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WJEC Further Unit 1 2023 June Q4
7 marks Standard +0.3
4. The transformation \(T\) in the plane consists of a translation in which the point \(( x , y )\) is transformed to the point ( \(x + 2 , y - 2\) ), followed by a reflection in the line \(y = x\).
  1. Determine the \(3 \times 3\) matrix which represents \(T\).
  2. Determine how many invariant points exist under the transformation \(T\).
WJEC Further Unit 1 2023 June Q5
6 marks Standard +0.3
5. The points \(A\) and \(B\) have coordinates \(( 3,4 , - 2 )\) and \(( - 2,0,7 )\) respectively. The equation of the plane \(\Pi\) is given by \(2 x + 3 y + 3 z = 27\).
  1. Show that the vector equation of the line \(A B\) may be expressed in the form $$\mathbf { r } = ( 3 - 5 \lambda ) \mathbf { i } + ( 4 - 4 \lambda ) \mathbf { j } + ( - 2 + 9 \lambda ) \mathbf { k }$$
  2. Find the coordinates of the point of intersection of the line \(A B\) and the plane \(\Pi\).
WJEC Further Unit 1 2023 June Q6
6 marks Standard +0.3
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + \mathrm { i } | = 2 | z - 5 - 2 \mathrm { i } |$$ show that the locus of \(P\) is a circle and write down the coordinates of its centre.
WJEC Further Unit 1 2023 June Q7
7 marks Standard +0.8
7. Using mathematical induction, prove that $$\left[ \begin{array} { l l } 2 & 5 \\ 0 & 2 \end{array} \right] ^ { n } = \left[ \begin{array} { c c } 2 ^ { n } & 2 ^ { n - 1 } \times 5 n \\ 0 & 2 ^ { n } \end{array} \right]$$ for all positive integers \(n\).
WJEC Further Unit 1 2023 June Q8
9 marks Challenging +1.2
8. The roots of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 2 x + 8 = 0\) are denoted by \(\alpha , \beta , \gamma\). Determine the cubic equation whose roots are \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.
WJEC Further Unit 1 2023 June Q9
12 marks Standard +0.8
9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).
  1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
  3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).
WJEC Further Unit 1 2023 June Q10
8 marks Challenging +1.2
10. Gareth is investigating a series involving cube numbers. His series is $$1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 } - 6 ^ { 3 } + 7 ^ { 3 } - \ldots$$ Gareth continues his series and ends with an odd number.
Find and simplify an expression for the sum of Gareth's series in terms of \(k\), where \(k\) is the number of odd numbers in his series.
WJEC Further Unit 1 2024 June Q1
5 marks Moderate -0.5
  1. The complex numbers \(z , v\) and \(w\) are related by the equation
$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
WJEC Further Unit 1 2024 June Q2
3 marks Moderate -0.5
2. Given that \(x ^ { 2 } + 4 x + 5\) is a factor of \(x ^ { 3 } + x ^ { 2 } - 7 x - 15\), solve the equation \(x ^ { 3 } + x ^ { 2 } - 7 x - 15 = 0\).
WJEC Further Unit 1 2024 June Q3
6 marks Standard +0.8
3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\). A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 2024 June Q4
10 marks Standard +0.8
  1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and
$$w = \frac { z } { 1 - z }$$
  1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).
WJEC Further Unit 1 2024 June Q5
7 marks Standard +0.8
5. Given that $$\sum _ { r = k } ^ { 76 } ( r - 31 ) = 980$$ show that there are two possible values of \(k\).
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 2024 June Q6
12 marks Standard +0.3
  1. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram.
Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), are given by: $$\begin{aligned} & L _ { 1 } : | z - 2 + \mathrm { i } | = | z + 2 - 3 \mathrm { i } | \\ & L _ { 2 } : | z - 2 + \mathrm { i } | = \sqrt { 10 } \end{aligned}$$
  1. Find the coordinates of the points of intersection of these loci.
  2. On the same Argand diagram, sketch the loci \(L _ { 1 }\) and \(L _ { 2 }\). Clearly label the coordinates of the points of intersection.
WJEC Further Unit 1 2024 June Q7
7 marks Standard +0.3
7. Prove, by mathematical induction, that \(13 ^ { ( 2 n - 1 ) } + 8\) is a multiple of 7 for all positive integers \(n\).
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 2024 June Q8
12 marks Challenging +1.8
  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
8 marks Standard +0.3
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 2 2019 June Q1
7 marks Standard +0.3
  1. 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
  2. Calculate Spearman's rank correlation coefficient for this dataset.
  3. 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
9 marks Challenging +1.2
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
9 marks Standard +0.3
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
15 marks Standard +0.3
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
11 marks Standard +0.3
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 marks Moderate -0.3
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
13 marks Moderate -0.5
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
7 marks Easy -1.8
  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
11 marks Standard +0.3
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?