WJEC Unit 4 (Unit 4) 2024 June

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.

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.

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 .

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] \end{figure} \begin{figure}[h] \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?
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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.

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]

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.

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.
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1. 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}

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 \(\mathbf { r } _ { 0 }\).

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 $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$

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