Pre-U Pre-U 9794/3 (Pre-U Mathematics Paper 3) 2017 June

1 Levels of nitrogen dioxide in the atmosphere are being monitored at the side of a road in a busy city centre. A sample of 18 measurements taken (in suitable units) is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 83 & 44 & 95 & 92 & 98 & 63 & 69 & 76 & 19 & 91 & 70 & 91 & 74 & 65 & 62 & 70 & 95 & 108 \end{array}$$

1. Find the mean and standard deviation of the sample.
2. Hence identify, with justification, any possible outliers.

2 The table shows the turnover, in millions of pounds, of a small company at 3-year intervals over a period of 15 years, starting in 2000.

1. For the information in the table find the equation of the least squares regression line of \(y\) on \(x\), where \(x\) is the year since 2000 and \(y\) is the turnover in millions of pounds.
2. Use the equation of the regression line to calculate the residual for 2009.
3. Use the equation of the regression line to estimate the turnover in 2024, and explain why it is inadvisable to rely on this estimate.

3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find the variance of \(X\).
3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).

4 The letters of the word 'STATISTICS' are to be rearranged.

1. How many distinct arrangements are there?
2. How many of the arrangements start and end with the letter S ?
3. What is the probability that, in a randomly chosen arrangement, the S's are all together?

5 The random variable \(X\) has a geometric distribution: \(X \sim \operatorname { Geo } ( p )\).

1. Show that \(\mathrm { P } ( X > n ) = q ^ { n }\), where \(q = 1 - p\). You are given that \(\mathrm { P } ( X \geqslant 4 ) = 0.216\).
2. Use the result given in part (i) to find the value of \(p\) and \(\mathrm { P } ( X \leqslant 8 )\).
3. Write down \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.

1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
   Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
2. Find the tension in the cable in each of the three stages.
3. Sketch the velocity-time graph for the complete downward motion of the crate.
4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.

7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.

1. Find the initial speed of the particle.
2. Find the magnitude and direction of the velocity of the particle when it hits the ground.

8 An object of weight 16 N is supported in equilibrium by a force of \(P \mathrm {~N}\) at \(30 ^ { \circ }\) to the vertical and by another of 10 N at \(\theta ^ { \circ }\) to the vertical as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85c5c346-8eb5-47ea-b94e-80b1a0038ce1-4_549_483_397_831}

1. Draw a triangle to show that the forces acting on the object are in equilibrium.
2. Find the two possible values of \(\theta\) and the corresponding values of \(P\).

9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.

1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
   1. Write down the values of \(t\) when \(v = 0\).
   2. Show that \(x = 5.6\) when \(t = 18\).
   3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.

10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }