Questions - Edexcel S2 - A-Level Maths

Find the median of $D$.
Find the mode of $D$. Justify your answer. The experiment is conducted on 80 children.
Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.

Edexcel S2 2014 June Q7

7. A piece of string $A B$ has length 9 cm . The string is cut at random at a point $P$ and the random variable $X$ represents the length of the piece of string $A P$.

Write down the distribution of $X$.
Find the probability that the length of the piece of string $A P$ is more than 6 cm . The two pieces of string $A P$ and $P B$ are used to form two sides of a rectangle. The random variable $R$ represents the area of the rectangle.
Show that $R = a X ^ { 2 } + b X$ and state the values of the constants $a$ and $b$.
Find $\mathrm { E } ( R )$.
Find the probability that $R$ is more than twice the area of a square whose side has the length of the piece of string $A P$.

Edexcel S2 2014 June Q1

Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
  1. exactly 7
  2. at least 10
A patient arrives at 11.30 a.m.
Find the probability that the next patient arrives before 11.45 a.m.

Edexcel S2 2014 June Q2

2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, $T$, with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9
0 & \text { otherwise } \end{array} \right.$$ where $c$ is a constant.

Show that the value of $c$ is $\frac { 1 } { 486 }$
Show that the cumulative distribution function $\mathrm { F } ( t )$ is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0
\frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9
1 & t > 9 \end{array} \right.$$
Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

Edexcel S2 2014 June Q3

A company claims that it receives emails at a mean rate of 2 every 5 minutes.
1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
2. Using a $5 \%$ level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
3. Find the actual level of significance of this test.
To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
Test, at the $10 \%$ level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men’s Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.

Edexcel S2 2014 June Q4

A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is $p$. When firing from a distance $d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )$. Each shot is fired independently.

The cadet fires 10 shots from a distance of 40 m .

1. Find the probability that exactly 6 shots hit the target.
2. Find the probability that at least 8 shots hit the target. The cadet fires 20 shots from a distance of $x \mathrm {~m}$.
Find, to the nearest integer, the value of $x$ if the cadet has an $80 \%$ chance of hitting the target at least once. The cadet fires 100 shots from 25 m .
Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.

Edexcel S2 2014 June Q5

5. (a) State the conditions under which the normal distribution may be used as an approximation to the binomial distribution. A company sells seeds and claims that 55\% of its pea seeds germinate.
(b) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. To test the company's claim, a random sample of 220 pea seeds was planted.
(c) State the hypotheses for a two-tailed test of the company's claim. Given that 135 of the 220 pea seeds germinated,
(d) use a normal approximation to test, at the $5 \%$ level of significance, whether or not the company’s claim is justified.

Edexcel S2 2014 June Q6

6. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { 9 } & 0 \leqslant x \leqslant 1
\frac { 2 } { 9 } & 1 < x < 4
\frac { 2 } { 3 } - \frac { x } { 9 } & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{array} \right.$$

Find $\mathrm { E } ( X )$.
Find the cumulative distribution function $\mathrm { F } ( x )$ for all values of $x$.
Find the median of $X$.
Describe the skewness. Give a reason for your answer.
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Edexcel S2 2015 June Q1

In a survey it is found that barn owls occur randomly at a rate of 9 per $1000 \mathrm {~km} ^ { 2 }$.
1. Find the probability that in a randomly selected area of $1000 \mathrm {~km} ^ { 2 }$ there are at least 10 barn owls.
2. Find the probability that in a randomly selected area of $200 \mathrm {~km} ^ { 2 }$ there are exactly 2 barn owls.
3. Using a suitable approximation, find the probability that in a randomly selected area of $50000 \mathrm {~km} ^ { 2 }$ there are at least 470 barn owls.
4. The proportion of houses in Radville which are unable to receive digital radio is $25 \%$. In a survey of a random sample of 30 houses taken from Radville, the number, $X$, of houses which are unable to receive digital radio is recorded.
5. Find $\mathrm { P } ( 5 \leqslant X < 11 )$
A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.
Test, at the $10 \%$ level of significance, the radio company's claim. State your hypotheses clearly.

