Questions S1 - A-Level Maths

$\mathrm { P } ( Y > 17 )$
$\mathrm { P } ( \mu < Y < 17 )$
$\mathrm { P } ( Y < \mu \mid Y < 17 )$

Edexcel S1 2014 June Q5

15 marks Standard +0.3

5. The discrete random variable $X$ has the following probability distribution

$x$	- 2	0	2	4
$\mathrm { P } ( X = x )$	$a$	$b$	$a$	$c$

where $a$, $b$ and $c$ are probabilities.
Given that $\mathrm { E } ( X ) = 0.8$

find the value of $c$. Given also that $\mathrm { E } \left( X ^ { 2 } \right) = 5$ find
the value of $a$ and the value of $b$,
$\operatorname { Var } ( X )$ The random variable $Y = 5 - 3 X$
Find
$\mathrm { E } ( Y )$
$\operatorname { Var } ( Y )$
$\mathrm { P } ( Y \geqslant 0 )$

Edexcel S1 2014 June Q6

12 marks Moderate -0.3

6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly’s restaurant.
$S =$ the event a customer has a starter.
$M =$ the event a customer has a main course.
$D =$ the event a customer has a dessert.
\includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events $S$ and $D$ are statistically independent

find the value of $p$.
Hence find the value of $q$.
Find
1. $\quad$ P( $D \mid M \cap S$ )
2. $\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)$ One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
Estimate the number of desserts that these 63 customers will have.

Edexcel S1 2014 June Q7

10 marks Standard +0.3

7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.

Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than $d$ metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
find the value of $d$. Three children are selected at random from those who take part in the throwing a tennis ball event.
Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.

Edexcel S1 2015 June Q1

4 marks Moderate -0.8

The discrete random variable $X$ can only take the values $1,2,3$ and 4 For these values the cumulative distribution function is defined by

$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$ where $k$ is a constant.

Find the value of $k$.
Find the probability distribution of $X$.

Edexcel S1 2015 June Q2

13 marks Moderate -0.3

2. Paul believes there is a relationship between the value and the floor size of a house. He takes a random sample of 20 houses and records the value, $\pounds v$, and the floor size, $s \mathrm {~m} ^ { 2 }$ The data were coded using $x = \frac { s - 50 } { 10 }$ and $y = \frac { v } { 100000 }$ and the following statistics obtained. $$\sum x = 441.5 , \quad \sum y = 59.8 , \quad \sum x ^ { 2 } = 11261.25 , \quad \sum y ^ { 2 } = 196.66 , \quad \sum x y = 1474.1$$

Find the value of $S _ { x y }$ and the value of $S _ { x x }$
Find the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$ The least squares regression line of $v$ on $s$ is $v = c + d s$
Show that $d = 1020$ to 3 significant figures and find the value of $c$
Estimate the value of a house of floor size $130 \mathrm {~m} ^ { 2 }$
Interpret the value $d$ Paul wants to increase the value of his house. He decides to add an extension to increase the floor size by $31 \mathrm {~m} ^ { 2 }$
Estimate the increase in the value of Paul's house after adding the extension.

Edexcel S1 2015 June Q3

8 marks Easy -1.3

A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below.

Length of time employed, $x$ years	Female	Male
$x < 4$	9	16
$4 \leqslant x < 10$	14	20
$10 \leqslant x$	7	24

One of the 90 administrators is selected at random.

Find the probability that the administrator is female.
Given that the administrator has been employed by the company for less than 4 years, find the probability that this administrator is male.
Given that the administrator has been employed by the company for less than 10 years, find the probability that this administrator is male.
State, with a reason, whether or not the event 'selecting a male' is independent of the event 'selecting an administrator who has been employed by the company for less than 4 years'.

Edexcel S1 2015 June Q4

9 marks Easy -1.3

A bag contains 19 red beads and 1 blue bead only.

