Questions Unit 2 - A-Level Maths

In a study, samples of soil were collected during the summer. Soil samples of dimensions $25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}$ were collected for analysis. The study found that there were, on average, 11 earthworms per sample. a) Explain briefly the conditions under which a Poisson distribution could be used to model the number of earthworms per sample.
b) In July, pupils at a primary school are asked to dig a smaller hole, $25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 10 \mathrm {~cm}$, and to count the number of earthworms they find. Calculate the probability that the pupils find exactly 5 earthworms.
c) In the autumn, the average number of earthworms per sample is greater than in the summer. The probability that, in the autumn, there are fewer than 13 earthworms in a soil sample of dimensions $25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}$ is close to $36 \%$. Find the mean number of earthworms, to the nearest whole number, per $25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}$ soil sample in the autumn.

Jessica is studying the relationship between hip girth, $h \mathrm {~cm}$, and thigh girth, $t \mathrm {~cm}$, for American adults who are physically active. She takes a random sample of 11 people from a very large dataset which she has downloaded into a spreadsheet software package. The results are shown below.

$h ( \mathrm {~cm} )$	$98 \cdot 6$	$112 \cdot 1$	$97 \cdot 9$	$110 \cdot 2$	$89 \cdot 2$	$111 \cdot 7$	$87 \cdot 0$	$94 \cdot 7$	$100 \cdot 4$	$104 \cdot 0$	$88 \cdot 4$
$t ( \mathrm {~cm} )$	$48 \cdot 3$	$87 \cdot 2$	$55 \cdot 2$	$68 \cdot 0$	$48 \cdot 5$	$63 \cdot 2$	$49 \cdot 5$	$55 \cdot 7$	$59 \cdot 1$	$64 \cdot 0$	$52 \cdot 4$

a) Jessica notes that, for the thigh girth data, the lower quartile is 49.5 and the upper quartile is $64 \cdot 0$.
i) Show that 87.2 should be classified as an outlier for $t$.
ii) Give a reason why Jessica might exclude the outlier.
iii) Give a reason why Jessica might include the outlier. Jessica decides to exclude the outlier and produces the following scatter diagram. \section*{Thigh girth versus Hip girth} \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-04_647_1250_1439_404}
b) Interpret, in context, the correlation in the data shown in the diagram. The equation of the regression line of $t$ on $h$ for this sample is $$t = 0.69 h - 11.26$$ c) Interpret the gradient of the regression line in this context.
d) Use your knowledge of large data sets and spreadsheet software packages to suggest a way in which Jessica could improve her investigation. A company, Run4Lyfe, sponsors an athletic event. The organisers of the event claim that $70 \%$ of the participants know the name of the sponsoring company. Run4Lyfe is concerned that the proportion, $p$, of participants knowing the name of the sponsoring company is less than $70 \%$. They decide to survey 60 randomly selected participants to carry out a significance test.
a) State suitable hypotheses for carrying out the test.
b) i) Explain what is meant by the critical region for this test.
ii) Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, $5 \%$.
iii) Given that 40 participants out of the 60 in the sample know the name of the company, complete the significance test.
c) State, with a reason, how you would advise Run4Lyfe with regards to sponsoring the event next year.

The fertility rate for a country is the average number of children that are born to a woman over her lifetime. The graphs and table below show some data on the fertility rates for 197 countries in the years 1914 and 2014. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Fertility rates in 1914} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_671_1483_593_283}

\end{figure} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Fertility rates in 2014} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_616_1219_1434_287}

\end{figure} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Decreases in fertility rates from 1914 to 2014} \includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_476_613_2270_388}

\end{figure}

Minimum value	- 0.71
Lower quartile	2.08
Median	3.19
Upper quartile	3.94
Maximum value	6.49

a) Comment on the shapes of the distributions of fertility rates for 1914 and 2014.
b) Interpret the minimum value, $- 0 \cdot 71$, in the boxplot. You are also given the following information:

Country

Fertility rate

for 1914

Fertility rate

for 2014

France

Between 2

and 3

1.98

Ethiopia

Between 6

and 7

4.4

c) i) Find the best possible estimate for the decrease in the fertility rate from 1914 to 2014 for France.
ii) Find the best possible estimate for the decrease in the fertility rate from 1914 to 2014 for Ethiopia.
iii) Give one possible reason why the answers to i) and ii) are so different.
iv) Explain why these estimates may not be very accurate. \section*{Section B: Mechanics}

$\mathbf { 0 }$

The diagram below shows a vehicle of mass 1300 kg towing a trailer of mass 500 kg by means of a light horizontal tow bar. The vehicle is moving forward along a straight horizontal road such that a constant resistance of magnitude 650 N acts on the vehicle and a constant resistance of magnitude 320 N acts on the trailer. The vehicle's engine produces a constant driving force of $F \mathrm {~N}$.

