 2 z = 5
& \Pi _ { 2 } : 6 x + y + z = 9 \end{aligned}$$
  1. Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Give your answer correct to three significant figures.
  2. Find the perpendicular distance from the point \(A ( 5 , - 2 , - 6 )\) to the plane \(\Pi _ { 1 }\).
    1. Show that the point \(B ( 5,5,0 )\) lies on \(\Pi _ { 1 }\) and that the point \(C ( 1,3,0 )\) lies on \(\Pi _ { 2 }\).
    2. State an equation of a plane that contains the points \(B\) and \(C\).
      Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 Specimen Q1
  1. Use mathematical induction to prove that \(4 ^ { n } + 2\) is divisible by 6 for all positive integers \(n\).
  2. Solve the equation \(2 z + i \bar { z } = \frac { - 1 + 7 i } { 2 + i }\).
    1. Give your answer in Cartesian form
    2. Give your answer in modulus-argument form.
    3. 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.
WJEC Further Unit 1 Specimen Q4
4. The roots of the equation $$x ^ { 3 } - 4 x ^ { 2 } + 14 x - 20 = 0$$ 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.
  2. Given that one of the roots is \(1 + 3 \mathrm { i }\), find the other two roots. Explain your method for each root.
WJEC Further Unit 1 Specimen Q5
5. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram and $$| z - 3 | = 2 | z + \mathrm { i } |$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre.
WJEC Further Unit 1 Specimen Q6
6. 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 \(90 ^ { \circ }\) about the origin.
  1. Find the \(3 \times 3\) matrix representing \(T\).
  2. Show that \(T\) has no fixed points.
WJEC Further Unit 1 Specimen Q7
7. 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 $$v = y ( 1 + 2 x )$$ and find an expression for \(u\) in terms of \(x\) and \(y\).
  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 \(v = a u ^ { 2 } + b u\), where \(a\) and \(b\) are constants whose values are to be determined.
WJEC Further Unit 1 Specimen Q8
8. The line \(L\) passes through the points \(\mathrm { A } ( 1,2,3 )\) and \(\mathrm { B } ( 2,3,1 )\).
    1. Find the vector \(\mathbf { A B }\).
    2. Write down the vector equation of the line \(L\).
  2. The plane \(\Pi\) has equation \(x + 3 y - 2 z = 5\).
    1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
    2. Find the acute angle between \(L\) and \(\Pi\).
WJEC Further Unit 2 2018 June Q1
  1. The random variable \(X\) has the binomial distribution \(\mathrm { B } ( 12,0 \cdot 3 )\). The independent random variable \(Y\) has the Poisson distribution \(\mathrm { Po } ( 4 )\). Find
    1. \(\mathrm { E } ( X Y )\),
    2. \(\quad \operatorname { Var } ( X Y )\).
    3. The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function
    $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
    \frac { x ^ { 3 } } { 432 } ( 8 - x ) & \text { for } 0 \leqslant x \leqslant 6
    1 & \text { for } x > 6 \end{cases}$$
  2. Find \(P ( X > 5 )\).
  3. 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.
  4. Show that the upper quartile of the distribution lies between \(4 \cdot 5\) and \(4 \cdot 6\).
  5. Find \(f ( x )\), the probability density function for \(X\).
  6. Find the mean lifetime of the batteries in hours.
  7. The graph of \(f ( x )\) is given below.
    \includegraphics[max width=\textwidth, alt={}, center]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-2_695_1463_1896_299}
WJEC Further Unit 2 2018 June Q3
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 \(\pounds 5\). If the number is between 2 and 13 inclusive, the player wins \(\pounds 1\). If the number is greater than 13 , the player loses.
  1. Find the probability distribution of the player's profit.
  2. Find the mean and standard deviation of the player's profit.
  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.
WJEC Further Unit 2 2018 June Q4
4. 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.
  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. One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement.
WJEC Further Unit 2 2018 June Q5
5. 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 sold
0123456
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.
  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.
  3. What do the parameters 6 and 0.6 mean in this context?
WJEC Further Unit 2 2018 June Q6
6. 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 £200001832202810108
£20000 to £60000509511215550462
More than £600003222935594
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.
