Questions - A-Level Maths

OCR MEI C4 2006 June Q3

8 marks Standard +0.3

3 Given that $\sin ( \theta + \alpha ) = 2 \sin \theta$, show that $\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }$. Hence solve the equation $\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C4 2006 June Q4

13 marks Moderate -0.3

4

The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating $x$, the number of bacteria, to the time $t$.
In another colony, the number of bacteria, $y$, after time $t$ minutes is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 10000 } { \sqrt { y } } .$$ Find $y$ in terms of $t$, given that $y = 900$ when $t = 0$. Hence find the number of bacteria after 10 minutes.

OCR MEI C4 2006 June Q5

11 marks Standard +0.3

5

Show that $\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c$. A vase is made in the shape of the volume of revolution of the curve $y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }$ about the $x$-axis between $x = 0$ and $x = 2$ (see Fig. 5). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
Show that this volume of revolution is $\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)$. Fig. 6 shows the arch ABCD of a bridge. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The section from B to C is part of the curve OBCE with parametric equations $$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ where $a$ is a constant.
Find, in terms of $a$,
(A) the length of the straight line OE,
(B) the maximum height of the arch.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. The straight line sections AB and CD are inclined at $30 ^ { \circ }$ to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the $x$-axis. BF is parallel to the $y$-axis.
Show that at the point B the parameter $\theta$ satisfies the equation $$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$ Verify that $\theta = \frac { 2 } { 3 } \pi$ is a solution of this equation.
Hence show that $\mathrm { BF } = \frac { 3 } { 2 } a$, and find OF in terms of $a$, giving your answer exactly.
Find BC and AF in terms of $a$. Given that the straight line distance AD is 20 metres, calculate the value of $a$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
Find the length AE .
Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30$$ Write down a vector normal to this plane.
Show that the vector $\left( \begin{array} { l } 4 \\ 3 \\ 5 \end{array} \right)$ is normal to the plane ABDE . Hence find the equation of the plane ABDE .
Find the angle between the planes ABC and ABDE . RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} Monday 12 JUNE 2006 Afternoon Up to 1 hour Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2) TIME Up to 1 hour
For Examiner's Use
Qu. Mark
1
2
3
4
5
6
Total
1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?
2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
4 Explain how the equation in line 49, $$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$ is consistent with Fig. 2
initially,
for large values of $t$.
$\_\_\_\_$ 5 A model for an athletics record has the form $$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
Sketch the graph of $R$ against $t$, showing $A$ and $B$ on your graph.
Name one event for which this might be an appropriate model.
\includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
$\_\_\_\_$

OCR MEI C4 2006 June Q6

21 marks Moderate -0.8

6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$

What is the value of $t$ for the year 2012?
What record time does this model predict for the year 2012?

$\_\_\_\_$
$\_\_\_\_$

OCR MEI C4 2008 June Q1

3 marks Easy -1.2

1 Express $\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }$ as a single fraction, simplifying your answer.

OCR MEI C4 2008 June Q2

4 marks Standard +0.3

2 Fig. 2 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The region bounded by the curve, the $x$-axis, the $y$-axis and the line $x = 1$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Show that the volume of the solid of revolution produced is $\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)$.

OCR MEI C4 2008 June Q3

7 marks Moderate -0.3

3 Solve the equation $\cos 2 \theta = \sin \theta$ for $0 \leqslant \theta \leqslant 2 \pi$, giving your answers in terms of $\pi$.

OCR MEI C4 2008 June Q4

3 marks Moderate -0.8

4 Given that $x = 2 \sec \theta$ and $y = 3 \tan \theta$, show that $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$.

OCR MEI C4 2008 June Q5

5 marks Moderate -0.8

5 A curve has parametric equations $x = 1 + u ^ { 2 } , y = 2 u ^ { 3 }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $u$.
Hence find the gradient of the curve at the point with coordinates $( 5,16 )$.

OCR MEI C4 2008 June Q6

8 marks Standard +0.3

6

Find the first three non-zero terms of the binomial series expansion of $\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }$, and state the set of values of $x$ for which the expansion is valid.
Hence find the first three non-zero terms of the series expansion of $\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }$.

