Questions S1 - A-Level Maths

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Edexcel S1 Q5

14 marks Moderate -0.8

5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.

Represent this information on a Venn diagram.
One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
1. works full-time and is female,
2. works part-time, given that he is male.
Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
1. all three teachers chosen work full-time,
2. at least one of the three teachers chosen is female.

Edexcel S1 Q6

14 marks Moderate -0.8

6. A physics student recorded the length, $l \mathrm {~cm}$, of a spring when different masses, $m$ grams, were suspended from it giving the following results.

$m ( \mathrm {~g} )$	50	100	200	300	400	500	600	700
$l ( \mathrm {~cm} )$	7.8	10.7	16.5	22.1	28.0	33.9	35.2	35.6

Represent these data on a scatter diagram with $l$ on the vertical axis. The student decides to find the equation of a regression line of the form $l = a + b m$ using only the data for $m \leq 500 \mathrm {~g}$.
Give a reason to support the fitting of such a regression line and explain why the student is excluding two of his values.
(2 marks)
You may use $$\Sigma m = 1550 , \quad \Sigma l = 119 , \quad \Sigma m ^ { 2 } = 552500 , \quad \Sigma l ^ { 2 } = 2869.2 , \quad \Sigma m l = 39540 .$$
Find the values of $a$ and $b$.
Explain the significance of the values of $a$ and $b$ in this situation.

Edexcel S1 Q1

10 marks Moderate -0.8

There are 16 competitors in a table-tennis competition, 5 of which come from Racknor Comprehensive School. Prizes are awarded to the competitors finishing in each of first, second and third place.

Assuming that all the competitors have an equal chance of success, find the probability that the students from Racknor Comprehensive

win no prizes,
win the $1 ^ { \text {st } }$ and $3 ^ { \text {rd } }$ place prizes but not the $2 ^ { \text {nd } }$ place prize,
win exactly one of the prizes.

Edexcel S1 Q2

10 marks Moderate -0.3

2. A statistics student gave a questionnaire to a random sample of 50 pupils at his school. The sample included pupils aged from 11 to 18 years old. The student summarised the data on age in completed years, $A$, and the number of hours spent doing homework in the previous week, $H$, giving the following: $$\Sigma A = 703 , \quad \Sigma H = 217 , \quad \Sigma A ^ { 2 } = 10131 , \quad \Sigma H ^ { 2 } = 1338.5 , \quad \Sigma A H = 3253.5$$

Calculate the product moment correlation coefficient for these data and explain what is shown by your result.
(6 marks)
The student also asked each pupil how many hours of paid work they had done in the previous week. He then calculated the product moment correlation coefficient for the data on hours doing homework and hours doing paid work, giving a value of $r = 0.5213$ The student concluded that paid work did not interfere with homework as pupils doing more paid work also tended to do more homework.
Explain why this conclusion may not be valid.
Explain briefly how the student could more effectively investigate the effect of paid work on homework.
(2 marks)

Edexcel S1 Q3

11 marks Moderate -0.3

3. A soccer fan collected data on the number of minutes of league football, $m$, played by each team in the four main divisions before first scoring a goal at the start of a new season. Her results are shown in the table below.

$m$ (minutes)	Number of teams
$0 \leq m < 40$	36
$40 \leq m < 80$	28
$80 \leq m < 120$	10
$120 \leq m < 160$	4
$160 \leq m < 200$	5
$200 \leq m < 300$	4
$300 \leq m < 400$	2
$400 \leq m < 600$	3

Calculate estimates of the mean and standard deviation of these data.
Explain why the mean and standard deviation might not be the best summary statistics to use with these data.
Suggest alternative summary statistics that would better represent these data.

Edexcel S1 Q4

13 marks Standard +0.3

4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.

Calculate the probability that on any one day Alan will run for less than 20 minutes.
Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.

Edexcel S1 Q5

14 marks Easy -1.2

5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.

Number of months		(2 \| 1 means 21 months)	Totals
0		11224446779	(11)
1		02355689	( )
2	1568		( )
3	079		( )
4	5		( )
5	27		(2)
6	3		(1)
7	0		(1)

Write down the values needed to complete the totals column on the stem and leaf diagram.
State the mode of these data.
Find the median and quartiles of these data. Given that any values outside of the limits $\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ and $\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ are to be regarded as outliers,
determine if there are any outliers in these data,
draw a box plot representing these data on graph paper,
describe the skewness of these data and suggest a reason for it.