Linda selects a bead at random from the bag. She notes its colour and replaces the bead in the bag. She then selects a second bead at random from the bag and notes its colour. Find the probability that

both beads selected are blue,
exactly one bead selected is red. In another bag there are 9 beads, 4 of which are green and the rest are yellow.
Linda selects 3 beads from this bag at random without replacement.
Find the probability that 2 of these beads are yellow and 1 is green. Linda replaces the 3 beads and then selects another 4 at random without replacement.
Find the probability that at least 1 of the beads is green.

Edexcel S1 2015 June Q5

12 marks Moderate -0.3

Police measure the speed of cars passing a particular point on a motorway. The random variable $X$ is the speed of a car.
$X$ is modelled by a normal distribution with mean 55 mph (miles per hour).
1. Draw a sketch to illustrate the distribution of $X$. Label the mean on your sketch.
The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police. The distribution of $X$ has a standard deviation of 20 mph .
Find the percentage of cars that are travelling faster than the speed limit. The fastest $1 \%$ of car drivers will be banned from driving.
Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly. Car drivers will just be given a caution if they are travelling at a speed $m$ such that $$\mathrm { P } ( 70 < X < m ) = 0.1315$$
Find the value of $m$. Show your working clearly.

Edexcel S1 2015 June Q6

9 marks Moderate -0.8

The random variable $X$ has a discrete uniform distribution and takes the values $1,2,3,4$ Find
1. $\mathrm { F } ( 3 )$, where $\mathrm { F } ( x )$ is the cumulative distribution function of $X$,
2. $\mathrm { E } ( X )$.
3. Show that $\operatorname { Var } ( X ) = \frac { 5 } { 4 }$
The random variable $Y$ has a discrete uniform distribution and takes the values $$3,3 + k , 3 + 2 k , 3 + 3 k$$ where $k$ is a constant.
Write down $\mathrm { P } ( Y = y )$ for $y = 3,3 + k , 3 + 2 k , 3 + 3 k$ The relationship between $X$ and $Y$ may be written in the form $Y = k X + c$ where $c$ is a constant.
Find $\operatorname { Var } ( Y )$ in terms of $k$.
Express $c$ in terms of $k$.

Edexcel S1 2015 June Q7

6 marks Easy -1.8

7. A doctor is investigating the correlation between blood protein, $p$, and body mass index, $b$. He takes a random sample of 8 patients and the data are shown in the table below.

Patient	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
$b$	32	36	40	44	42	21	27	37
$p$	18	21	31	39	21	12	19	70

Draw a scatter diagram of these data on the axes provided.
\includegraphics[max width=\textwidth, alt={}, center]{36cf6341-1957-45b9-9f7d-0914506f5919-13_938_673_785_614} The doctor decides to leave out patient $H$ from his calculations.
Give a reason for the doctor's decision. For the 7 patients $A , B , C , D , E , F$ and $G$, $$S _ { b p } = 369 , \quad S _ { p p } = 490 \text { and } S _ { b b } = 423 \frac { 5 } { 7 }$$
Find the product moment correlation coefficient, $r$, for these 7 patients.
Without any further calculations, state how $r$ would differ from your answer in part (c) if it was calculated for all 8 patients. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cf6341-1957-45b9-9f7d-0914506f5919-15_1322_1593_207_173} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The histogram in Figure 1 summarises the times, in minutes, that 200 people spent shopping in a supermarket.
Give a reason to justify the use of a histogram to represent these data. Given that 40 people spent between 11 and 21 minutes shopping in the supermarket, estimate
the number of people that spent between 18 and 25 minutes shopping in the supermarket,
the median time spent shopping in the supermarket by these 200 people. The mid-point of each bar is represented by $x$ and the corresponding frequency by f .
Show that $\sum \mathrm { f } x = 6390$ Given that $\sum \mathrm { f } x ^ { 2 } = 238430$
for the data shown in the histogram, calculate estimates of
1. the mean,
2. the standard deviation. A coefficient of skewness is given by $\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$
Calculate this coefficient of skewness for these data. The manager of the supermarket decides to model these data with a normal distribution.
Comment on the manager's decision. Give a justification for your answer.