  2. The table below shows the expected values. Calculate the value of \(k\).
    Expected valuesFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20000\(k\)24.2326•1935.4610.57
    £20000 to £6000049.40103.67112.02151.68\(45 \cdot 23\)
    More than £6000010.0521.09\(22 \cdot 79\)30.869.20
  3. The following computer output is obtained. Calculate the values of \(m\) and \(n\).
    Chi Squared ContributionsFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £200003.604530799\(m\)1.461651.56860.03098
    £20000 to £600000.0072727350.725354E-060.072640.50396
    More than £60000\(4 \cdot 946619863\)0.03897169081\(0 \cdot 55498\)\(n\)
    X-squared \(= 19 \cdot 61301 , d f = 8 , p\)-value \(= 0 \cdot 0119\)
    1. Without carrying out any further calculations, explain how X-squared \(= 19 \cdot 61301\) (the \(\chi ^ { 2 }\) test statistic) was calculated.
    2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c).
  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.
WJEC Further Unit 2 2018 June Q7
7. 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. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Job satisfaction versus Interesting work} \includegraphics[alt={},max width=\textwidth]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-6_647_1330_959_340}
\end{figure}
  1. Calculate the equation of the least squares regression line of Job satisfaction ( \(y\) ) on Interesting work ( \(x\) ), given the following summary statistics. $$\begin{array} { l l l } \sum x = 46 \cdot 2 , & \sum y = 898 , & S _ { x x } = 3 \cdot 48
    S _ { x y } = 49 \cdot 45 , & S _ { y y } = 1437 \cdot 714 , & n = 14 \end{array}$$
  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.
WJEC Further Unit 2 2019 June Q1
  1. (a) Sketch a scatter diagram of a dataset for which Spearman's rank correlation coefficient is + 1 , but the product moment correlation coefficient is less than 1 .
Two judges were judging cheese at the UK Cheese Festival. There were 8 blue cheeses in a particular category. The rankings are shown below.
CheeseABCDEFGH
Judge 115876432
Judge 213852467
(b) Calculate Spearman's rank correlation coefficient for this dataset.
(c) By sketching a scatter diagram of the rankings, or otherwise, comment on the extent to which the judges agree.
WJEC Further Unit 2 2019 June Q2
2. The probability of winning a certain game at a funfair is \(p\). Aman plays the game 5 times and Boaz plays the game 8 times. The independent random variables \(X\) and \(Y\) denote the number of wins for Aman and Boaz respectively.
  1. Given that \(\mathrm { E } ( X Y ) = 6 \cdot 4\), calculate \(p\).
  2. Find \(\operatorname { Var } ( X Y )\).
WJEC Further Unit 2 2019 June Q3
3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.
  1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
  2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
  3. What is the probability that her father gives her the car?
WJEC Further Unit 2 2019 June Q4
4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1
k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}
  1. Show that \(k = \frac { 4 } { 17 }\).
  2. Determine \(\mathrm { E } ( X )\).
  3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
WJEC Further Unit 2 2019 June Q5
5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.
ABcD
1Birth MonthObservedExpectedChi-Squared Contributions
2Jan-Mar259217.258.023302647
3Apr-June232217.251.001438435
4Jul-Sept200217.251.369677791
5Oct-Dec178217.257.091196778
6Total86986917.48561565
7
8p value
90.000561458
  1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.
    JanFebMarAprMayJunJulAugSepOctNovDec
    37114122665856
  2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.
WJEC Further Unit 2 2019 June Q6
6. The University of Arizona surveyed a large number of households. One purpose of the survey was to determine if annual household income could be predicted from size of family home. The graph of Annual household income, \(y\), versus Size of family home, \(x\), is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_616_1257_566_365}
  1. State the limitations of using the regression line above with reference to the scatter diagram. The data for size of family homes between 2000 and 3000 square feet are shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_652_1244_1516_360} Summary statistics for these data are as follows. $$\begin{array} { r c c } \sum x = 93160 & \sum y = 3907142 & n = 37
    S _ { x x } = 2869673.03 & S _ { y y } = 44312797167 & S _ { x y } = 348512820 \cdot 6 \end{array}$$
  2. Calculate the equation of the least squares regression line to predict Annual household income from Size of family home for these data.