Edexcel S2 2015 June Q3

3. A random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 2
k \left( 1 - \frac { x } { 6 } \right) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where $k$ is a constant.

Show that $k = \frac { 1 } { 4 }$
Write down the mode of $X$.
Specify fully the cumulative distribution function $\mathrm { F } ( x )$.
Find the upper quartile of $X$.

Edexcel S2 2015 June Q4

The continuous random variable $L$ represents the error, in metres, made when a machine cuts poles to a target length. The distribution of $L$ is a continuous uniform distribution over the interval $[ 0,0.5 ]$
1. Find $\mathrm { P } ( L < 0.4 )$.
2. Write down $\mathrm { E } ( L )$.
3. Calculate $\operatorname { Var } ( L )$.
A random sample of 30 poles cut by this machine is taken.
Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. When a new machine cuts poles to a target length, the error, $X$ metres, is modelled by the cumulative distribution function $\mathrm { F } ( x )$ where $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
4 x - 4 x ^ { 2 } & 0 \leqslant x \leqslant 0.5
1 & \text { otherwise } \end{array} \right.$$
Using this model, find $\mathrm { P } ( X > 0.4 )$ A random sample of 100 poles cut by this new machine is taken.
Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.

Edexcel S2 2015 June Q5

Liftsforall claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.
1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to $2.5 \%$ as possible.
Over a randomly selected 1 month period the lift broke down 3 times.
Test, at the $5 \%$ level of significance, whether Liftsforall's claim is correct. State your hypotheses clearly.
State the actual significance level of this test.
! The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay Liftsforall $\pounds 500$ for every quarter ( 3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. Liftsforall installs a new lift in the block of flats.
Given that the new lift breaks down at a mean rate of 2 times per month,
find the probability that the residents do not pay more than $\pounds 500$ to Liftsforall in the next year.

Edexcel S2 2015 June Q6

A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ where

$$f ( x ) = \left\{ \begin{array} { c c } k x ^ { n } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$ where $k$ and $n$ are positive integers.

Find $k$ in terms of $n$.
Find $\mathrm { E } ( X )$ in terms of $n$.
Find $\mathrm { E } \left( X ^ { 2 } \right)$ in terms of $n$. Given that $n = 2$
find $\operatorname { Var } ( 3 X )$.

Edexcel S2 2015 June Q7

A bag contains a large number of $10 \mathrm { p } , 20 \mathrm { p }$ and 50 p coins in the ratio $1 : 2 : 2$

A random sample of 3 coins is taken from the bag.
Find the sampling distribution of the median of these samples.

Edexcel S2 2016 June Q1

A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.

Number of cherries	0	1	2	3	4	5	$\geqslant 6$
Frequency	24	37	21	12	4	2	0

Calculate the mean and the variance of these data.
Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake. The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains
1. exactly 2 cherries,
2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places. Twelve packets of Rays fruit cakes are selected at random.
Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

Edexcel S2 2016 June Q2

In a region of the UK, $5 \%$ of people have red hair. In a random sample of size $n$, taken from this region, the expected number of people with red hair is 3
1. Calculate the value of $n$.
A random sample of 20 people is taken from this region. Find the probability that
1. exactly 4 of these people have red hair,
2. at least 4 of these people have red hair. Patrick claims that Reddman people have a probability greater than $5 \%$ of having red hair. In a random sample of 50 Reddman people, 4 of them have red hair.
Stating your hypotheses clearly, test Patrick's claim. Use a $1 \%$ level of significance.
\href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

Edexcel S2 2016 June Q3

The random variable $R$ has a continuous uniform distribution over the interval [5,9]
1. Specify fully the probability density function of $R$.
2. Find $\mathrm { P } ( 7 < R < 10 )$
The random variable $A$ is the area of a circle radius $R \mathrm {~cm}$.
Find $\mathrm { E } ( \mathrm { A } )$

Edexcel S2 2016 June Q4

4. A continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 2
k \left( a x + b x ^ { 2 } - x ^ { 3 } \right) & 2 \leqslant x \leqslant 3
1 & x > 3 \end{array} \right.$$ Given that the mode of $X$ is $\frac { 8 } { 3 }$

show that $b = 8$
find the value of $k$.