Edexcel S1 2004 January Q1

13 marks Moderate -0.8

An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, $t { } ^ { \circ } \mathrm { C }$, was recorded $m$ minutes after 7.00 a.m. The results are shown in the table below.

$m$	0	10	20	30	40	50
$t$	6.0	8.9	11.8	13.5	15.3	16.1

Calculate the exact values of $S _ { m t }$ and $S _ { m m }$.
Calculate the equation of the regression line of $t$ on $m$ in the form $t = a + b m$.
Use your equation to estimate the value of $t$ at 7.35 a.m.
State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
1. at 9.00 a.m. that day,
2. at 7.15 a.m. one month later.

Edexcel S1 2004 January Q2

7 marks Easy -1.2

2. The random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$.

Write down 3 properties of the distribution of $X$. Given that $\mu = 27$ and $\sigma = 10$
find $\mathrm { P } ( 26 < X < 28 )$.

Edexcel S1 2004 January Q3

10 marks Easy -1.3

3. A discrete random variable $X$ has the probability function shown in the table below.

$x$	0	1	2	3
$\mathrm { P } ( X = x )$	$\frac { 1 } { 3 }$	$\frac { 1 } { 2 }$	$\frac { 1 } { 12 }$	$\frac { 1 } { 12 }$

Find

$\mathrm { P } ( 1 < X \leq 3 )$,
$\mathrm { F } ( 2.6 )$,
$\mathrm { E } ( X )$,
$\mathrm { E } ( 2 X - 3 )$,
$\operatorname { Var } ( X )$

Edexcel S1 2004 January Q4

11 marks Moderate -0.8

4. $\quad$ The events $A$ and $B$ are such that $\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }$ and $\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }$.

Find
1. $\mathrm { P } \left( A \cap B ^ { \prime } \right)$,
2. $\mathrm { P } ( A \cap B )$,
3. $\mathrm { P } ( A \cup B )$,
4. $\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)$.
State, with a reason, whether or $\operatorname { not } A$ and $B$ are
1. mutually exclusive,
2. independent.

Edexcel S1 2004 January Q5

18 marks Moderate -0.3

5. The values of daily sales, to the nearest $\pounds$, taken at a newsagents last year are summarised in the table below.

Sales	Number of days
$1 - 200$	166
$201 - 400$	100
$401 - 700$	59
$701 - 1000$	30
$1001 - 1500$	5

Draw a histogram to represent these data.
Use interpolation to estimate the median and inter-quartile range of daily sales.
Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
(2)

Edexcel S1 2004 January Q6

16 marks Moderate -0.3

6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is $\frac { 2 } { 3 }$, at the second stage is $\frac { 1 } { 2 }$, and at the third stage is $\frac { 1 } { 4 }$.

Draw a tree diagram to represent the 3 stages of the game.
Find the probability of collecting all 3 keys.
Find the probability of collecting exactly one key in a game.
Calculate the probability that keys are not collected on at least 2 successive stages in a game.

CAIE S1 2020 Specimen Q1

5 marks Easy -1.2

1 Th fb low ing b ck te b ck stem-ad leaf il ag am sw stb a lsalaries $\mathbf { 6 }$ agp $\mathbf { 6 } \mathbf { 9 }$ females adgn ales.

Females			Males
(4)	5200	0	3	(1
(9)	988764000	2	007	( $\mathcal { B }$
(8	87533100	2	004566	( $\varnothing$
( )	642100	3	002335677	(9)
( ( )	754000	4	0112556889	(1)
(4)	9500	8	3457789	$( \gamma$
(2)	50	8	046	(3

Key 4 Q 3 m eas ( st $\mathbf { o }$ females an of $\mathbf { o }$ males.

