Questions Unit 4 (17 questions)

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WJEC Unit 4 2018 June Q1
1 \(\mathbf { 0 }\) A particle of mass 2 kg moves under the action of a constant force \(\mathbf { F N }\), where \(\mathbf { F }\) is given by $$\mathbf { F } = - 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k }$$ a) Find the magnitude of the acceleration of the particle.
b) Given that at time \(t = 0\) seconds, the position vector of the particle is \(2 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k }\) and it is moving with velocity \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\), find the position vector of the particle when \(t = 2\) seconds. \section*{END OF PAPER}
WJEC Unit 4 2019 June Q1
\(\mathbf { 1 }\) | \(\mathbf { 0 }\) A tennis ball is projected with velocity vector \(( 30 \mathbf { i } - 1 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from a point \(P\) which is at a height of 2.4 m vertically above a horizontal tennis court. The ball then passes over a net of height 0.9 m , before hitting the ground after \(\frac { 4 } { 7 } \mathrm {~s}\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. The origin \(O\) lies on the ground directly below the point \(P\). The base of the net is \(x \mathrm {~m}\) from \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{7d864f3d-0bb8-41ff-8e98-fee9776c037e-8_456_1655_689_203}
a) Find the speed of the ball when it first hits the ground, giving your answer correct to one decimal place.
b) After \(\frac { 2 } { 5 } \mathrm {~s}\), the ball is directly above the net.
i) Find the position vector of the ball after \(\frac { 2 } { 5 } \mathrm {~s}\).
ii) Hence determine the value of \(x\) and show that the ball clears the net by approximately 16 cm .
c) In fact, the ball clears the net by only 4 cm .
i) Explain why the observed value is different from the value calculated in (b)(ii).
ii) Suggest a possible improvement to this model.
WJEC Unit 4 2024 June Q1
  1. The table below shows the destination from school of 180 year 11 pupils. Most pupils either continued education, in school or college, or went into some form of employment.
\cline { 2 - 6 } \multicolumn{1}{c|}{}SchoolCollegeEmploymentOtherTotal
Boys334982\(\mathbf { 9 2 }\)
Girls404071\(\mathbf { 8 8 }\)
Total\(\mathbf { 7 3 }\)\(\mathbf { 8 9 }\)\(\mathbf { 1 5 }\)\(\mathbf { 3 }\)\(\mathbf { 1 8 0 }\)
A reporter selects two pupils at random to interview. Given that the first pupil is in school or college, find the probability that both pupils are girls.
WJEC Unit 4 2024 June Q2
2. The smallest angle \(\theta\), in degrees, of a right-angled triangle with hypotenuse 8 cm , is uniformly distributed across all possible values.
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-04_419_696_479_687}
  1. Find the mean and standard deviation of \(\theta\).
  2. The shortest side of the triangle is of length \(X \mathrm {~cm}\). Find the probability that \(X\) is greater than 5 .
WJEC Unit 4 2024 June Q3
3. Awena has a large data set of body measurements, and she wants to investigate relationships between body dimensions. In this particular investigation, she is testing for a correlation between forearm girth and bicep girth. The diagrams below show how to measure these. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-06_499_590_534_276} \captionsetup{labelformat=empty} \caption{Forearm girth}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-06_499_591_534_1194} \captionsetup{labelformat=empty} \caption{Bicep girth}
\end{figure}
  1. Why is it appropriate for Awena to use a one-tailed test?
    Awena takes a random sample of size 11 from her data set and plots the following scatter diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-07_937_1431_420_312}
  2. Using the computer output above, carry out a one-tailed significance test on the sample product moment correlation coefficient at the \(0 \cdot 5 \%\) level.
  3. Blodwen also has access to the same large data set. She decides to do the same test using all of the 507 available data points. Her results are shown below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Forearm girth versus Bicep girth} \includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-08_933_1504_477_276}
    \end{figure}
    1. State the problem Blodwen will encounter when attempting to use statistical tables for her test.
    2. How should Blodwen deal with this problem?
      \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 4 2024 June Q4
  1. Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean \(2 \cdot 2\) and standard deviation \(0 \cdot 3\).
    1. Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .
    Jake delivers the lightest \(80 \%\) of parcels on his bike. The rest he puts in his car and delivers by car.
  2. Find the mass of the heaviest parcel he would deliver by bike.
  3. He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .
  4. In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the \(5 \%\) level of significance. Jake delivers each parcel to one of three areas, \(A , B\) or \(C\). The probabilities that a parcel has destination area \(A , B\) and \(C\) are \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. All parcels are considered to be independent.
  5. On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.
  6. On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.
WJEC Unit 4 2024 June Q5
7 marks
  1. The diagram below shows four coplanar horizontal forces of magnitude \(F \mathrm {~N} , 12 \mathrm {~N} , 16 \mathrm {~N}\) and 20 N acting at a point \(P\) in the directions shown.
    \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-14_792_862_593_607}
Given that the forces are in equilibrium, calculate the value of \(F\) and the size of the angle \(\alpha\). [7]
WJEC Unit 4 2024 June Q6
6. A ball is projected with velocity \(( 4 w \mathbf { i } + 7 w \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the top of a vertical tower. After 5 seconds, the ball hits the ground at a point that is 60 m horizontally from the foot of the tower. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.
  1. Find the value of \(w\) and hence determine the height of the tower.
  2. Determine the proportion of the 5 seconds for which the ball is on its way down.
WJEC Unit 4 2024 June Q7
3 marks
7. As part of a design for a new building, an architect wants to support a wooden beam in a horizontal position. The beam is suspended using a vertical steel cable and a smooth fixed support on its underside. The diagram below shows the architect's diagram and the adjacent table shows the categories of steel cable available.
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-18_504_1699_559_191} You may use the following modelling assumptions.
  • The wooden beam is a rigid uniform rod of mass 100 kg .
  • The force exerted on the beam by the support is vertical.
  • The steel cable is inextensible.
\section*{SAFETY REQUIREMENT} Both the steel cable and the support must be capable of withstanding forces of at least four times those present in the architect's diagram above. The wooden beam is held in horizontal equilibrium.
[0pt]
    1. Given that the support is capable of withstanding loads of up to 2000 N , show that the force exerted on the beam by the support satisfies the safety requirement. [3]
    2. Determine which categories of steel cable in the table opposite could meet the safety requirements.
  2. State how you have used the modelling assumption that the beam is a uniform rod. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 4 2024 June Q8
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are acting on an object of mass 3 kg such that
$$\begin{aligned} & \mathbf { F } _ { 1 } = ( \mathbf { i } + 8 c \mathbf { j } + 11 c \mathbf { k } ) \mathrm { N } ,
& \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } - c \mathbf { j } - 12 \mathbf { k } ) \mathrm { N } ,
& \mathbf { F } _ { 3 } = ( ( 15 c + 1 ) \mathbf { i } + 2 c \mathbf { j } - 5 c \mathbf { k } ) \mathrm { N } , \end{aligned}$$ where \(c\) is a constant. The acceleration of the object is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).
  1. Find the value of the constant \(c\) and hence show that the acceleration of the object is \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\).
  2. When \(t = 0\) seconds, the object has position vector \(\mathbf { r } _ { 0 } \mathrm {~m}\) and is moving with velocity \(( - 17 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When \(t = 4\) seconds, the object has position vector \(( - 13 \mathbf { i } + 84 \mathbf { j } ) \mathrm { m }\). Find the vector \(\mathbf { r } _ { 0 }\).
WJEC Unit 4 2024 June Q9
1 marks
9. The diagram below shows a parcel, of mass \(m \mathrm {~kg}\), sliding down a rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 7 } { 25 }\).
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-24_394_906_497_584} The coefficient of friction between the parcel and the slope is \(\frac { 1 } { 12 }\). In addition to friction, the parcel experiences a variable resistive force of \(m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the parcel at time \(t\) seconds.
  1. Show that the motion of the parcel satisfies the differential equation $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$
    numberAdditional page, if required.Examiner only
    \multirow{6}{*}{}
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 4 Specimen Q1
  1. It is known that \(4 \%\) of a population suffer from a certain disease. When a diagnostic test is applied to a person with the disease, it gives a positive response with probability 0.98 . When the test is applied to a person who does not have the disease, it gives a positive response with probability 0.01 .