OCR MEI C4 2008 June Q7

6 marks Standard +0.3

7 Express $\sqrt { 3 } \sin x - \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Express $\alpha$ in the form $k \pi$. Find the exact coordinates of the maximum point of the curve $y = \sqrt { 3 } \sin x - \cos x$ for which $0 < x < 2 \pi$.

OCR MEI C4 2008 June Q8

18 marks Standard +0.3

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-03_1004_1397_493_374} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Relative to axes $\mathrm { O } x$ (due east), $\mathrm { O } y$ (due north) and $\mathrm { O } z$ (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

Verify that the cartesian equation of the plane ABC is $3 x + 2 y + 20 z + 300 = 0$.
Find the vectors $\overrightarrow { \mathrm { DE } }$ and $\overrightarrow { \mathrm { DF } }$. Show that the vector $2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point $\mathrm { R } ( 15,34,0 )$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
Write down a vector equation of the line RS. Calculate the coordinates of S.

OCR MEI C4 2008 June Q9

18 marks Standard +0.3

9 A skydiver drops from a helicopter. Before she opens her parachute, her speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ after time $t$ seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When $t = 0 , v = 0$.

Find $v$ in terms of $t$.
According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Her speed $t$ seconds after this is $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
Express $\frac { 1 } { ( w - 4 ) ( w + 5 ) }$ in partial fractions.
Using this result, show that $\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }$.
According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
WEDNESDAY 21 MAY 2008
Afternoon
Time: Up to 1 hour
Additional materials: Rough paper
MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
\includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
This document consists of $\mathbf { 6 }$ printed pages, $\mathbf { 2 }$ blank pages and an insert. 1 Complete these Latin square puzzles.

2 1 3
3
\includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square
1 2 3 4
3 1 4 2
2 4 1 3
4 3 2 1
could not be the solution to a Sudoku puzzle.
Explain this briefly.
3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.
1 2 3 4
4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.
4 2 3 1
2 4
4 2
2 4 1 3
5 The number of $9 \times 9$ Sudokus is given in line 121 .
Without doing any calculations, explain why you would expect 9! to be a factor of this number.
6 In the table below, $M$ represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). $s$ is the side length of the Sudoku grid and $b$ is the side length of its blocks.
Block side
length, $b$
Sudoku,
$s \times s$
$M$
1 $1 \times 1$ -
2 $4 \times 4$ 12
3 $9 \times 9$
4 $16 \times 16$
5

Complete the table.
Give a formula for $M$ in terms of $b$.
7 A man is setting a Sudoku puzzle and starts with this solution.
1 2 3 4 5 6 7 8 9
4 5 6 8 9 7 3 1 2
7 8 9 3 1 2 5 6 4
2 3 1 5 6 4 8 9 7
5 6 4 9 7 8 1 2 3
8 9 7 1 2 3 6 4 5
3 1 2 6 4 5 9 7 8
6 4 5 7 8 9 2 3 1
9 7 8 2 3 1 4 5 6
He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
[0pt] [2,2]

1 2 4 6 9
4 8 9 1
8 6
2 1 4 7
6 4 7 8 1 2
8 9 2 4
1 6 4 9 7
6 4 7 9 1
9 8 2 1 4 6
1 2 3 4 5 6 7 8 9
4 5 6 8 9 7 3 1 2
7 8 9 5 6 4
2 3 1 5 6 4 8 9 7
5 6 4 9 7 8 1 2 3
8 9 7 6 4 5
3 1 2 6 4 5 9 7 8
6 4 5 7 8 9 2 3 1
9 7 8 4 5 6

$\_\_\_\_$
$\_\_\_\_$

AQA S1 2005 January Q1

7 marks Moderate -0.3

1 Each Monday, Azher has a stall at a town's outdoor market. The table below shows, for each of a random sample of 10 Mondays during 2003, the air temperature, $x ^ { \circ } \mathrm { C }$, at 9 am and Azher's takings, £y.