Fid b med ara d b ɛ rtiles $\mathbf { 6 }$ th females' salaries. Yo are gie $n$th $t$ th med an salary $\mathbf { 6 }$ th males is $\boldsymbol { \otimes } \rho$ th lw er $\mathbf { q }$ rtile is $\boldsymbol { \$ } \boldsymbol { \theta }$ ad th $\mathbf { p }$ r e rtile is \$50
Drawap ir d ad wh sk rpos in a sig ed ag amo to g id b lw to rep esen th d ta. [β
\includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-02_997_1589_1736_310}

CAIE S1 2020 Specimen Q2

4 marks Easy -1.2

2 A sm mary $\mathbf { 6 }$ th sp esl, $x \mathrm { k }$ lm etres $\boldsymbol { \rho } \mathbf { r b }$, $\mathbf { 0 } 2$ cars $\boldsymbol { \rho }$ ssig a certain $\dot { \mathrm { p } } \mathrm { ng }$ th fb low ig if o matin $$\Sigma ( x \oplus ) = 3 \mathrm { a } \quad \mathrm {~d} \quad \Sigma ( x \oplus ) ^ { 2 } = \mathbb { t }$$ Fid b riance 6 th sp ed ad $n$ e fid b vale $6 \Sigma x ^ { 2 }$. [4]

CAIE S1 2020 Specimen Q3

7 marks Moderate -0.5

3 A b clb sed 6 p p rb ck ad 2 h r ck b to Mrs Ho . Sb cb es 4 6 tb se b at rach to take with b r o b id y. Th rach \& riable $X$ rep esen s tb m br $\mathbf { b }$ p $\mathbf { p }$ rb ck b sh cb es.

Sth that th p b b lityt $\mathbf { h }$ tsb cb es extlye perb clb is $\frac { 3 } { 14 }$. [R
Draw up b pb b lityd strib in tab e fo $X$.
Yu reg it h t $\mathrm { E } ( X ) = 3$ Fid $\operatorname { Var } ( X )$.

CAIE S1 2020 Specimen Q4

10 marks Moderate -0.5

4 A $\boldsymbol { \rho }$ trb station fid th tits $\mathbf { d }$ ily sales，in litres，are $\mathbf { n }$ mally $\dot { \mathbf { d } }$ strib ed with mean ad stad rd d $\dot { \mathbf { v } }$ atin $\quad 0$
（a）Fid 0 may dy 6 th $\mathbf { y }$ ar（B d $\mathbf { y }$ ）th d ily sales can b eq cted to e区 eed $\boldsymbol { \theta }$ litres． Th d ily sales at an $\mathbf { b } r \mathbf { p }$ trb station are $X$ litres，we re $X$ is $\mathbf { n }$ mally $\dot { \mathbf { d } }$ strib ed with mean $m$ ad stad rd iv atird $\quad t$ is g it h $\mathrm { t } \mathrm { P } ( X > 0 = \mathbb { 0 }$
（b）Fid by le $6 m$ ．
(c) Fid th p b b lity th t d ily sales at th s p trb station ex eed $\theta$ litres $\mathbf { n }$ fewer th n 266 rach lyc $b$ end $y$.
[0pt] [ $\beta$

CAIE S1 2020 Specimen Q5

7 marks Moderate -0.5

5 A fair six sid dl e,w itlf aces mark dress s ther im imes.

Use ara $p$ in matin of id b pb b lity b ta 3 s ob ain of ewer th rㅇs imes. [4]
Js tifys se 6 th ap ox matin pe rt (a). Ora $\mathbf { h }$ b roccasity he same $\dot { \mathbf { d } }$ e is th $\boldsymbol { w }$ ep ated y il a $\mathbf { 3 } \mathrm { sb }$ aie d
Fid b pb b lity b tb ain g ʒ eq res fewer th $n$ st $h$ s.

CAIE S1 2020 Specimen Q6

7 marks Standard +0.3

6 Ag $\mathbf { \Phi }$ of ries trac ls to b airp t irt wd axis, $P$ ad Q.E acht ax cart ak $\boldsymbol { \mathcal { C } }$ sseg rs.

