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WJEC Unit 2 Specimen Q1
6 marks Moderate -0.8
The events \(A, B\) are such that \(P(A) = 0.2, P(B) = 0.3\). Determine the value of \(P(A \cup B)\) when
  1. \(A,B\) are mutually exclusive, [2]
  2. \(A,B\) are independent, [3]
  3. \(A \subset B\). [1]
WJEC Unit 2 Specimen Q2
9 marks Standard +0.3
Dewi, a candidate in an election, believes that 45% of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,
  1. state hypotheses to be used to resolve this difference of opinion. [1]
They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.
    1. Determine the significance level of this critical region.
    2. If in fact \(p\) is actually 0.35, calculate the probability of a Type II error.
    3. Explain in context the meaning of a Type II error.
    4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level? [8]
WJEC Unit 2 Specimen Q3
7 marks Moderate -0.8
Cars arrive at random at a toll bridge at a mean rate of 15 per hour.
  1. Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval. [1]
  2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive. [3]
  3. Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3. [3]
WJEC Unit 2 Specimen Q4
7 marks Easy -1.3
A researcher wishes to investigate the relationship between the amount of carbohydrate and the number of calories in different fruits. He compiles a list of 90 different fruits, e.g. apricots, kiwi fruits, raspberries. As he does not have enough time to collect data for each of the 90 different fruits, he decides to select a simple random sample of 14 different fruits from the list. For each fruit selected, he then uses a dieting website to find the number of calories (kcal) and the amount of carbohydrate (g) in a typical adult portion (e.g. a whole apple, a bunch of 10 grapes, half a cup of strawberries). He enters these data into a spreadsheet for analysis.
  1. Explain how the random number function on a calculator could be used to select this sample of 14 different fruits. [3]
  2. The scatter graph represents 'Number of calories' against 'Carbohydrate' for the sample of 14 different fruits.
    1. Describe the correlation between 'Number of calories' and 'Carbohydrate'. [1]
    2. Interpret the correlation between 'Number of calories' and 'Carbohydrate' in this context. [1]
    \includegraphics{figure_1}
  3. The equation of the regression line for this dataset is: 'Number of calories' = 12.4 + 2.9 × 'Carbohydrate'
    1. Interpret the gradient of the regression line in this context. [1]
    2. Explain why it is reasonable for the regression line to have a non-zero intercept in this context. [1]
WJEC Unit 2 Specimen Q5
12 marks Easy -1.2
Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. In the pop industry most songs on the radio are not longer than three minutes.
  1. He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.) Comment on the suitability of this sample to investigate the magazine's claim. [1]
  2. Gareth recorded the data in the table below.
    Length of singles for top 50 UK Official Chart singles, 17 June 2016
    2.5-(3.0)3.0-(3.5)3.5-(4.0)4.0-(4.5)4.5-(5.0)5.0-(5.5)5.5-(6.0)6.0-(6.5)6.5-(7.0)7.0-(7.5)
    317227000001
    He used these data to produce a graph of the distributions of the lengths of singles \includegraphics{figure_2} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table. [2]
  3. Gareth also produced a box plot of the lengths of singles. \includegraphics{figure_3} He sees that there is one obvious outlier.
    1. What will happen to the mean if the outlier is removed?
    2. What will happen to the standard deviation if the outlier is removed? [2]
  4. Gareth decided to remove the outlier. He then produced a table of summary statistics.
    1. Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.
      Summary statistics
      Length of single for top 50 UK Official Singles Chart (minutes)
      Length of singleNMeanStandard deviationMinimumLower quartileMedianUpper quartileMaximum
      493.570.3932.773.263.603.894.38
    2. State, with a reason, whether these statistics support the magazine's claim. [4]
  5. Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.
    Summary statistics
    Length of single for Gareth's random sample of 30 singles (minutes)
    Length of singleNMeanStandard deviationMinimumLower quartileMedianUpper quartileMaximum
    303.130.3642.582.732.923.223.95
    Compare and contrast the distribution of lengths of singles in Gareth's personal collection with the distribution in the top 50 UK Official Singles Chart. [3]
WJEC Unit 2 Specimen Q6
7 marks Easy -1.2
A small object, of mass 0.02 kg, is dropped from rest from the top of a building which is160 m high.
