Questions - Edexcel S2 - A-Level Maths

Sketch $\mathrm { f } ( x )$ for all values of $x$
Show that $k = \frac { 1 } { 6 }$
Specify fully the cumulative distribution function $\mathrm { F } ( x )$ of $X$ Given that $\mathrm { E } ( X ) = \frac { 61 } { 12 }$
find $\mathrm { P } ( X > \mathrm { E } ( X ) )$
Describe the skewness of the distribution, giving a reason for your answer.

Edexcel S2 2022 January Q5

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of $n$ applicants.
State one assumption necessary for this distribution to be a suitable model of this situation.
Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
Using a suitable approximation, test Jaymini's claim at the $5 \%$ level of significance. State your hypotheses clearly.

Edexcel S2 2022 January Q6

Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio $6 : 3 : 2$ respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
Find the sampling distribution for the total number, $T$, of breakfast items bought by these 2 customers. Show your working clearly.
Find $\mathrm { E } ( T )$

Edexcel S2 2022 January Q7

7 The sides of a square are each of length $L \mathrm {~cm}$ and its area is $A \mathrm {~cm} ^ { 2 }$ Given that $A$ is uniformly distributed on the interval [10,30]

find $\mathrm { P } ( L \geqslant 4.5 )$
find $\operatorname { Var } ( L )$

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Edexcel S2 2023 January Q1

A shop sells shoes at a mean rate of 4 pairs of shoes per hour on a weekday.
1. Suggest a suitable distribution for modelling the number of sales of pairs of shoes made per hour on a weekday.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Find the probability that on a weekday the shop sells
  1. more than 4 pairs of shoes in a one-hour period,
  2. more than 4 pairs of shoes in each of 3 consecutive one-hour periods.
The area manager visits the shop on a weekday, the day after an advert for the shop appears in a local paper. In a one-hour period during the manager's visit, the shop sells 7 pairs of shoes. This leads the manager to believe that the advert has increased the shop's sales of pairs of shoes.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of an increase in sales of pairs of shoes following the appearance of the advert.

Edexcel S2 2023 January Q2

A bag contains a large number of coins. It only contains 20 p and 50 p coins. A random sample of 3 coins is taken from the bag.
1. List all the possible combinations of 3 coins that might be taken.
Let $\bar { X }$ represent the mean value of the 3 coins taken.
Part of the sampling distribution of $\bar { X }$ is given below.
$\bar { x }$ 20 $a$ $b$ 50
$\mathrm { P } ( \bar { X } = \bar { x } )$ $\frac { 4913 } { 8000 }$ $c$ $d$ $\frac { 27 } { 8000 }$
Write down the value of $a$ and the value of $b$ The probability of taking a 20p coin at random from the bag is $p$ The probability of taking a 50p coin at random from the bag is $q$
Find the value of $p$ and the value of $q$
Hence, find the value of $c$ and the value of $d$ Let $M$ represent the mode of the 3 coins taken at random from the bag.
Find the sampling distribution of $M$

Edexcel S2 2023 January Q3

Superbounce is a manufacturer of tennis balls.

It knows from past records that 10\% of its tennis balls fail a bounce test.

Find the probability that from a random sample of 10 of these tennis balls
1. at least 4 fail the bounce test
2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
Test, at the $5 \%$ significance level, whether or not this result supports the managing director's claim. In a second random sample of $n$ tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
Find the smallest possible value of $n$

Edexcel S2 2023 January Q4

The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$, shown in the diagram, where $k$ is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{f4fa6add-5860-4c88-bb70-f3edd9b22211-12_511_1096_351_351}
1. Find $\mathrm { P } ( X < 10 k )$
2. Show that $k = \frac { 1 } { \pi }$
3. Find, in terms of $\pi$, the values of
  1. $\mathrm { E } ( X )$
  2. $\operatorname { Var } ( X )$
Circles are drawn with area $A$, where $$A = \pi \left( X + \frac { 2 } { \pi } \right) ^ { 2 }$$
Find $\mathrm { E } ( A )$

Edexcel S2 2023 January Q5

A company produces steel cable.

Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.

Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable $x$ metres long.
Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
Find the value of $x$

Edexcel S2 2023 January Q6

The continuous random variable $X$ has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
a x + b x ^ { 2 } & 0 \leqslant x \leqslant k
1 & x > k \end{array} \right.$$ where $a , b$ and $k$ are positive constants.

Show that $a k = 1 - b k ^ { 2 }$ Using part (a) and given that $\mathrm { E } ( X ) = \frac { 6 } { 5 }$
show that $5 b k ^ { 3 } = 36 - 15 k$ Using part (a) and given that $\mathrm { E } ( X ) = \frac { 6 } { 5 }$ and $\operatorname { Var } ( X ) = \frac { 22 } { 75 }$
show that $5 b k ^ { 4 } = 52 - 10 k ^ { 2 }$ Given that $k < 3$
find the value of $k$
Hence find the value of $a$ and the value of $b$

Edexcel S2 2024 January Q1

The manager of a supermarket is investigating the number of complaints per day received from customers.

A random sample of 180 days is taken and the results are shown in the table below.

Number of complaints per day	0	1	2	3	4	5	6	$\geqslant 7$
Frequency	12	28	37	38	29	17	19	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 complaints per day. The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
For a randomly selected day find, using the manager's model, the probability that there are
1. at least 3 complaints,
2. more than 4 complaints but less than 8 complaints. A week consists of 7 consecutive days.
Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
Show your working clearly.
(Solutions relying on calculator technology are not acceptable.) A period of 13 weeks is selected at random.
Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
Show your working clearly.

Edexcel S2 2024 January Q2

The length of pregnancy for a randomly selected pregnant sheep is $D$ days where

$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,

find the value of $\sigma$ Qiang selects 25 pregnant sheep at random from a large flock.
Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.

Edexcel S2 2024 January Q3

Rowan believes that $35 \%$ of type $A$ vacuum tubes shatter when exposed to alternating high and low temperatures.

Rowan takes a random sample of 15 of these type $A$ vacuum tubes and uses a two-tailed test, at the $5 \%$ level of significance, to test his belief.

Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type $A$ vacuum tubes that shatter when exposed to alternating high and low temperatures.
Using a binomial distribution, find the critical region for the test. You should state the probability of rejection in each tail, which should be as close as possible to 0.025
Find the actual level of significance of the test based on your critical region from part (b) Rowan records that in the latest batch of 15 type $A$ vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
With reference to part (b), comment on Rowan’s belief. Give a reason for your answer. Rowan changes to type $B$ vacuum tubes. He takes a random sample of 40 type $B$ vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
Test, at the $5 \%$ level of significance, whether or not there is evidence that the proportion of type $B$ vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than $35 \%$
You should state your hypotheses clearly.

Edexcel S2 2024 January Q4

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

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2
\frac { 3 } { 20 } & 2 < g \leqslant 4
0 & \text { otherwise } \end{cases}$$

Sketch the graph of $\mathrm { f } ( \mathrm { g } )$
Find $\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )$ The continuous random variable $H$ is such that $\mathrm { E } ( H ) = 12$ and $\operatorname { Var } ( H ) = 2.4$
Find $\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)$ Show your working clearly.
(Solutions relying on calculator technology are not acceptable.)

Edexcel S2 2024 January Q5

The random variable $W$ has a continuous uniform distribution over the interval $[ - 6 , a ]$ where $a$ is a constant.

Given that $\operatorname { Var } ( W ) = 27$

show that $a = 12$ Given that $\mathrm { P } ( W > b ) = \frac { 3 } { 5 }$
1. find the value of $b$
2. find $\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)$ A piece of wood $A B$ has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
Calculate the probability that the area of the rectangle is greater than $975 \mathrm {~cm} ^ { 2 }$

Edexcel S2 2024 January Q6

A bag contains a large number of counters with an odd number or an even number written on each.