WJEC Further Unit 2 2019 June Q7
7. An article published in a medical journal investigated sports injuries in adolescents' ball games: football, handball and basketball. In a study of 906 randomly selected adolescent players in the three ball games, 379 players incurred injuries over the course of one year of playing the sport. Rhian wants to test whether there is an association between the site of injury and the sport played. A summary of the injuries is shown in the table below.
\multirow{2}{*}{}Site of injury
Observed valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOtherTotal
\multirow{3}{*}{Sport}Football834536513612191
Handball14266154266115
Basketball428442211073
Total265755551154328379
  1. Calculate the values of \(A , B , C\) in the tables below.
    \multirow{2}{*}{}Site of injury
    Expected valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodod}Football13.102928.725627.717727.717757.955121.670214.1108
    Handball7.889217.295516.688716.6887A13.04758.4960
    Basketball5.007910.978910.593710.593722.15048.28235.3931
    \multirow{2}{*}{}\multirow[b]{2}{*}{Chi-Squared Contributions}Site of injury
    Shoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodoct}Football1.9873223.03890\(10 \cdot 77575\)2.47484\(B\)9.475860.31575
    Handball4.733334.38079C0.170871.446903.806640.73331
    Basketball0.2028626.388654.104004.104000.001026.403063.93521
  2. Given that the test statistic, \(X ^ { 2 }\), is 116.16, carry out the significance test at the \(5 \%\) level.
  3. Which site of injury most affects the conclusion of this test? Comment on your answer. Rhian also analyses the data on the type of contact that caused the injuries and the sport in which they occur, shown in the table below.
    Observed valuesBallOpponentSurfaceNoneTotal
    Football17681792194
    Handball23341938114
    Basketball2817121471
    Total6811948144379
    The chi-squared test statistic is 46.0937 . Rhian notes that this value is smaller than 116.16 , the test statistic in part (b). She concludes that there is weaker evidence for association in this case than there was in part (b).
  4. State Rhian's misconception and explain what she should consider instead. \section*{END OF PAPER}
WJEC Further Unit 2 2022 June Q1
  1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.
\(x\)021001000
\(\mathrm { P } ( X = x )\)0.90.09\(p\)0.0001
  1. Calculate the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
    2. Suggest why, in practice, people would be prepared to pay more than this minimum price.
WJEC Further Unit 2 2022 June Q2
2. An economist suggested the rate of unemployment and the rate of wage inflation are independent. Amy sets about investigating this suggestion. She collects unemployment data and wage inflation data from a random sample of regions in the UK and decides that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient. Amy's summary statistics for percentage unemployment, \(x\), and percentage wage inflation, \(y\), are shown below. $$\begin{array} { l l l } \sum x = 62 \cdot 8 & \sum y = 19 \cdot 4 & n = 10
\sum x ^ { 2 } = 413 \cdot 44 & \sum y ^ { 2 } = 46 \cdot 16 & \sum x y = 113 \cdot 16 \end{array}$$
  1. Calculate Pearson's product moment correlation coefficient for these data.
  2. Carry out Amy's test at the \(5 \%\) level of significance and state whether the economist's suggestion is reasonable. Amy also collects unemployment data and wage inflation data from a random sample of 10 regions in Spain and calculates Pearson's product moment correlation coefficient to be - 0.2525 .
  3. Should this change Amy's opinion on the economist's suggestion above? What could she do to improve her investigation?
  4. What assumption has Amy made in deciding that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient?
WJEC Further Unit 2 2022 June Q3
3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.
  1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
  2. A quarter of a game lasts 12 minutes.
    1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
    2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
  3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.
WJEC Further Unit 2 2022 June Q4
4. The continuous random variable \(R\) has probability density function \(f ( r )\) given by $$f ( r ) = \begin{cases} k r ( b - r ) & \text { for } 1 \leqslant r \leqslant 4 ,
0 & \text { otherwise } , \end{cases}$$ where \(k\) and \(b\) are positive constants.
  1. Explain why \(b \geqslant 4\).
  2. Given that \(b = 4\),
    1. show that \(k = \frac { 1 } { 9 }\),
    2. find an expression for \(F ( r )\), valid for \(1 \leqslant r \leqslant 4\), where \(F\) denotes the cumulative distribution function of \(R\),
    3. find the probability that \(R\) lies between 2 and 3 .