Edexcel AS Paper 2 (AS Paper 2) 2018 June

1. A company is introducing a job evaluation scheme. Points ( \(x\) ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( \(\pounds y\) ) will then be allocated to each job according to the number of points awarded.

Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was \(y = 4.5 x - 47\)

1. Describe the correlation between points and pay.
2. Give an interpretation of the gradient of this regression line.
3. Explain why this model might not be appropriate for all jobs in the company.

1. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty.

Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".

1. Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\)
   Naasir and his friends play the game 15 times.
   1. Find the probability that Naasir wins
      1. exactly 2 games,
      2. more than 5 games.
   Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.

1. Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.

\begin{table}[h] \end{table}

1. Calculate the mean for these data.
2. Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]

   Month	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
   Mean	7.58	8.26	8.57	8.57	11.57
   Standard Deviation	2.93	3.89	3.46	3.87	4.64

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
3. Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
4. 1. State the meaning of the * symbol on some of the box plots.
   2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}

5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)

1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).

1. A train travels along a straight horizontal track between two stations, \(A\) and \(B\).

In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. For this model of the motion of the train between \(A\) and \(B\),
   1. state the value of the constant velocity of the train,
   2. state the time for which the train is decelerating,
   3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.

1. A particle, \(P\), moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement, \(x\) metres, of \(P\) from the origin \(O\), is given by \(x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)\)
   1. Find the times when \(P\) is instantaneously at rest.
   2. Find the total distance travelled by \(P\) in the time interval \(0 \leqslant t \leqslant 2\)
   3. Show that \(P\) will never move along the negative \(x\)-axis.

9. \begin{figure}[h] \end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

1. find, in terms of \(m\) and \(g\), the tension in the string,
2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
3. find the value of \(k\).
4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).