Questions — WJEC (504 questions)

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WJEC Further Unit 6 2024 June Q1
Standard +0.8
  1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.
Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Find the velocity of \(A\) immediately after the collision.
  2. Calculate the coefficient of restitution between \(A\) and \(B\).
  3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
  4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q2
Challenging +1.2
  1. An object, of mass 1.8 kg , is falling vertically downwards under gravity. During the motion, it experiences a variable resistance of \(0 \cdot 2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the object at time \(t\) seconds.
    1. Show that \(v\) satisfies the differential equation
    $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 9 g - v ^ { 2 } } { 9 }$$ At time \(t = 0\), the object passes a point \(A\) with a speed of \(\sqrt { g } \mathrm {~ms} ^ { - 1 }\). The object then hits the ground with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
  2. Calculate the time taken for the object to hit the ground.
    …………………………………………………………………………………………………………………………………………………... \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_67_1614_639_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_76_1614_717_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_79_1614_801_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_72_1609_895_267} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_75_1614_979_264}
  3. Given that the distance of the object from \(A\) at time \(t\) is \(x\) metres, form another differential equation to find an expression for \(x\) in terms of \(v\). Hence, find the height of \(A\) above the ground.
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q3
Standard +0.8
  1. \(A C B\) is the diameter of a semi-circular lamina of radius \(2 a\) and centre \(C\). Another semi-circular lamina, having \(A C\) as its diameter, is added to form a uniform lamina, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}
    1. (i) Show that the distance of the centre of mass of the lamina from \(A B\) is \(\frac { 28 } { 15 \pi } a\).
      (ii) Calculate the distance of the centre of mass of the lamina from a line drawn through \(A\) that is perpendicular to \(A B\).
      □
      □
    2. Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at \(A\) and \(B\) so that \(A B\) is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at \(B\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q4
Standard +0.3
  1. The diagram below shows part of a game at a funfair that consists of a target moving along a straight horizontal line \(A B\). The centre of the target may be modelled as a particle moving with Simple Harmonic Motion about centre \(O\), where \(O\) is the midpoint of \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-14_245_1145_525_452}
When the target is at a distance of 84 cm from \(O\), its speed is \(52 \mathrm { cms } ^ { - 1 }\) and the magnitude of its acceleration is \(1344 \mathrm { cms } ^ { - 2 }\).
  1. Show that the period of the motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
    (b) Determine the maximum speed of the target.Examiner
  2. During a game, players fire a ball at the target. A timer is started when the target is at \(A\). Players must wait for the target to complete at least one full cycle before firing. Given that the target is hit when it is at a distance of 67 cm from \(O\), calculate the two earliest possible times taken to hit the target.
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 6 2024 June Q5
Standard +0.8
  1. The diagram below shows a uniform rod \(A B\) of weight \(W N\) and length \(2 l\), with its lower end \(A\) resting on a rough horizontal floor. A light cable is attached to the other end \(B\). The rod is in equilibrium when it is inclined at an angle of \(\theta\) to the floor, where \(0 ^ { \circ } < \theta \leqslant 45 ^ { \circ }\). The tension in the cable is \(T \mathrm {~N}\) acting at an angle of \(2 \theta\) to the rod, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-18_508_1105_559_479}
    1. (i) Show that \(T = \frac { W } { 4 } \operatorname { cosec } \theta\).
      (ii) Hence determine the normal reaction of the floor on the rod at \(A\), giving your answer in terms of \(W\).
    2. Given that the coefficient of friction between the floor and the rod is \(\frac { \sqrt { 3 } } { 3 }\), calculate the minimum possible value for \(\theta\).
    3. The region \(R\), shown in the diagram below, is bounded by the coordinate axes and the curve
    $$y = \frac { a } { b } \sqrt { b ^ { 2 } - x ^ { 2 } }$$ where \(a , b\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-21_451_1116_644_468} The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 3 } \pi a ^ { 2 } b\).
  2. Use integration to show that the distance of the centre of mass of \(S\) from the \(y\)-axis is \(\frac { 3 b } { 8 }\).
