Questions S2 - A-Level Maths

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Edexcel S2 2021 October Q3

10 marks Standard +0.8

3. A continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ 4 a x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ where $a$ and $b$ are positive constants.

Show that $b = 4$
Find the exact value of $a$
Find $\mathrm { P } ( X > 2.25 )$
Showing your working clearly,
1. sketch the probability density function of $X$
2. calculate the mode of $X$

Edexcel S2 2021 October Q4

15 marks Challenging +1.2

The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6
1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
2. Find the smallest value of $n$ such that the probability that at least $n$ cars enter the safari park in 10 minutes is less than 0.05
The probability that no cars enter the safari park in $m$ minutes, where $m$ is an integer, is less than 0.05
Find the smallest value of $m$ A car enters the safari park.
Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
Determine the critical region for the test at the $5 \%$ level of significance.

Edexcel S2 2021 October Q5

8 marks Standard +0.3

A bag contains a large number of counters.

40\% of the counters are numbered 1
$35 \%$ of the counters are numbered 2
$25 \%$ of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.

Show that Alif gets a score of 8 with probability 0.0875
Find the sampling distribution of Alif's score.
Calculate Alif's expected score.

Edexcel S2 2021 October Q6

17 marks Standard +0.3

6. The continuous random variable $Y$ has probability density function $\mathrm { f } ( y )$ given by $$f ( y ) = \begin{cases} \frac { 1 } { 14 } ( y + 2 ) & - 1 < y \leqslant 1 \\ \frac { 3 } { 14 } & 1 < y \leqslant 3 \\ \frac { 1 } { 14 } ( 6 - y ) & 3 < y \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

Sketch the probability density function $\mathrm { f } ( \mathrm { y } )$ Given that $\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 131 } { 21 }$
find $\operatorname { Var } ( 2 Y - 3 )$ The cumulative distribution function of $Y$ is $\mathrm { F } ( y )$
Show that $\mathrm { F } ( y ) = \frac { 1 } { 14 } \left( \frac { y ^ { 2 } } { 2 } + 2 y + \frac { 3 } { 2 } \right)$ for $- 1 < y \leqslant 1$
Find $\mathrm { F } ( y )$ for all values of $y$
Find the exact value of the 30th percentile of $Y$
Find $\mathrm { P } ( 4 Y \leqslant 5 \mid Y \leqslant 3 )$

Edexcel S2 2022 October Q1

11 marks Standard +0.3

Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.
1. Find the probability that in 15 metres of her cloth there are
  1. less than 3 faults,
  2. at least 6 faults.
Each roll contains 100 metres of Bhavna's cloth.
She selects 15 rolls at random.
Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
Each piece of her cloth is 4 metres long.
The cost to make each piece is $\pounds 5.00$
She sells each piece of her cloth that contains no faults for $\pounds 7.40$
She sells each piece of her cloth that contains faults for $\pounds 2.00$
Find the expected profit that Bhavna will make on each piece of her cloth that she sells.

Edexcel S2 2022 October Q2

15 marks Moderate -0.3

A random variable $X$ has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & - \frac { 1 } { 2 } \leqslant x < \frac { 1 } { 2 } \\ 2 x - \frac { 3 } { 4 } & \frac { 1 } { 2 } \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ is a positive constant.

Sketch the graph of $\mathrm { f } ( x )$
By forming and solving an equation in $k$, show that $k = 1.25$
Use calculus to find $\mathrm { E } ( X )$
Calculate the interquartile range of $X$

Edexcel S2 2022 October Q3

10 marks Standard +0.3

A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

Using a $5 \%$ level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to $2.5 \%$, for each tail of your critical region.
Write down the actual significance level of this test. Past records suggest that $2.8 \%$ of the company's sunflower seeds grow to a height of more than 3 metres.
A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
Using a suitable approximation test, at the $5 \%$ level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than $2.8 \%$
State your hypotheses clearly.