Odd and even numbered counters occur in the ratio $4 : 1$
In a game a player takes a random sample of 4 counters from the bag.
The player scores
5 points for each counter taken that has an even number written on it
2 points for each counter taken that has an odd number written on it
The random variable $X$ represents the total score, in points, from the 4 counters.

Find the sampling distribution of $X$ A random sample of $n$ sets of 4 counters is taken. The random variable $Y$ represents the number of these $n$ sets that have a total score of exactly 14
Calculate the minimum value of $n$ such that $\mathrm { P } ( Y \geqslant 1 ) > 0.95$

Edexcel S2 2024 January Q7

A continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where $a$, $b$ and $k$ are non-zero constants.
Given that the mode of $X$ is 1.5

show that $b = 3$
Hence show that $a = 2$
Show that the median of $X$ lies between 1.4 and 1.5

Edexcel S2 2014 June Q1

(a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A farmer supplies a bakery with eggs. The manager of the bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009
(b) State suitable hypotheses for testing these claims. In a batch of 500 eggs the baker records 9 eggs with a double yolk.
(c) Using a suitable approximation, test at the $5 \%$ level of significance whether or not this supports the farmer's claim.

Edexcel S2 2014 June Q2

2. The amount of flour used by a factory in a week is $Y$ thousand kg where $Y$ has probability density function $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k \left( 4 - y ^ { 2 } \right) & 0 \leqslant y \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

Show that the value of $k$ is $\frac { 3 } { 16 }$ Use algebraic integration to find
the mean number of kilograms of flour used by the factory in a week,
the standard deviation of the number of kilograms of flour used by the factory in a week,
the probability that more than 1500 kg of flour will be used by the factory next week.

Edexcel S2 2014 June Q3

The continuous random variable $T$ is uniformly distributed on the interval $[ \alpha , \beta ]$ where $\beta > \alpha$

Given that $\mathrm { E } ( T ) = 2$ and $\operatorname { Var } ( T ) = \frac { 16 } { 3 }$, find

the value of $\alpha$ and the value of $\beta$,
$\mathrm { P } ( T < 3.4 )$

Edexcel S2 2014 June Q4

4. Pieces of ribbon are cut to length $L \mathrm {~cm}$ where $L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)$

Given that $30 \%$ of the pieces of ribbon have length more than 100 cm , find the value of $\mu$ to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .

Edexcel S2 2014 June Q5

5. A company claims that $35 \%$ of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.

1. State suitable hypotheses for a two-tailed test of this claim.
2. Using a $5 \%$ level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to $2.5 \%$ as possible.
Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
State the actual significance level of this test.

Edexcel S2 2014 June Q6

6. A continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < 0
\frac { x ^ { 2 } } { 20 } ( 9 - 2 x ) & 0 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$

Verify that the median of $X$ lies between 1.23 and 1.24
Specify fully the probability density function $\mathrm { f } ( x )$.
Find the mode of $X$.
Describe the skewness of this distribution. Justify your answer.

Edexcel S2 2014 June Q7

7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .

Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length $x \mathrm {~m}$ is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
Find the value of $x$.

Edexcel S2 2015 June Q1

A continuous random variable $X$ has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 2
\frac { 1 } { 20 } \left( x ^ { 2 } - 4 \right) & 2 \leqslant x \leqslant 4
\frac { 1 } { 5 } ( 2 x - 5 ) & 4 < x \leqslant 5
1 & x > 5 \end{array} \right.$$

Calculate $\mathrm { P } ( X > 4 )$
Find the probability density function of $X$, specifying it for all values of $x$.
Find the value of $a$ such that $\mathrm { P } ( 3 < X < a ) = 0.642$
Find the probability density function of $X$, specifying it for all values of $x$.

\(\bar { x }\)	20	\(a\)	\(b\)	50
\(\mathrm { P } ( \bar { X } = \bar { x } )\)	\(\frac { 4913 } { 8000 }\)	\(c\)	\(d\)	\(\frac { 27 } { 8000 }\)

Number of complaints per day	0	1	2	3	4	5	6	\(\geqslant 7\)
Frequency	12	28	37	38	29	17	19	0

Questions — Edexcel S2 (494 questions)

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