Th 8 fried dive th msele s in o two gp 6,4 日 gp fo tax $P$ ad o gp fo tax $Q$,w ithlo il aralt rae llig it te same tax. Fid b m brd dl fferen way inw hick his carb de .
\includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_298_492_226_447}
\includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_301_478_223_1142} Each tax can tak $1 \boldsymbol { \rho }$ sseg r in th fro ad $3 \boldsymbol { \rho }$ sseg rs in th $\mathbf { b }$ ck (see $\dot { \mathbf { d } }$ ag am). Mark sits in th
\includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_51_1227_598_242}
Fid b m brd d fferen seatig rrag men s th tare $\mathbf { w }$ sibefo th of ried . [4]

CAIE S1 2020 Specimen Q7

10 marks Standard +0.3

7 Bag $A$ ch ais $4 \mathbf { b }$ lls $\mathrm { m } \quad \mathbf { b }$ red $2,4,58$ Bag $B$ ch ais $5 \mathbf { b }$ lls $\mathrm { m } \quad \mathbf { b }$ red 1,3688 Bag $C$ co ais 7 b lls m b redram a b $l l$ is selected $t$ rach frm eaclb $g$

Ed $n X$ is 'ed ctlyt wo th selecteb lls $\mathbf { h }$ th same m br'.
Ed n $Y$ is 'tb b ll selected rm bag $A \mathbf { h }$ sm br4.
1. FidP (X).
2. Fid ( $X \cap Y$ ) aid $\mathbf { n }$ ed termin wh ther or $\mathbf { n }$ even $\mathrm { s } X$ ad $Y$ are id $\mathbf { p } \mathbf { d } \quad \mathrm { h }$. [B
3. Fid the p b b lity th t two $\mathbf { b }$ lls are $\mathrm { m } \quad \mathbf { b }$ red $2 \dot { \mathrm {~g} }$ n th t ex ctly two $\mathbf { 6 }$ th selected $\mathbf { b }$ lls h \& th same m br.

If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin $\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )$ ms tb clearlys n n

OCR S1 2009 January Q1

8 marks Easy -1.2

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.

Number	Probability
0	0.7
1	0.2
2	0.1

The spinner is spun twice. The total of the two numbers on which it lands is denoted by $X$.

Show that $\mathrm { P } ( X = 2 ) = 0.18$. The probability distribution of $X$ is given in the table.
$x$ 0 1 2 3 4
$\mathrm { P } ( X = x )$ 0.49 0.28 0.18 0.04 0.01
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

\(x\)	- 2	0	2	4
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(a\)	\(c\)

\(x\)	0	1	2	3
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 12 }\)

Length of time employed, \(x\) years	Female	Male
\(x < 4\)	9	16
\(4 \leqslant x < 10\)	14	20
\(10 \leqslant x\)	7	24

Sales	Number of days
\(1 - 200\)	166
\(201 - 400\)	100
\(401 - 700\)	59
\(701 - 1000\)	30
\(1001 - 1500\)	5

Patient	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(b\)	32	36	40	44	42	21	27	37
\(p\)	18	21	31	39	21	12	19	70

\(x\)	0	1	2	3	4
\(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01

Questions S1 (1967 questions)

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Edexcel S1 2014 June Q4

Edexcel S1 2014 June Q5

Edexcel S1 2014 June Q6

Edexcel S1 2014 June Q7

Edexcel S1 2015 June Q1

Edexcel S1 2015 June Q2

Edexcel S1 2015 June Q3

Edexcel S1 2015 June Q4

Edexcel S1 2015 June Q5

Edexcel S1 2015 June Q6

Edexcel S1 2015 June Q7

Edexcel S1 2004 January Q1

Edexcel S1 2004 January Q2

Edexcel S1 2004 January Q3

Edexcel S1 2004 January Q4

Edexcel S1 2004 January Q5

Edexcel S1 2004 January Q6

CAIE S1 2020 Specimen Q1

CAIE S1 2020 Specimen Q2

CAIE S1 2020 Specimen Q3

CAIE S1 2020 Specimen Q4

CAIE S1 2020 Specimen Q5

CAIE S1 2020 Specimen Q6

CAIE S1 2020 Specimen Q7

OCR S1 2009 January Q1