Edexcel S2 2022 October Q4

9 marks Standard +0.3

The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, $X$, represents the number of people who complete the task in less than 15 minutes.
1. Find $\mathrm { P } ( 5 \leqslant X < 8 )$
Mia takes a random sample of 140 people.
Using a normal approximation, the probability that fewer than $n$ of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
Find the value of $n$ Show your working clearly.

Edexcel S2 2022 October Q5

9 marks Standard +0.3

The continuous random variable $X$ has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3 \\ \frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4 \\ \frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c \\ 1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7 \\ 1 & x \geqslant 7 \end{array} \right.$$ where $c$ and $d$ are constants.

Show that $c = 6$
Find $\mathrm { P } ( X > 3.5 )$
Find $\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )$

Edexcel S2 2022 October Q6

9 marks Standard +0.8

A bag contains a large number of counters with one of the numbers 5 , 10 or 20 written on each of them in the ratio $5 : 2 : a$

A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio $1 : 3$ One counter is selected at random from the bag and then two counters are selected at random from the jar.
The random variable $R$ represents the range of the numbers on the 3 counters.
Given that $\mathrm { P } ( R = 15 ) = \frac { 63 } { 256 }$

by forming and solving an equation in $a$, show that $a = 9$
find the sampling distribution of $R$

Edexcel S2 2022 October Q7

12 marks Standard +0.3

(i) The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$

Given that $\mathrm { P } ( 5 < X < 13 ) = \frac { 1 } { 5 }$ and $\mathrm { E } ( X ) = 9$, find $\mathrm { P } ( 3 X > a + b )$
(ii) The continuous random variable $Y$ is uniformly distributed over the interval $[ 1 , c ]$ Given that $\operatorname { Var } ( Y ) = 0.48$, find the exact value of $\mathrm { E } \left( Y ^ { 2 } \right)$
(iii) A wire of length 20 cm is cut into 2 pieces at a random point. The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether. Find the probability that the length of the shortest piece of wire is less than 6 cm .

Edexcel S2 2023 October Q1

10 marks Moderate -0.8

Sam is a telephone sales representative.

For each call to a customer

Sam either makes a sale or does not make a sale
sales are made independently

Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2

Find the probability that Sam makes
1. exactly 2 sales in 14 calls,
2. more than 3 sales in 25 calls. Sam makes $n$ calls each day.
Find the minimum value of $n$
1. so that the expected number of sales each day is at least 6
2. so that the probability of at least 1 sale in a randomly selected day exceeds 0.95

Edexcel S2 2023 October Q2

8 marks Standard +0.8

The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by

$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant d \\ 0 & \text { otherwise } \end{cases}$$ where $a$, $b$, $c$ and $d$ are constants such that

$b x + c = a x ^ { 3 }$ at $x = 4$
$b x + c$ is a straight line segment with end coordinates ( $4,64 a$ ) and ( $d , 0$ )
1. State the mode of $X$

Given that the mode of $X$ is equal to the median of $X$

use algebraic integration to show that $a = \frac { 1 } { 128 }$

Find the value of $d$

Hence find the value of $b$ and the value of $c$

Edexcel S2 2023 October Q3

9 marks Moderate -0.8

Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15

It always takes Navtej 3 minutes to walk to the bus stop
Buses run every 15 minutes and Navtej catches the first bus that arrives
Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work

The total time, $T$ minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval $[ \alpha , \beta ]$

1. Show that $\alpha = 32$
2. Show that $\beta = 47$
State fully the probability density function for this distribution.
Find the value of
1. $\mathrm { E } ( T )$
2. $\operatorname { Var } ( T )$
Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.

Edexcel S2 2023 October Q4

10 marks Standard +0.3

A manufacturer makes t -shirts in 3 sizes, small, medium and large.

20\% of the t -shirts made by the manufacturer are small and sell for $\pounds 10 30 \%$ of the t -shirts made by the manufacturer are medium and sell for $\pounds 12$ The rest of the t -shirts made by the manufacturer are large and sell for $\pounds 15$

Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
List all the possible combinations of the individual selling prices of these 3 t-shirts.
Find the sampling distribution of the median selling price of these 3 t-shirts.