Monday	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$	$\mathbf { 9 }$	$\mathbf { 1 0 }$
$\boldsymbol { x }$	2	6	9	18	1	3	7	12	13	4
$\boldsymbol { y }$	97	103	136	245	121	78	145	128	141	312

A scatter diagram of these data is shown below. \includegraphics[max width=\textwidth, alt={}, center]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-2_901_1068_1078_447} Give two distinct comments, in context, on what this diagram reveals.
One of the Mondays is found to be Easter Monday, the busiest Monday market of the year. Identify which Monday this is most likely to be.
Removing the data for the Monday you identified in part (b), calculate the value of the product moment correlation coefficient for the remaining 9 pairs of values of $x$ and $y$.
Name one other variable that would have been likely to affect Azher's takings at this town's outdoor market.
(l mark)

AQA S1 2005 January Q2

9 marks Moderate -0.3

2 The volume, in millilitres, of lemonade in mini-cans may be assumed to be normally distributed with a standard deviation of 3.5. The volumes, in millilitres, of lemonade in a random sample of 12 mini-cans were as follows.

155	148	156	149	147	156
157	156	150	154	148	154

Construct a $98 \%$ confidence interval for the mean volume of lemonade in a mini-can, giving the limits to one decimal place.
On each mini-can is printed " 150 ml ". Comment on this, using the given sample and your confidence interval in part (a).
State why, in part (a), use of the Central Limit Theorem was not necessary.

AQA S1 2005 January Q3

12 marks Moderate -0.8

3 [Figure 1, printed on the insert, is provided for use in this question.]
A parcel delivery company has a depot on the outskirts of a town. Each weekday, a van leaves the depot to deliver parcels across a nearby area. The table below shows, for a random sample of 10 weekdays, the number, $x$, of parcels to be delivered and the total time, $y$ minutes, that the van is out of the depot.

$\boldsymbol { x }$	9	16	22	11	19	26	14	10	11	17
$\boldsymbol { y }$	79	127	172	109	152	214	131	80	94	148

On Figure 1, plot a scatter diagram of these data.
Calculate the equation of the least squares regression line of $y$ on $x$ and draw your line on Figure 1.
Use your regression equation to estimate the total time that the van is out of the depot when delivering:
1. 15 parcels;
2. 35 parcels. Comment on the likely reliability of each of your estimates.
The time that the van is out of the depot delivering parcels may be thought of as the time needed to travel to and from the area plus an amount of time proportional to the number of parcels to be delivered. Given that the regression line of $y$ on $x$ is of the form $y = a + b x$, give an interpretation, in context, for each of your values of $a$ and $b$.
(2 marks)

AQA S1 2005 January Q4

15 marks Moderate -0.3

4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, $X$ grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean $\mu$ and standard deviation 4.

Assuming that $\mu = 106$, determine the probability that a randomly selected bag of chopped lettuce:
1. weighs less than 110 grams;
2. is underweight.
Determine the minimum value of $\mu$ so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by $\bar { X }$. Given that $\mu = 108.5$ :
1. write down values for the mean and variance of $\bar { X }$;
2. determine the probability that $\bar { X }$ exceeds 110 .

AQA S1 2005 January Q5

15 marks Standard +0.3

5 Each evening Aaron sets his alarm for 7 am. He believes that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.

Assuming that Aaron's belief is correct, determine the probability that, during a week (7 mornings), he wakes before his alarm rings:
1. on 2 or fewer mornings;
2. on more than 1 but fewer than 5 mornings.
Assuming that Aaron's belief is correct, calculate the probability that, during a 4 -week period, he wakes before his alarm rings on exactly 7 mornings.
Assuming that Aaron's belief is correct, calculate values for the mean and standard deviation of the number of mornings in a week when Aaron wakes before his alarm rings.
(2 marks)
During a 50-week period, Aaron records, each week, the number of mornings on which he wakes before his alarm rings. The results are as follows.
Number of mornings 0 1 2 3 4 5 6 7
Frequency 10 8 7 7 5 5 4 4
1. Calculate the mean and standard deviation of these data.
2. State, giving reasons, whether your answers to part (d)(i) support Aaron's belief that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
  (3 marks)

AQA S1 2005 January Q6

14 marks Easy -1.2

6 The table below shows the numbers of males and females in each of three employment categories at a university on 31 July 2003.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Employment category
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Managerial	Academic	Support
Male	38	369	303
Female	26	275	643