Edexcel S2 2016 June Q5

5. In a large school, $20 \%$ of students own a touch screen laptop. A random sample of $n$ students is chosen from the school. Using a normal approximation, the probability that more than 55 of these $n$ students own a touch screen laptop is 0.0401 correct to 3 significant figures. Find the value of $n$.
(8)
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Edexcel S2 2016 June Q6

6. A bag contains a large number of counters with one of the numbers 4,6 or 8 written on each of them in the ratio $5 : 3 : 2$ respectively. A random sample of 2 counters is taken from the bag.

List all the possible samples of size 2 that can be taken. The random variable $M$ represents the mean value of the 2 counters.
Given that $\mathrm { P } ( M = 4 ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( M = 8 ) = \frac { 1 } { 25 }$
find the sampling distribution for $M$. A sample of $n$ sets of 2 counters is taken. The random variable $Y$ represents the number of these $n$ sets that have a mean of 8
Calculate the minimum value of $n$ such that $\mathrm { P } ( Y \geqslant 1 ) > 0.9$

Edexcel S2 2016 June Q7

7. The weight, $X \mathrm {~kg}$, of staples in a bin full of paper has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 9 x - 3 x ^ { 2 } } { 10 } & 0 \leqslant x < 2
0 & \text { otherwise } \end{array} \right.$$ Use integration to find

$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$
$\mathrm { P } ( X > 1.5 )$ Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for $\pounds 50$ but if the weight of the staples exceeds 1.5 kg it sells for $\pounds 25$
Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have $20 \%$ fewer bins full of paper to sell.
Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.
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Edexcel S2 2017 June Q1

A potter believes that $20 \%$ of pots break whilst being fired in a kiln. Pots are fired in batches of 25 .
1. Let $X$ denote the number of broken pots in a batch. A batch is selected at random. Using a 10\% significance level, find the critical region for a two tailed test of the potter's belief. You should state the probability in each tail of your critical region.
The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50 . He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.
Test, at the $5 \%$ level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly.

Edexcel S2 2017 June Q2

A company receives telephone calls at random at a mean rate of 2.5 per hour.
1. Find the probability that the company receives
  1. at least 4 telephone calls in the next hour,
  2. exactly 3 telephone calls in the next 15 minutes.
2. Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2
The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10
Test at the 5\% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.

Edexcel S2 2017 June Q3

3. The lifetime, $X$, in tens of hours, of a battery is modelled by the probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 9 } x ( 4 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$ Use algebraic integration to find

$\mathrm { E } ( X )$
$\mathrm { P } ( X > 2.5 )$ A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.
Find the probability that the radio will be working after 25 hours of use. Given that the radio is working after 16 hours of use,
find the probability that the radio will be working after being used for another 9 hours.

Edexcel S2 2017 June Q4

4. The continuous random variable $X$ is uniformly distributed over the interval $[ \alpha , \beta ]$ Given that $\mathrm { E } ( X ) = 3.5$ and $\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }$

find the value of $\alpha$ and the value of $\beta$ Given that $\mathrm { P } ( X < c ) = \frac { 2 } { 3 }$
1. find the value of $c$
2. find $\mathrm { P } ( c < X < 9 )$ A rectangle has a perimeter of 200 cm . The length, $S \mathrm {~cm}$, of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
Find the probability that the length of the shorter side of the rectangle is less than 45 cm .

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