Edexcel S1 Q6

17 marks Easy -1.8

6. In a game two spinners are used. The score on the first spinner is given by the random variable $A$, which has the following probability distribution:

$a$	1	2	3
$\mathrm { P } ( A = a )$	$\frac { 1 } { 3 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 3 }$

State the name of this distribution.
Write down $\mathrm { E } ( A )$. The score on the second spinner is given by the random variable $B$, which has the following probability distribution:
$b$ 1 2 3
$\mathrm { P } ( B = b )$ $\frac { 1 } { 2 }$ $\frac { 1 } { 4 }$ $\frac { 1 } { 4 }$
Find $\mathrm { E } ( B )$. On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable $C$.
Show that $\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }$.
Find the probability distribution of $C$.
Show that $\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )$.

Edexcel S1 Q1

7 marks Moderate -0.8

A shop recorded the number of pairs of gloves, $n$, that it sold and the average daytime temperature, $T ^ { \circ } \mathrm { C }$, for each month over a 12-month period.

The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$

Calculate the product moment correlation coefficient for these data.
Comment on what your value shows and suggest a reason for this.

Edexcel S1 Q2

8 marks Standard +0.3

2. Events $A$ and $B$ are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find

$\mathrm { P } ( A \cap B )$,
$\mathrm { P } ( B )$,
$\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)$.

Edexcel S1 Q3

10 marks Easy -1.2

3. The random variable $X$ is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of $a$ and $b$ for

$\mathrm { E } ( 2 X + 3 )$,
$\quad \operatorname { Var } ( 2 X + 3 )$,
$\mathrm { E } \left( X ^ { 2 } \right)$.
Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

Edexcel S1 Q4

11 marks Standard +0.3

An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, $n$, that formed and the wind speed, $v$ metres per second, for 8 different values of $v$ with all other conditions remaining constant. He then coded the data using $x = v - 700$ and $y = n - 20$ and calculated the following summary statistics.

$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$

Find an equation of the regression line of $y$ on $x$.
Hence, find an equation of the regression line of $n$ on $v$.
Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
(2 marks)

Edexcel S1 Q5

12 marks Moderate -0.3

5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.

Value of goods stolen (£)	Number of weeks
0-199	31
200-399	6
400-599	3
600-799	4
800-999	5
1000-1999	2
2000-2999	1

Letting $x$ represent the mid-point of each group and using the coding $y = \frac { x - 699.5 } { 200 }$,

find $\sum$ fy.
estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for $\sum f y$ and $\sum f y ^ { 2 } = 424$.
(6 marks)
The median for these data is $\pounds 82$.
Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
(3 marks)

Edexcel S1 Q6

12 marks Moderate -0.3

6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.

Show that the probability that the first two contestants to be eliminated are both male is $\frac { 2 } { 15 }$.
Find the probability that more females than males are eliminated in the first three rounds of the game.
Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
(3 marks)

Edexcel S1 Q7

15 marks Moderate -0.8

7. A cyber-cafe recorded how long each user stayed during one day giving the following results.

Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
Find the median and quartiles that this model would predict.
Comment on the suitability of the suggested model in the light of the new results.

CAIE S1 2006 June Q6

9 marks Easy -1.2

How many teams play in only 1 match?
How many teams play in exactly 2 matches?
Draw up a frequency table for the numbers of matches which the teams play.
Calculate the mean and variance of the numbers of matches which the teams play.

OCR S1 2011 January Q5

10 marks Moderate -0.8

The number of free gifts that Jan receives in a week is denoted by $X$. Name a suitable probability distribution with which to model $X$, giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
Find
1. $\mathrm { P } ( X \leqslant 2 )$,
2. $\mathrm { P } ( X = 2 )$.
3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. {}
  3
  2
  2
  2
  2
  
  \multirow[t]{21}{*}{3
  }
  \multirow[t]{4}{*}{3
  }
  3
  {}

OCR S1 2012 January Q7

8 marks Moderate -0.8

State a suitable distribution that can be used as a model for $X$, giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
$\mathrm { P } ( X = 4 )$,
$\mathrm { P } ( X \geqslant 4 )$.