An employee is selected at random. Determine the probability that the employee is:
1. female;
2. a female academic;
3. either female or academic or both;
4. female, given that the employee is academic.
Three employees are selected at random, without replacement. Determine the probability that:
1. all three employees are male;
2. exactly one employee is male.
The event "employee selected is academic" is denoted by $A$. The event "employee selected is female" is denoted by $F$. Describe in context, as simply as possible, the events denoted by:
1. $F \cap A$;
2. $F ^ { \prime } \cup A$.
  Surname Other Names
  Centre Number Candidate Number
  Candidate Signature
  General Certificate of Education
  January 2005
  Advanced Subsidiary Examination MS/SS1B AQA
  459:5EMLM
  : 11 P וPII " 1 : : ר
  ALLI.ub c \section*{STATISTICS} Unit Statistics 1B Insert for use in Question 3.
  Fill in the boxes at the top of this page.
  Fasten this insert securely to your answer book. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Scatter diagram for parcel deliveries by a van} \includegraphics[alt={},max width=\textwidth]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-8_2420_1664_349_175}
  \end{figure} Figure 1 (for Question 3)

AQA S1 2007 January Q1

9 marks Easy -1.2

1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were

17	19	22	26	28	31	34	36	38	39
41	42	43	47	50	51	53	55	57	58

Calculate the mean and the standard deviation of these times.
In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people.
(4 marks)
For the times taken by all 23 people, explain why:
1. the mode is not an appropriate numerical measure;
2. the range is not an appropriate numerical measure.

AQA S1 2007 January Q2

12 marks Moderate -0.8

2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are $0.45,0.25$ and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.

On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
1. at most 12 of these guests require a continental breakfast;
2. more than 10 but fewer than 20 of these guests require no breakfast.
When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.

AQA S1 2007 January Q3

5 marks Easy -1.3

3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables $x$ and $y$ in each of the three scatter diagrams.

\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_631_659_516_301}
\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_620_647_525_1119}
\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_624_655_1279_303}
(5 marks)

AQA S1 2007 January Q4

7 marks Moderate -0.3

4 A very popular play has been performed at a London theatre on each of 6 evenings per week for about a year. Over the past 13 weeks ( 78 performances), records have been kept of the proceeds from the sales of programmes at each performance. An analysis of these records has found that the mean was $\pounds 184$ and the standard deviation was $\pounds 32$.

Assuming that the 78 performances may be considered to be a random sample, construct a $90 \%$ confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play.
Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of:
1. this particular play;
2. any play.

AQA S1 2007 January Q5

10 marks Moderate -0.8

5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are $0.6,0.7$ and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:

none of the three cyclists takes part;
Fabio is the only one of the three cyclists to take part;
exactly one of the three cyclists takes part;
either one or two of the three cyclists take part.

AQA S1 2007 January Q6

17 marks Moderate -0.3

6 When Monica walks to work from home, she uses either route A or route B.

Her journey time, $X$ minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
1. $\mathrm { P } ( X < 45 )$;
2. $\mathrm { P } ( 30 < X < 45 )$.
Her journey time, $Y$ minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of $\sigma$. Given that $\mathrm { P } ( Y > 45 ) = 0.12$, calculate the value of $\sigma$.
If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
When Monica travels to work from home by car, her journey time, $W$ minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

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1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9	3	1	2	5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7	1	2	3	6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8	2	3	1	4	5	6

1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9				5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7				6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8				4	5	6

Monday	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)	\(\mathbf { 9 }\)	\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)	2	6	9	18	1	3	7	12	13	4
\(\boldsymbol { y }\)	97	103	136	245	121	78	145	128	141	312

Surname	Other Names
Centre Number		Candidate Number
Candidate Signature

For Examiner's Use
Qu.	Mark
1
2
3
4
5
6
Total

Number of mornings	0	1	2	3	4	5	6	7
Frequency	10	8	7	7	5	5	4	4

1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9	3	1	2	5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7	1	2	3	6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8	2	3	1	4	5	6

1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9				5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7				6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8				4	5	6

1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9	3	1	2	5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7	1	2	3	6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8	2	3	1	4	5	6

1	2	3	4	5	6	7	8	9
4	5	6	8	9	7	3	1	2
7	8	9				5	6	4
2	3	1	5	6	4	8	9	7
5	6	4	9	7	8	1	2	3
8	9	7				6	4	5
3	1	2	6	4	5	9	7	8
6	4	5	7	8	9	2	3	1
9	7	8				4	5	6