Questions S1 - A-Level Maths

Write down the median weight of the bags of popcorn.
Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
Find the probability that a customer will complain.

Edexcel S1 2008 January Q7

7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered $0,1,2$, and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable $R$ is the score on the red die and the random variable $B$ is the score on the blue die.

Find $\mathrm { P } ( R = 3$ and $B = 0 )$. The random variable $T$ is $R$ multiplied by $B$.
Complete the diagram below to represent the sample space that shows all the possible values of $T$.
\includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of $T$}
The table below represents the probability distribution of the random variable $T$.
$t$ 0 1 2 3 4 6 9
$\mathrm { P } ( T = t )$ $a$ $b$ $1 / 8$ $1 / 8$ $c$ $1 / 8$ $d$
Find the values of $a , b , c$ and $d$. Find the values of
$\mathrm { E } ( T )$,
$\operatorname { Var } ( T )$.

Edexcel S1 2009 January Q1

A teacher is monitoring the progress of students using a computer based revision course. The improvement in performance, $y$ marks, is recorded for each student along with the time, $x$ hours, that the student spent using the revision course. The results for a random sample of 10 students are recorded below.

$$\text { [You may use } \sum x = 21.4 , \quad \sum y = 96 , \quad \sum x ^ { 2 } = 57.22 , \quad \sum x y = 313.7 \text { ] }$$

Calculate $S _ { x x }$ and $S _ { x y }$.
Find the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$.
Give an interpretation of the gradient of your regression line. Rosemary spends 3.3 hours using the revision course.
Predict her improvement in marks. Lee spends 8 hours using the revision course claiming that this should give him an improvement in performance of over 60 marks.
Comment on Lee's claim.

Edexcel S1 2009 January Q2

2. A group of office workers were questioned for a health magazine and $\frac { 2 } { 5 }$ were found to take regular exercise. When questioned about their eating habits $\frac { 2 } { 3 }$ said they always eat breakfast and, of those who always eat breakfast $\frac { 9 } { 25 }$ also took regular exercise. Find the probability that a randomly selected member of the group

always eats breakfast and takes regular exercise,
does not always eat breakfast and does not take regular exercise.
Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.

Edexcel S1 2009 January Q3

3. When Rohit plays a game, the number of points he receives is given by the discrete random variable $X$ with the following probability distribution.

$x$	0	1	2	3
$\mathrm { P } ( X = x )$	0.4	0.3	0.2	0.1

Find $\mathrm { E } ( X )$.
Find $\mathrm { F } ( 1.5 )$.
Show that $\operatorname { Var } ( X ) = 1$
Find $\operatorname { Var } ( 5 - 3 X )$. Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
Find the probability that Rohit wins the prize.

Edexcel S1 2009 January Q4

4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, $y$ minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$

Find the median and quartiles for these data. A value that is greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ or smaller than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ is defined as an outlier.
Show that 110 is the only outlier.
Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
Show that $S _ { y y }$ for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, $x$, they sent in the same week. The values of $S _ { x x }$ and $S _ { x y }$ for these 10 students are $S _ { x x } = 3463.6$ and $S _ { x y } = - 18.3$.
Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}

Edexcel S1 2009 January Q5

5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.

Number of hours	Mid-point	Frequency
0-5	2.75	20
6-7	6.5	16
8-10	9	18
11-15	13	25
16-25	20.5	15
26-50	38	10

A histogram was drawn and the group ( $8 - 10$ ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

Calculate the width and height of the rectangle representing the group (16-25) hours.
Use linear interpolation to estimate the median and interquartile range.
Estimate the mean and standard deviation of the number of hours spent shopping.
State, giving a reason, the skewness of these data.
State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

Edexcel S1 2009 January Q6

6. The random variable $X$ has a normal distribution with mean 30 and standard deviation 5 .

Find $\mathrm { P } ( X < 39 )$.
Find the value of $d$ such that $\mathrm { P } ( X < d ) = 0.1151$
Find the value of $e$ such that $\mathrm { P } ( X > e ) = 0.1151$
Find $\mathrm { P } ( d < X < e )$.