  1. Calculate the speed of the object as it hits the ground. [3]
  2. Determine the time taken for the object to reach the ground. [3]
  3. State one assumption you have made in your solution. [1]
WJEC Unit 2 Specimen Q7
7 marks Moderate -0.8
The diagram below shows two particles \(A\) and \(B\), of mass 2 kg and 5 kg respectively, which are connected by a light inextensible string passing over a fixed smooth pulley. Initially, \(B\) is held at rest with the string just taut. It is then released. \includegraphics{figure_4}
  1. Calculate the magnitude of the acceleration of \(A\) and the tension in the string. [6]
  2. What assumption does the word 'light' in the description of the string enable you to make in your solution? [1]
WJEC Unit 2 Specimen Q8
5 marks Moderate -0.8
A particle \(P\), of mass 3 kg, moves along the horizontal \(x\)-axis under the action of a resultant force \(F\) N. Its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given by $$v = 12t - 3t^2.$$
  1. Given that the particle is at the origin \(O\) when \(t = 1\), find an expression for the displacement of the particle from \(O\) at time \(t\) s. [3]
  2. Find an expression for the acceleration of the particle at time \(t\) s. [2]
WJEC Unit 2 Specimen Q9
8 marks Standard +0.3
A truck of mass 180 kg runs on smooth horizontal rails. A light inextensible rope is attached to the front of the truck. The rope runs parallel to the rails until it passes over a light smooth pulley. The rest of the rope hangs down a vertical shaft. When the truck is required to move, a load of \(M\) kg is attached to the end of the rope in the shaft and the brakes are then released.
  1. Find the tension in the rope when the truck and the load move with an acceleration of magnitude 0.8 ms\(^{-2}\) and calculate the corresponding value of \(M\). [5]
  2. In addition to the assumptions given in the question, write down one further assumption that you have made in your solution to this problem and explain how that assumption affects both of your answers. [3]
WJEC Unit 2 Specimen Q10
7 marks Moderate -0.3
Two forces \(\mathbf{F}\) and \(\mathbf{G}\) acting on an object are such that $$\mathbf{F} = \mathbf{i} - 8\mathbf{j},$$ $$\mathbf{G} = 3\mathbf{i} + 11\mathbf{j}.$$ The object has a mass of 3 kg. Calculate the magnitude and direction of the acceleration of the object. [7]
WJEC Unit 3 2018 June Q1
4 marks Standard +0.3
Solve the equation $$|2x + 1| = 3|x - 2|.$$ [4]
WJEC Unit 3 2018 June Q2
4 marks Moderate -0.8
The diagram below shows a circle centre O, radius 4 cm. Points A and B lie on the circumference such that arc AB is 5 cm. \includegraphics{figure_2}
  1. Calculate the angle subtended at O by the arc AB. [2]
  2. Determine the area of the sector OAB. [2]
WJEC Unit 3 2018 June Q3
6 marks Moderate -0.8
The diagram below shows a sketch of the graph of \(y = f(x)\). The graph passes through the points \((-2, 0)\), \((0, 8)\), \((4, 0)\) and has a maximum point at \((1, 9)\). \includegraphics{figure_3}
  1. Sketch the graph of \(y = 2f(x + 3)\). Indicate the coordinates of the stationary point and the points where the graph crosses the \(x\)-axis. [3]
  2. Sketch the graph of \(y = 5 - f(x)\). Indicate the coordinates of the stationary point and the point where the graph crosses the \(y\)-axis. [3]
WJEC Unit 3 2018 June Q4
5 marks Standard +0.8
Solve the equation $$2\tan^2\theta + 2\tan\theta - \sec^2\theta = 2,$$ for values of \(\theta\) between \(0°\) and \(360°\). [5]
WJEC Unit 3 2018 June Q5
8 marks Moderate -0.3
