WJEC Unit 4 (Unit 4) 2018 June

An architect bids for two construction projects. He estimates the probability of winning bid \(A\) is \(0 \cdot 6\), the probability of winning bid \(B\) is \(0 \cdot 5\) and the probability of winning both is \(0 \cdot 2\).

1. Show that the probability that he does not win either bid is \(0 \cdot 1\). [2]
2. Find the probability that he wins exactly one bid. [2]
3. Given that he does not win bid \(A\), find the probability that he wins bid \(B\). [3]

1. Marie is an athlete who competes in the high jump. In a certain competition she is allowed two attempts to clear each height, but if she is successful with the first attempt she does not jump again at this height. The probability that she is successful with her first jump at a height of \(1 \cdot 7\) m is \(p\). The probability that she is successful with her second jump is also \(p\). The probability that she clears \(1 \cdot 7\) m is \(0 \cdot 64\). Find the value of \(p\). [4]
2. The following table shows the numbers of male and female athletes competing for Wales in track and field events at a competition.

   	Track	Field
   Male	13	9
   Female	7	4

   Two athletes are chosen at random to participate in a drugs test. Given that the first athlete is male, find the probability that both are field athletes. [3]

Antonio arrives at a train station at a random point in time. The trains to his desired destination are scheduled to depart at 12-minute intervals.

1. Assume that Antonio gets on the next train.
   1. Suggest an appropriate distribution to model his waiting time and give the parameters.
   2. State the mean and the variance of this distribution.
   3. State an assumption you have made in suggesting this distribution. [4]
2. Now assume that the probability that Antonio misses the next available train because he is distracted by his smartphone is \(0 \cdot 12\). If he misses the next available train, he is sure to get on the one after that.
   1. Find the probability that he waits between 9 and 19 minutes.
   2. Given that he waits between 9 and 19 minutes, find the probability that he gets on the first train. [6]

Arwyn collects data about household expenditure on food. He records the weekly expenditure on food for 80 randomly selected households from across Wales.

1. Explain why a normal distribution may be an appropriate model for the weekly expenditure on food for this sample. [1]

Arwyn uses the distribution N(64, 15²) to model expenditure on food.

1. Find the number of households in the sample that this model would predict to have weekly food expenditure in the range
   1. \(60 \leqslant x < 70\),
   2. \(x \geqslant 90\). [4]
2. Use your answers to part (b)
   1. to comment on the suitability of this model,
   2. to explain how Arwyn could improve the model by changing one of its parameters. [2]
3. Arwyn's friend Colleen wishes to use the improved model to predict household expenditure on food in Northern Ireland. Comment on this plan. [1]

Rebecca is a farmer who is monitoring prices for products to use on her farm. She records the prices of two products made from different grains, wheat and oats, at random points in time, to investigate whether there is any correlation. \includegraphics{figure_1} The product moment correlation coefficient for the data is \(0 \cdot 244\). There are 12 data points, and the \(p\)-value is \(0 \cdot 4447\).

1. Comment on the correlation between the prices of Feed Wheat and Feed Oats. [2]

Rebecca also records the prices of two wheat products at random points in time, to investigate whether there is any correlation. \includegraphics{figure_2} The product moment correlation coefficient for the data is \(0 \cdot 653\). There are 12 data points.

1. Stating your hypotheses clearly, test at the 5% level of significance whether there is any evidence of correlation between the prices of these two products. [5]
2. Without referring to the positioning of the points on the graphs, suggest why the product moment correlation coefficient is higher for the second set of data. [1]

The diagram shows a uniform plank \(AB\) of length 4 m supported in horizontal equilibrium by means of a central pivot. On the plank there are three objects of masses 8 kg, 2 kg and 15 kg placed in positions \(C\), \(D\) and \(E\) respectively. The distance \(AC\) is \(0 \cdot 6\) m and the distance \(AE\) is \(2 \cdot 8\) m. \includegraphics{figure_3} Find the distance \(AD\). [4]

An object of mass \(0 \cdot 5\) kg is thrown vertically upwards with initial speed \(24\) ms\(^{-1}\). The velocity of the object at time \(t\) seconds is \(v\) ms\(^{-1}\). During the upward motion, the object experiences a resistance to motion \(RN\), where \(R\) is proportional to \(v\). When the velocity of the object is \(0 \cdot 2\) ms\(^{-1}\) the resistance to motion is \(0 \cdot 08\) N.

1. Show that the upward motion of the object satisfies the differential equation $$\frac{\mathrm{d}v}{\mathrm{d}t} = -9 \cdot 8 - 0 \cdot 8\,v.$$ [3]
2. Find an expression for \(v\) at time \(t\). [6]
3. Determine the value of \(t\) when the object is at the highest point of the motion. [2]

An object of mass 60 kg is on a rough plane inclined at an angle of 20° to the horizontal. The coefficient of friction between the object and the plane is \(0 \cdot 3\). Initially, the object is held at rest. A force which is parallel to the plane and of magnitude \(T\) N is applied to the object in an upward direction along the line of greatest slope. The object is then released.

1. Given that \(T = 15\), calculate the acceleration of the object down the plane. [6]
2. Given that \(T = 350\), determine whether or not the object moves up the plane. Give a reason for your answer. [3]

Points \(A\) and \(B\) lie on horizontal ground. At time \(t = 0\) seconds, an object \(P\) is projected from \(A\) towards \(B\) such that \(AB\) is the range of \(P\). The speed of projection is \(24 \cdot 5\) ms\(^{-1}\) in a direction which is 30° above the horizontal.

1. Calculate the range \(AB\) of the object \(P\). [5]

At time \(t = 1\) second, another object \(Q\) is projected from \(B\) towards \(A\) with the same speed of projection \(24 \cdot 5\) ms\(^{-1}\) and in a direction which is also 30° above the horizontal.

1. Determine the height above the ground at which \(P\) and \(Q\) collide. [5]

A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the magnitude of the acceleration of the particle. [3]
2. 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. [3]