Edexcel S1 2010 January Q1

A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly.
2. Find the probability that a blue bead and a green bead are drawn from the jar.
3. The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below.
$2 \mid 6$ means a score of 26
0 7 $( 1 )$
1 88 $( 2 )$
2 4468 $( 4 )$
3 2333459 $( 7 )$
4 00000 $( 5 )$
Find
the median score,
the interquartile range. The company director decides that any employees whose scores are so low that they are outliers will undergo retraining. An outlier is an observation whose value is less than the lower quartile minus 1.0 times the interquartile range.
Explain why there is only one employee who will undergo retraining.
On the graph paper on page 5, draw a box plot to illustrate the employees' scores. \includegraphics[max width=\textwidth, alt={}, center]{a0058e3c-046f-4271-aee4-33a74c719e2a-04_611_1596_2006_185}

Edexcel S1 2010 January Q3

3. The birth weights, in kg, of 1500 babies are summarised in the table below.

Weight (kg)	Midpoint, xkg	Frequency, f
0.0-1.0	0.50	1
1.0-2.0	1.50	6
2.0-2.5	2.25	60
2.5-3.0		280
3.0-3.5	3.25	820
3.5-4.0	3.75	320
4.0-5.0	4.50	10
5.0-6.0		3

$$\text { [You may use } \sum \mathrm { f } x = 4841 \text { and } \sum \mathrm { f } x ^ { 2 } = 15889.5 \text { ] }$$

Write down the missing midpoints in the table above.
Calculate an estimate of the mean birth weight.
Calculate an estimate of the standard deviation of the birth weight.
Use interpolation to estimate the median birth weight.
Describe the skewness of the distribution. Give a reason for your answer.

Edexcel S1 2010 January Q4

4. There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. \begin{displayquote} 112 take systems support,
70 take developing software,
81 take networking,
35 take developing software and systems support,
28 take networking and developing software,
40 take systems support and networking,
4 take all three extra options.

In the space below, draw a Venn diagram to represent this information. \end{displayquote} A student from the course is chosen at random. Find the probability that this student takes
none of the three extra options,
networking only. Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,
find the probability that this student takes all three extra options.

Edexcel S1 2010 January Q5

5. The probability function of a discrete random variable $X$ is given by $$\mathrm { p } ( x ) = k x ^ { 2 } \quad x = 1,2,3$$ where $k$ is a positive constant.

Show that $k = \frac { 1 } { 14 }$ Find
$\mathrm { P } ( X \geqslant 2 )$
$\mathrm { E } ( X )$
$\operatorname { Var } ( 1 - X )$

Edexcel S1 2010 January Q6

The blood pressures, $p$ mmHg, and the ages, $t$ years, of 7 hospital patients are shown in the table below.

Patient	A	B	C	D	E	F	G
$t$	42	74	48	35	56	26	60
$p$	98	130	120	88	182	80	135

$$\left[ \sum t = 341 , \sum p = 833 , \sum t ^ { 2 } = 18181 , \sum p ^ { 2 } = 106397 , \sum t p = 42948 \right]$$

Find $S _ { p p } , S _ { t p }$ and $S _ { t t }$ for these data.
Calculate the product moment correlation coefficient for these data.
Interpret the correlation coefficient.
On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients.
Find the equation of the regression line of $p$ on $t$.
Plot your regression line on your scatter diagram.
Use your regression line to estimate the blood pressure of a 40 year old patient. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a0058e3c-046f-4271-aee4-33a74c719e2a-12_2071_1729_386_157}
\end{figure}

Edexcel S1 2010 January Q7

The heights of a population of women are normally distributed with mean $\mu \mathrm { cm }$ and standard deviation $\sigma \mathrm { cm }$. It is known that $30 \%$ of the women are taller than 172 cm and $5 \%$ are shorter than 154 cm .
1. Sketch a diagram to show the distribution of heights represented by this information.
2. Show that $\mu = 154 + 1.6449 \sigma$.
3. Obtain a second equation and hence find the value of $\mu$ and the value of $\sigma$.
A woman is chosen at random from the population.
Find the probability that she is taller than 160 cm .

Edexcel S1 2011 January Q1

A random sample of 50 salmon was caught by a scientist. He recorded the length $l \mathrm {~cm}$ and weight $w \mathrm {~kg}$ of each salmon.