  1. Show that $$\frac{3x}{(x-1)(x-4)^2} = \frac{A}{(x-1)} + \frac{B}{(x-4)} + \frac{C}{(x-4)^2},$$ where \(A\), \(B\) and \(C\) are constants to be found. [3]
  2. Evaluate \(\int_5^7 \frac{3x}{(x-1)(x-4)^2} \, dx\), giving your answer correct to 3 decimal places. [5]
WJEC Unit 3 2018 June Q6
5 marks Moderate -0.3
Write down the first three terms in the binomial expansion of \((1-4x)^{-\frac{1}{2}}\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac{1}{13}\) in your expansion, find an approximate value for \(\sqrt{13}\) in the form \(\frac{a}{b}\), where \(a\), \(b\) are integers. [5]
WJEC Unit 3 2018 June Q7
3 marks Standard +0.3
Use small angle approximations to find the small negative root of the equation $$\sin x + \cos x = 0.5.$$ [3]
WJEC Unit 3 2018 June Q8
5 marks Moderate -0.3
Find seven numbers which are in arithmetic progression such that the last term is 71 and the sum of all of the numbers is 329. [5]
WJEC Unit 3 2018 June Q9
10 marks Moderate -0.3
  1. Explain why the sum to infinity of a geometric series with common ratio \(r\) only converges when \(|r| < 1\). [1]
  2. A geometric progression \(V\) has first term 2 and common ratio \(r\). Another progression \(W\) is formed by squaring each term in \(V\). Show that \(W\) is also a geometric progression. Given that the sum to infinity of \(W\) is 3 times the sum to infinity of \(V\), find the value of \(r\). [6]
  3. At the beginning of each year, a man invests £5000 in a savings account earning compound interest at the rate of 3% per annum. The interest is added at the end of each year. Find the total amount of his savings at the end of the 20th year correct to the nearest pound. [3]
WJEC Unit 3 2018 June Q10
14 marks Standard +0.3
The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2\theta, \quad y = \cos\theta.$$
  1. Find the Cartesian equation of \(C\). [2]
  2. Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac{\pi}{3}\), and at the point \(Q\), where \(\theta = \frac{\pi}{2}\). Write down the coordinates of \(P\) and \(Q\). [5]
  3. Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. [7]
WJEC Unit 3 2018 June Q11
4 marks Standard +0.8
Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac{\pi}{2}\), $$\sin x + \cos x \geqslant 1.$$ [4]
WJEC Unit 3 2018 June Q12
10 marks Moderate -0.8
  1. Given that \(f\) is a function,
    1. state the condition for \(f^{-1}\) to exist,
    2. find \(ff^{-1}(x)\). [2]
  2. The functions \(g\) and \(h\), are given by $$g(x) = x^2 - 1,$$ $$h(x) = e^x + 1.$$
    1. Suggest a domain for \(g\) such that \(g^{-1}\) exists.
    2. Given the domain of \(h\) is \((-\infty, \infty)\), find an expression for \(h^{-1}(x)\) and sketch, using the same axes, the graphs of \(h(x)\) and \(h^{-1}(x)\). Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
    3. Determine an expression for \(gh(x)\) in its simplest form. [8]
WJEC Unit 3 2018 June Q13
8 marks Standard +0.3
  1. Express \(8\sin\theta - 15\cos\theta\) in the form \(R\sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$8\sin\theta - 15\cos\theta - 7 = 0.$$ [3]
  3. Determine the greatest value and the least value of the expression $$\frac{1}{8\sin\theta - 15\cos\theta + 23}.$$ [2]
WJEC Unit 3 2018 June Q14
12 marks Standard +0.3
Evaluate
  1. \(\int_0^2 x^3 \ln x \, dx\). [6]
  2. \(\int_0^1 \frac{2+x}{\sqrt{4-x^2}} \, dx\). [6]
WJEC Unit 3 2018 June Q15
5 marks Moderate -0.3
The variable \(y\) satisfies the differential equation $$2\frac{dy}{dx} = 5 - 2y, \quad \text{where } x \geqslant 0.$$ Given that \(y = 1\) when \(x = 0\), find an expression for \(y\) in terms of \(x\). [5]