    The diagram below shows a small tree growing in a pot. The uniform solid \(S\) described on the previous page may be used to model the part of the tree above the pot. This part of the tree has height \(h \mathrm {~cm}\) and base radius \(\frac { h } { 4 } \mathrm {~cm}\). The pot, including its contents, may be modelled as a solid cylinder of height 50 cm and radius 25 cm . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-22_846_839_1596_612} You may assume that the density of the pot, including its contents, is equal to 20 times the density of the part of the tree above the pot.
  3. A gardener suggests that a tree is said to have outgrown its pot if the centre of mass, of both the tree and its pot, lies above the height of the pot. Determine the maximum value of \(h\) before the tree outgrows its pot.
  4. Identify one possible limitation of the model used that could affect your answer to part (b). \section*{END OF PAPER} Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 1 2018 June Q1
Moderate -0.8
Showing all your working, simplify a) \(\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }\),
b) \(\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }\).
WJEC Unit 1 2018 June Q2
Standard +0.3
The points \(A\) and \(B\) have coordinates \(( - 1,10 )\) and \(( 5,1 )\) respectively. The straight line \(L\) has equation \(2 x - 3 y + 6 = 0\). a) The line \(L\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
b) Determine the ratio in which the line \(L\) divides the line \(A B\).
c) The line \(L\) crosses the \(x\)-axis at the point \(D\). Find the coordinates of \(D\).
d) i) Show that \(L\) is perpendicular to \(A B\).
ii) Calculate the area of the triangle \(A C D\).
WJEC Unit 1 2018 June Q3
Moderate -0.8
Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$
04
a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\).
05
The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
□
0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\).
07
Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$
WJEC Unit 1 2018 June Q8
Moderate -0.8
Given that \(( x - 2 )\) and \(( x + 2 )\) are factors of the polynomial \(2 x ^ { 3 } + p x ^ { 2 } + q x - 12\), a) find the values of \(p\) and \(q\),
b) determine the other factor of the polynomial.
WJEC Unit 1 2018 June Q9
5 marks Moderate -0.3
The triangle \(A B C\) is such that \(A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}\) and \(A \widehat { B C } = 32 ^ { \circ }\). Find two possible values for the area of the triangle \(A B C\).
10
a) Use the binomial theorem to expand \(( a + \sqrt { b } ) ^ { 4 }\).
b) Hence, deduce an expression in terms of \(a\) and \(b\) for \(( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }\).
11
a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }\) and \(\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }\). Determine the range of values for \(\mu\) such that \(\mu | \mathbf { v } | > | \mathbf { u } |\).
b) The point \(A\) has position vector \(11 \mathbf { i } - 4 \mathbf { j }\) and the point \(B\) has position vector \(21 \mathbf { i } + \mathbf { j }\). Determine the position vector of the point \(C\), which lies between \(A\) and \(B\), such that \(A C : C B\) is \(2 : 3\).
12
Find the values of \(m\) for which the equation \(4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )\) has real roots. [5]
WJEC Unit 1 2018 June Q13
Moderate -0.8
A curve \(C\) has equation \(y = x ^ { 3 } - 3 x ^ { 2 }\). a) Find the stationary points of \(C\) and determine their nature.
b) Draw a sketch of \(C\), clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\) is positive or
negative, giving a reason for your answer.
14
In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\ & \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true.
15
The value of a car, \(\pounds V\), may be modelled as a continuous variable. At time \(t\) years, the value of the car is given by \(V = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. When the car is new, it is worth \(\pounds 30000\). When the car is two years old, it is worth \(\pounds 20000\). Determine the value of the car when it is six years old, giving your answer correct to the nearest \(\pounds 100\).
16
The curve \(C\) has equation \(y = 7 + 13 x - 2 x ^ { 2 }\). The point \(P\) lies on \(C\) and is such that the tangent to \(C\) at \(P\) has equation \(y = x + c\), where \(c\) is a constant. Find the coordinates of \(P\) and the value of \(c\).
17
a) Solve \(2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2\).
b) Solve \(3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }\).
c) Express \(4 ^ { x } - 10 \times 2 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\). Hence solve the equation \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\).
WJEC Unit 1 2018 June Q18
Moderate -0.5
The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.
WJEC Unit 2 2022 June Q2
Standard +0.3
The probability distribution for \(X\), the lifetime of a light bulb, in hours, is given below.