The following summary statistics were calculated from these data.
$\sum l = 4027 \quad \sum l ^ { 2 } = 327754.5 \quad \sum w = 357.1 \quad \sum l w = 29330.5 \quad S _ { w w } = 289.6$

Find $S _ { l l }$ and $S _ { l w }$
Calculate, to 3 significant figures, the product moment correlation coefficient between $l$ and $w$.
Give an interpretation of your coefficient.

Edexcel S1 2011 January Q2

Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
0.0
0.5
1.8
2.8
2.3
5.6
9.4

Jenny then records the amount of rainfall, $x \mathrm {~mm}$, at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$

Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
State, giving your reason, the effect this will have on the mean.

Edexcel S1 2011 January Q3

3. Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.

	Amount received in sales (£1000s)
Two lowest values	3,4
Lower quartile	7
Median	12
Upper quartile	14
Two highest values	20,25

An outlier is an observation that falls
either $1.5 \times$ interquartile range above the upper quartile or $1.5 \times$ interquartile range below the lower quartile.

On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.
(5)
\includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-05_933_1226_1283_367}
State the skewness of the distribution of the amount of sales received. Justify your answer.
The company claims that for $75 \%$ of the months, the amount received per month is greater than $\pounds 10000$. Comment on this claim, giving a reason for your answer.
(2)

Edexcel S1 2011 January Q4

A farmer collected data on the annual rainfall, $x \mathrm {~cm}$, and the annual yield of peas, $p$ tonnes per acre.

The data for annual rainfall was coded using $v = \frac { x - 5 } { 10 }$ and the following statistics were found. $$S _ { v v } = 5.753 \quad S _ { p v } = 1.688 \quad S _ { p p } = 1.168 \quad \bar { p } = 3.22 \quad \bar { v } = 4.42$$

Find the equation of the regression line of $p$ on $v$ in the form $p = a + b v$.
Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm .

Edexcel S1 2011 January Q5

5. On a randomly chosen day, each of the 32 students in a class recorded the time, $t$ minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.

Time, $t$	Number of students
10-19	2
20-29	4
30-39	8
40-49	11
50-69	5
70-79	2

Use interpolation to estimate the value of the median. Given that $$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
find the mean and the standard deviation of the times spent by the students on their homework.
Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.

Edexcel S1 2011 January Q6

The discrete random variable $X$ has the probability distribution

$x$	1	2	3	4
$\mathrm { P } ( X = x )$	$k$	$2 k$	$3 k$	$4 k$

Show that $k = 0.1$ Find
$\mathrm { E } ( X )$
$\mathrm { E } \left( X ^ { 2 } \right)$
$\operatorname { Var } ( 2 - 5 X )$ Two independent observations $X _ { 1 }$ and $X _ { 2 }$ are made of $X$.
Show that $\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1$
Complete the probability distribution table for $X _ { 1 } + X _ { 2 }$
$y$ 2 3 4 5 6 7 8
$\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)$ 0.01 0.04 0.10 0.25 0.24
Find $\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)$

Edexcel S1 2011 January Q7

The bag $P$ contains 6 balls of which 3 are red and 3 are yellow.

The bag $Q$ contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag $P$ and placed in bag $Q$. A second ball is drawn at random from bag $P$ and placed in bag $Q$.
A third ball is then drawn at random from the 9 balls in bag $Q$. The event $A$ occurs when the 2 balls drawn from bag $P$ are of the same colour. The event $B$ occurs when the ball drawn from bag $Q$ is red.

Complete the tree diagram shown below.
(4)
\includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-12_1201_1390_753_269}
Find $\mathrm { P } ( A )$
Show that $\mathrm { P } ( B ) = \frac { 5 } { 9 }$
Show that $\mathrm { P } ( A \cap B ) = \frac { 2 } { 9 }$
Hence find $\mathrm { P } ( A \cup B )$
Given that all three balls drawn are the same colour, find the probability that they are all red.
(3)

Edexcel S1 2011 January Q8

The weight, $X$ grams, of soup put in a tin by machine $A$ is normally distributed with a mean of 160 g and a standard deviation of 5 g .
A tin is selected at random.
1. Find the probability that this tin contains more than 168 g .
The weight stated on the tin is $w$ grams.
Find $w$ such that $\mathrm { P } ( X < w ) = 0.01$ The weight, $Y$ grams, of soup put into a carton by machine $B$ is normally distributed with mean $\mu$ grams and standard deviation $\sigma$ grams.
Given that $\mathrm { P } ( Y < 160 ) = 0.99$ and $\mathrm { P } ( Y > 152 ) = 0.90$ find the value of $\mu$ and the value of $\sigma$.