    1. Using a tree diagram, or otherwise, show that the probability of a person who does not have the disease giving a negative response is 0.9504 .
    The test is applied to a randomly selected member of the population.
  2. Find the probability that a positive response is obtained.
  3. Given that a positive response is obtained, find the probability that the person has the disease.
WJEC Unit 4 Specimen Q2
2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.
  1. Determine the probability that Jeff wins the game
    i) with his first shot,
    ii) with his second shot.
  2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
  3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.
WJEC Unit 4 Specimen Q3
3. A string of length 60 cm is cut a random point.
  1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
  2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).
WJEC Unit 4 Specimen Q4
4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.
  1. The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
    \end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
  2. Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
  3. A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.
    Summary statistics
    Mean
    Standard
    deviation
    		Minimum
    Lower
    quartile
    		Median
    Upper
    quartile
    		Maximum
    6.890.2966.456.636.887.087.48
    i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
    ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.
WJEC Unit 4 Specimen Q5
5. A hotel owner in Cardiff is interested in what factors hotel guests think are important when staying at a hotel. From a hotel booking website he collects the ratings for 'Cleanliness', 'Location', 'Comfort' and 'Value for money' for a random sample of 17 Cardiff hotels.
(Each rating is the average of all scores awarded by guests who have contributed reviews using a scale from 1 to 10 , where 10 is 'Excellent'.) The scatter graph shows the relationship between 'Value for money' and 'Cleanliness' for the sample of Cardiff hotels.
\includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-4_693_1033_749_516}
  1. The product moment correlation coefficient for 'Value for money' and 'Cleanliness' for the sample of 17 Cardiff hotels is 0.895 . Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether this correlation is significant. State your conclusion in context.
  2. The hotel owner also wishes to investigate whether 'Value for money' has a significant correlation with 'Cost per night'. He used a statistical analysis package which provided the following output which includes the Pearson correlation coefficient of interest and the corresponding \(p\)-value.
    Value for moneyCost per night
    Value for money1
    Cost per night
    0.047
    \(( 0.859 )\)
    		1
    Comment on the correlation between 'Value for money' and 'Cost per night'.
WJEC Unit 4 Specimen Q6
  1. An object of mass 4 kg is moving on a horizontal plane under the action of a constant force \(4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}\). At time \(t = 0 \mathrm {~s}\), its position vector is \(7 \mathbf { i } - 26 \mathbf { j }\) with respect to the origin \(O\) and its velocity vector is \(- \mathbf { i } + 4 \mathbf { j }\).
    1. Determine the velocity vector of the object at time \(t = 5 \mathrm {~s}\).
    2. Calculate the distance of the object from the origin when \(t = 2 \mathrm {~s}\).
    3. The diagram below shows an object of weight 160 N at a point \(C\), supported by two cables \(A C\) and \(B C\) inclined at angles of \(23 ^ { \circ }\) and \(40 ^ { \circ }\) to the horizontal respectively.
      \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}
    4. Find the tension in \(A C\) and the tension in \(B C\).
    5. State two modelling assumptions you have made in your solution.
    6. The rate of change of a population of a colony of bacteria is proportional to the size of the population \(P\), with constant of proportionality \(k\). At time \(t = 0\) (hours), the size of the population is 10 .
    7. Find an expression, in terms of \(k\), for \(P\) at time \(t\).
    8. Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
    9. A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .
      Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
    10. A body is projected at time \(t = 0 \mathrm {~s}\) from a point \(O\) with speed \(V \mathrm {~ms} ^ { - 1 }\) in a direction inclined at an angle of \(\theta\) to the horizontal.
    11. Write down expressions for the horizontal and vertical components \(x \mathrm {~m}\) and \(y \mathrm {~m}\) of its displacement from \(O\) at time \(t \mathrm {~s}\).
    12. Show that the range \(R \mathrm {~m}\) on a horizontal plane through the point of projection is given by
    $$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
  2. Given that the maximum range is 392 m , find, correct to one decimal place,
    i) the speed of projection,
    ii) the time of flight,
    iii) the maximum height attained.