OCR MEI S1 2005 June Q5

6 marks Moderate -0.8

On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
[0pt] [1]
\multirow{2}{*}{} Second integer
1 2 3 4 5 6
\multirow{6}{*}{First integer} 1 1 2 3 4 5 6
2 2 2 6 4 10 6
3 3 6 3 12 15 6
4 4 4 12 12
5 5 10 15
6 6 6 6 12
Two fair dice are thrown and the lowest common multiple of the two scores is found.
Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5 .
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

Edexcel S1 Q2

9 marks Moderate -0.8

Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using $Y$ for the number of years since 1993 and $T$ for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
Calculate the product moment correlation coefficient for these data.
Comment on your result in relation to the student's enquiry.

OCR MEI S1 Q3

6 marks Easy -1.8

On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .

Second integer

\cline { 2 - 8 } \multicolumn{2}{|c|}{}

Two fair dice are thrown and the lowest common multiple of the two scores is found.

Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5.
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

OCR MEI S1 2005 June Q6

15 marks Standard +0.3

6 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.

The probability of passing the first game is 0.9
Players who pass any game have probability 0.9 of passing the next game
Players who fail any game have probability 0.5 of failing the next game
1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{668963b4-994d-475a-a1c8-c3e3a252e4e6-4_691_1329_978_397}
2. Find the probability that a randomly selected player
  (A) is invited to join the first team squad,
  (B) is invited to join the second team squad.
3. Hence write down the probability that a randomly selected player is asked to leave.
4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.

Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that

at least one of the three is asked to leave,

they pass a total of 7 games between them.

OCR MEI S1 Q3

8 marks Easy -1.3

3 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

OCR MEI S1 Q4

18 marks Standard +0.8

4 Answer part (i) of this question on the insert provided. Mancaster Hockey Club invite prospective new players to take part in a series of three trial games. At the end of each game the performance of each player is assessed as pass or fail. Players who achieve a pass in all three games are invited to join the first team squad. Players who achieve a pass in two games are invited to join the second team squad. Players who fail in two games are asked to leave. This may happen after two games.

The probability of passing the first game is 0.9
Players who pass any game have probability 0.9 of passing the next game
Players who fail any game have probability 0.5 of failing the next game
1. On the insert, complete the tree diagram which illustrates the information above. \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-4_643_1239_942_417}
2. Find the probability that a randomly selected player
  (A) is invited to join the first team squad,
  (B) is invited to join the second team squad.
3. Hence write down the probability that a randomly selected player is asked to leave.
4. Find the probability that a randomly selected player is asked to leave after two games, given that the player is asked to leave.

Angela, Bryony and Shareen attend the trials at the same time. Assuming their performances are independent, find the probability that

at least one of the three is asked to leave,

they pass a total of 7 games between them.

OCR MEI S1 Q4

9 marks Easy -1.8

4 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

\(m\) (minutes)	Number of teams
\(0 \leq m < 40\)	36
\(40 \leq m < 80\)	28
\(80 \leq m < 120\)	10
\(120 \leq m < 160\)	4
\(160 \leq m < 200\)	5
\(200 \leq m < 300\)	4
\(300 \leq m < 400\)	2
\(400 \leq m < 600\)	3

\(a\)	1	2	3
\(\mathrm { P } ( A = a )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)

\(b\)	1	2	3
\(\mathrm { P } ( B = b )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

\(m ( \mathrm {~g} )\)	50	100	200	300	400	500	600	700
\(l ( \mathrm {~cm} )\)	7.8	10.7	16.5	22.1	28.0	33.9	35.2	35.6

\multirow{2}{*}{}		Second integer
		1	2	3	4	5	6
\multirow{6}{*}{First integer}	1	1	2	3	4	5	6
	2	2	2	6	4	10	6
	3	3	6	3	12	15	6
	4	4	4	12			12
	5	5	10	15
	6	6	6	6	12

\multirow[t]{21}{*}{3 }




















\multirow[t]{4}{*}{3 }



3

Questions S1 (2020 questions)

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Edexcel S1 Q5

Edexcel S1 Q6

Edexcel S1 Q1

Edexcel S1 Q2

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q5

Edexcel S1 Q6

Edexcel S1 Q1

Edexcel S1 Q2

Edexcel S1 Q3

Edexcel S1 Q4

Edexcel S1 Q5

Edexcel S1 Q6

Edexcel S1 Q7

CAIE S1 2006 June Q6

OCR S1 2011 January Q5

OCR S1 2012 January Q7

OCR MEI S1 2005 June Q5

Edexcel S1 Q2

OCR MEI S1 Q3

OCR MEI S1 2005 June Q6

OCR MEI S1 Q3

OCR MEI S1 Q4

OCR MEI S1 Q4