Edexcel S2 2023 October Q5

16 marks Standard +0.3

A supermarket receives complaints at a mean rate of 6 per week.
1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
  1. fewer than 3 complaints received by the supermarket,
  2. at least 6 complaints received by the supermarket.
In a randomly selected week, the supermarket received 12 complaints.
Test, at the $5 \%$ level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

Edexcel S2 2023 October Q6

12 marks Challenging +1.2

The continuous random variable $Y$ has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

Find $\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)$
Find the value of $k$
Use algebraic calculus to find $\mathrm { E } ( Y )$

Edexcel S2 2023 October Q20

Moderate -0.3

Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
List all the possible combinations of the individual selling prices of these 3 t-shirts.
Find the sampling distribution of the median selling price of these 3 t-shirts.
1. A supermarket receives complaints at a mean rate of 6 per week.
2. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
3. Find the probability that, in a given week, there are
  1. fewer than 3 complaints received by the supermarket,
  2. at least 6 complaints received by the supermarket.
In a randomly selected week, the supermarket received 12 complaints.
Test, at the $5 \%$ level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
1. The continuous random variable $Y$ has cumulative distribution function given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$
Find $\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)$
Find the value of $k$
Use algebraic calculus to find $\mathrm { E } ( Y )$
1. The discrete random variable $X$ is given by
$$X \sim \mathrm {~B} ( n , p )$$ The value of $n$ and the value of $p$ are such that $X$ can be approximated by a normal random variable $Y$ where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$
show that $\sigma = 6$
Hence find the value of $n$ and the value of $p$

Edexcel S2 2018 Specimen Q1

16 marks Standard +0.8

The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera.
A car has been caught speeding by this camera.
Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined £60
Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than $\pounds 5000$

Edexcel S2 2018 Specimen Q2

11 marks Moderate -0.3

2. A continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$

Find $\mathrm { P } ( X > 4 )$
Write down the value of $\mathrm { P } ( X \neq 4 )$
Find the probability density function of $X$, specifying it for all values of $x$
Write down the value of $\mathrm { E } ( X )$
Find $\operatorname { Var } ( X )$
Hence or otherwise find $\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)$

Edexcel S2 2018 Specimen Q3

11 marks Moderate -0.3

3. Explain what you understand by

a statistic,
a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and $Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ denote the number of screws in each of these packets.
List all the possible samples.
Find the sampling distribution of $\bar { Y }$

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Edexcel S2 2018 Specimen Q4

7 marks Standard +0.3

4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, $X$, which occur at the crossroads in a year.

Find $\mathrm { P } ( 5 \leqslant X < 7 )$ A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
Test, at the $5 \%$ level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.
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Edexcel S2 2018 Specimen Q5

9 marks Standard +0.8

5. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ and $a$ are constants.
Given that $\mathrm { E } ( X ) = \frac { 17 } { 12 }$

find the value of $k$ and the value of $a$
Write down the mode of $X$
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Edexcel S2 2018 Specimen Q6

13 marks Standard +0.3

6. The Headteacher of a school claims that $30 \%$ of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, $X$, who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

the probability that $X = 5$
the mean and the standard deviation of $X$ The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than $30 \%$. Given that in the survey of 20 parents 8 do not support the new curriculum,
test, at the $5 \%$ level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only $25 \%$ of the parents are in support of the new curriculum. A second random sample, of size $2 n$, is taken and exactly half of this sample supports the new curriculum. A test is carried out at a $10 \%$ level of significance of the teachers' belief using this sample of size $2 n$ Using the hypotheses $\mathrm { H } _ { 0 } : p = 0.25$ and $\mathrm { H } _ { 1 } : p > 0.25$
find the minimum value of $n$ for which the outcome of the test is that the teachers' belief is rejected.