\(X\)\(256 \leqslant x < 259\)\(259 \leqslant x < 262\)\(262 \leqslant x < 265\)\(265 \leqslant x < 267\)\(267 \leqslant x < 300\)
Probability0.050.250.450.200.05
a) Suppose that a random sample of 40 light bulbs is tested, and a histogram is drawn of their lifetimes. Calculate the expected height of the bar for the interval \(262 \leqslant x < 265\).
b) Now suppose that the last two intervals are changed to \(265 \leqslant x < 268\) and \(268 \leqslant x < 300\). Explain why it is not possible to tell what will happen to the expected heights of the last two bars.
c) Celyn collects a different random sample of 40 light bulbs to test. She draws a histogram of their lifetimes and finds that it is different to the histogram referred to in part (a). Should Celyn be concerned that the two histograms are different?
WJEC Unit 2 2022 June Q3
Moderate -0.3
In a study, samples of soil were collected during the summer. Soil samples of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) were collected for analysis. The study found that there were, on average, 11 earthworms per sample. a) Explain briefly the conditions under which a Poisson distribution could be used to model the number of earthworms per sample.
b) In July, pupils at a primary school are asked to dig a smaller hole, \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 10 \mathrm {~cm}\), and to count the number of earthworms they find. Calculate the probability that the pupils find exactly 5 earthworms.
c) In the autumn, the average number of earthworms per sample is greater than in the summer. The probability that, in the autumn, there are fewer than 13 earthworms in a soil sample of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) is close to \(36 \%\). Find the mean number of earthworms, to the nearest whole number, per \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) soil sample in the autumn.
04
Jessica is studying the relationship between hip girth, \(h \mathrm {~cm}\), and thigh girth, \(t \mathrm {~cm}\), for American adults who are physically active. She takes a random sample of 11 people from a very large dataset which she has downloaded into a spreadsheet software package. The results are shown below.
\(h ( \mathrm {~cm} )\)\(98 \cdot 6\)\(112 \cdot 1\)\(97 \cdot 9\)\(110 \cdot 2\)\(89 \cdot 2\)\(111 \cdot 7\)\(87 \cdot 0\)\(94 \cdot 7\)\(100 \cdot 4\)\(104 \cdot 0\)\(88 \cdot 4\)
\(t ( \mathrm {~cm} )\)\(48 \cdot 3\)\(87 \cdot 2\)\(55 \cdot 2\)\(68 \cdot 0\)\(48 \cdot 5\)\(63 \cdot 2\)\(49 \cdot 5\)\(55 \cdot 7\)\(59 \cdot 1\)\(64 \cdot 0\)\(52 \cdot 4\)
a) Jessica notes that, for the thigh girth data, the lower quartile is 49.5 and the upper quartile is \(64 \cdot 0\).
i) Show that 87.2 should be classified as an outlier for \(t\).
ii) Give a reason why Jessica might exclude the outlier.
iii) Give a reason why Jessica might include the outlier. Jessica decides to exclude the outlier and produces the following scatter diagram. \section*{Thigh girth versus Hip girth} \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-04_647_1250_1439_404}
b) Interpret, in context, the correlation in the data shown in the diagram. The equation of the regression line of \(t\) on \(h\) for this sample is $$t = 0.69 h - 11.26$$ c) Interpret the gradient of the regression line in this context.
d) Use your knowledge of large data sets and spreadsheet software packages to suggest a way in which Jessica could improve her investigation. A company, Run4Lyfe, sponsors an athletic event. The organisers of the event claim that \(70 \%\) of the participants know the name of the sponsoring company. Run4Lyfe is concerned that the proportion, \(p\), of participants knowing the name of the sponsoring company is less than \(70 \%\). They decide to survey 60 randomly selected participants to carry out a significance test.
a) State suitable hypotheses for carrying out the test.
b) i) Explain what is meant by the critical region for this test.
ii) Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, \(5 \%\).
iii) Given that 40 participants out of the 60 in the sample know the name of the company, complete the significance test.
c) State, with a reason, how you would advise Run4Lyfe with regards to sponsoring the event next year.