Edexcel S1 2012 January Q1

The histogram in Figure 1 shows the time, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bc8ef6c7-a321-4ecf-962d-f469a95fc8c8-02_1312_673_349_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Complete the table.
Delay (minutes) Number of motorists
4-6 6
7-8
9 21
10-12 45
13-15 9
16-20
Estimate the number of motorists who were delayed between 8.5 and 13.5 minutes by the roadworks.

Edexcel S1 2012 January Q2

(a) State in words the relationship between two events $R$ and $S$ when $\mathrm { P } ( R \cap S ) = 0$

The events $A$ and $B$ are independent with $\mathrm { P } ( A ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( A \cup B ) = \frac { 2 } { 3 }$ Find
(b) $\mathrm { P } ( B )$
(c) $\mathrm { P } \left( A ^ { \prime } \cap B \right)$
(d) $\mathrm { P } \left( B ^ { \prime } \mid A \right)$

Edexcel S1 2012 January Q3

3. The discrete random variable $X$ can take only the values $2,3,4$ or 6 . For these values the probability distribution function is given by

$x$	2	3	4	6
$\mathrm { P } ( X = x )$	$\frac { 5 } { 21 }$	$\frac { 2 k } { 21 }$	$\frac { 7 } { 21 }$	$\frac { k } { 21 }$

where $k$ is a positive integer.

Show that $k = 3$ Find
$\mathrm { F } ( 3 )$
$\mathrm { E } ( X )$
$\mathrm { E } \left( X ^ { 2 } \right)$
$\operatorname { Var } ( 7 X - 5 )$

\(t\)	0	1	2	3	4	6	9
\(\mathrm { P } ( T = t )\)	\(a\)	\(b\)	\(1 / 8\)	\(1 / 8\)	\(c\)	\(1 / 8\)	\(d\)

	\(2 \mid 6\) means a score of 26
0	7	\(( 1 )\)
1	88	\(( 2 )\)
2	4468	\(( 4 )\)
3	2333459	\(( 7 )\)
4	00000	\(( 5 )\)

\(x\)	1	2	3	4
\(\mathrm { P } ( X = x )\)	\(k\)	\(2 k\)	\(3 k\)	\(4 k\)

\(x\)	2	3	4	6
\(\mathrm { P } ( X = x )\)	\(\frac { 5 } { 21 }\)	\(\frac { 2 k } { 21 }\)	\(\frac { 7 } { 21 }\)	\(\frac { k } { 21 }\)

Time, \(t\)	Number of students
10-19	2
20-29	4
30-39	8
40-49	11
50-69	5
70-79	2

\(y\)	2	3	4	5	6	7	8
\(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)	0.01	0.04	0.10		0.25	0.24

Delay (minutes)	Number of motorists
4-6	6
7-8
9	21
10-12	45
13-15	9
16-20

Questions S1 (1967 questions)

Browse by module

Browse by board

Edexcel S1 2008 January Q6

Edexcel S1 2008 January Q7

Edexcel S1 2009 January Q1

Edexcel S1 2009 January Q2

Edexcel S1 2009 January Q3

Edexcel S1 2009 January Q4

Edexcel S1 2009 January Q5

Edexcel S1 2009 January Q6

Edexcel S1 2010 January Q1

Edexcel S1 2010 January Q3

Edexcel S1 2010 January Q4

Edexcel S1 2010 January Q5

Edexcel S1 2010 January Q6

Edexcel S1 2010 January Q7

Edexcel S1 2011 January Q1

Edexcel S1 2011 January Q2

Edexcel S1 2011 January Q3

Edexcel S1 2011 January Q4

Edexcel S1 2011 January Q5

Edexcel S1 2011 January Q6

Edexcel S1 2011 January Q7

Edexcel S1 2011 January Q8

Edexcel S1 2012 January Q1

Edexcel S1 2012 January Q2

Edexcel S1 2012 January Q3