06
The fertility rate for a country is the average number of children that are born to a woman over her lifetime. The graphs and table below show some data on the fertility rates for 197 countries in the years 1914 and 2014. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fertility rates in 1914} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_671_1483_593_283}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fertility rates in 2014} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_616_1219_1434_287}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Decreases in fertility rates from 1914 to 2014} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_476_613_2270_388}
\end{figure}
Minimum value- 0.71
Lower quartile2.08
Median3.19
Upper quartile3.94
Maximum value6.49
a) Comment on the shapes of the distributions of fertility rates for 1914 and 2014.
b) Interpret the minimum value, \(- 0 \cdot 71\), in the boxplot. You are also given the following information:
Country
Fertility rate
for 1914
Fertility rate
for 2014
France
Between 2
and 3
1.98
Ethiopia
Between 6
and 7
4.4
c) i) Find the best possible estimate for the decrease in the fertility rate from 1914 to 2014 for France.
ii) Find the best possible estimate for the decrease in the fertility rate from 1914 to 2014 for Ethiopia.
iii) Give one possible reason why the answers to i) and ii) are so different.
iv) Explain why these estimates may not be very accurate. \section*{Section B: Mechanics}
\(\mathbf { 0 }\)7
The diagram below shows a vehicle of mass 1300 kg towing a trailer of mass 500 kg by means of a light horizontal tow bar. The vehicle is moving forward along a straight horizontal road such that a constant resistance of magnitude 650 N acts on the vehicle and a constant resistance of magnitude 320 N acts on the trailer. The vehicle's engine produces a constant driving force of \(F \mathrm {~N}\).
\includegraphics[max width=\textwidth, alt={}]{77c62e6d-58e4-42d3-9982-5a8325e8e826-08_158_851_781_609}
Given that the acceleration of the vehicle and trailer is \(0.85 \mathrm {~ms} ^ { - 2 }\), show that \(F = 2500\) and determine the tension in the tow bar.
WJEC Unit 2 2022 June Q8
Easy -1.2
An aircraft moves along a straight horizontal runway with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\). Points \(A\) and \(B\) lie on the runway. The aircraft passes \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and its speed at \(B\) must be at least \(78 \mathrm {~ms} ^ { - 1 }\) if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes \(A\).
b) Determine the minimum value of the distance \(A B\) for the aircraft to take off successfully. The diagram below shows an object \(A\), of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box \(B\) by a light inextensible string which passes over a smooth pulley \(P\), fixed at the edge of the surface, so that box \(B\) hangs freely. An object \(C\) lies on the horizontal floor of box \(B\) so that the combined mass of \(B\) and \(C\) is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to \(A\) in the direction \(P A\) so that box \(B\) is raised.
a) Find the magnitude of the acceleration of \(A\) and the tension in the string.
b) Given that object \(C\) has mass 4 kg , calculate the reaction of the floor of the box on object \(C\).
□
1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:
  • the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
  • the park is 2 km away on a bearing of \(060 ^ { \circ }\).
    a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
    b) Determine the total distance walked by Sarah from her house to the park.
    c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.
WJEC Unit 2 2022 June Q11
Standard +0.3
A particle \(P\) moves along the \(x\)-axis so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds \(( t \geqslant 0 )\) is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of \(t\) when \(P\) is instantaneously at rest.
b) Calculate the total distance travelled by the particle \(P\) whilst its velocity is decreasing.
WJEC Unit 3 2019 June Q1
Moderate -0.3
a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).
WJEC Unit 3 2019 June Q2
Standard +0.8
Expand \(\frac { 4 - x } { \sqrt { 1 + 2 x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). State the range of values of \(x\) for which the expansion is valid.
WJEC Unit 3 2019 June Q3
Moderate -0.8
The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
05
a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.
WJEC Unit 3 2019 June Q6
Moderate -0.3
A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\).
\(\mathbf { 0 }\)7
The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).
WJEC Unit 3 2019 June Q8
Standard +0.3
a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.
WJEC Unit 3 2019 June Q9
Standard +0.3
a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$
WJEC Unit 3 2019 June Q10
Moderate -0.3
a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
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1
The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
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1
A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.
WJEC Unit 3 2019 June Q14
Moderate -0.3
a) Find \(\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x\). b) Find \(\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x\).
c) Find \(\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x\).
d) Use the substitution \(u = 2 \cos x + 1\) to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$
WJEC Unit 3 2019 June Q15
Standard +0.3
Use proof by contradiction to show that \(\sqrt { 6 }\) is irrational.