\includegraphics[max width=\textwidth, alt={}]{77c62e6d-58e4-42d3-9982-5a8325e8e826-08_158_851_781_609}

Given that the acceleration of the vehicle and trailer is $0.85 \mathrm {~ms} ^ { - 2 }$, show that $F = 2500$ and determine the tension in the tow bar.

WJEC Unit 2 2022 June Q8

Easy -1.2

An aircraft moves along a straight horizontal runway with a constant acceleration of $1.5 \mathrm {~ms} ^ { - 2 }$. Points $A$ and $B$ lie on the runway. The aircraft passes $A$ with speed $4 \mathrm {~ms} ^ { - 1 }$ and its speed at $B$ must be at least $78 \mathrm {~ms} ^ { - 1 }$ if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes $A$.
b) Determine the minimum value of the distance $A B$ for the aircraft to take off successfully. The diagram below shows an object $A$, of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box $B$ by a light inextensible string which passes over a smooth pulley $P$, fixed at the edge of the surface, so that box $B$ hangs freely. An object $C$ lies on the horizontal floor of box $B$ so that the combined mass of $B$ and $C$ is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to $A$ in the direction $P A$ so that box $B$ is raised.
a) Find the magnitude of the acceleration of $A$ and the tension in the string.
b) Given that object $C$ has mass 4 kg , calculate the reaction of the floor of the box on object $C$.
□
1 In this question, $\mathbf { i }$ and $\mathbf { j }$ represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

the shop has position vector $\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }$,
the park is 2 km away on a bearing of $060 ^ { \circ }$.
a) Show that the position vector of the park relative to the house is $( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }$.
b) Determine the total distance walked by Sarah from her house to the park.
c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

Explain what $M \cap D'$ means in this context. [1]
One of these students is chosen at random. Find the probability that this student studies
1. exactly two of these subjects,
2. Mathematics or French or both. [3]
Determine whether studying Mathematics and studying Drama are statistically independent for these students. [3]

WJEC Unit 2 2018 June Q03

6 marks Moderate -0.8

Naomi produces oak tabletops, each of area 4·8 m². Defects in the oak tabletops occur randomly at a rate of 0·25 per m².

Find the probability that a randomly chosen tabletop will contain at most 2 defects. [3]
Find the probability that, in a random sample of 7 tabletops, exactly 4 will contain at most 2 defects each. [3]

WJEC Unit 2 2018 June Q04

9 marks Moderate -0.3

Edward can correctly identify 20% of types of wild flower. He studies some books to see if he can improve how often he can correctly identify types of wild flower. He collects a random sample of 10 types of wild flower in order to test whether or not he has improved.

1. Write suitable hypotheses for this test.
2. State a suitable test statistic that he could use. [2]
Using a 5% level of significance, find the critical region for this test. [3]
State the probability of a Type I error for this test and explain what it means in this context. [2]
Edward correctly identifies 4 of the 10 types of wild flower he collected. What conclusion should Edward reach? [2]

WJEC Unit 2 2018 June Q05

6 marks Easy -1.2

A baker is aware that the pH of his sourdough, $y$, and the hydration, $x$, affect the taste and texture of the final product. The hydration is measured in ml of water per 100 g of flour (ml/100 g). The baker researches how the pH of his sourdough changes as the hydration changes. The results of his research are shown in the diagram below. \includegraphics{figure_5}

Describe the relationship between pH and hydration. [2]
The equation of the regression line for $y$ on $x$ is $$y = 5.4 - 0.02x.$$
1. Interpret the gradient and intercept of the regression line in this context.
2. Estimate the pH of the sourdough when the hydration is 20 ml/100 g. Comment on the reliability of this estimate. [4]

WJEC Unit 2 2018 June Q06

10 marks Moderate -0.8

Basel is a keen learner of languages. He finds a website on which a large number of language tutors offer their services. Basel records the cost, in dollars, of a one hour lesson from a random sample of tutors. He puts the data into a computer program which gives the following summary statistics. Cost per 1 hour lesson Min. :10.0 1st Qu. :16.0 Median :17.2 Mean :19.8 3rd Qu. :21.0 Max. :40.0

Showing all calculations, comment on any outliers for the cost of a one hour lesson with a language tutor. [4]
Describe the skewness of the data and explain what it means in this context. [2]

Dafydd is also a keen learner of languages. He takes his own random sample of the cost, in dollars, for a one hour lesson. He produces the following box plot. \includegraphics{figure_6}

1. What will happen to the mean if the outlier is removed?
2. What will happen to the median if the outlier is removed? [2]
Compare and contrast the distributions of the cost of one hour language lessons for Dafydd's sample and Basel's sample. [2]

WJEC Unit 2 2018 June Q07

3 marks Moderate -0.8

A particle moves along the horizontal $x$-axis so that its velocity $v$ ms$^{-1}$ at time $t$ seconds is given by $$v = 6t^2 - 8t - 5.$$ At time $t = 1$, the particle's displacement from the origin is $-4$ m. Find an expression for the displacement of the particle at time $t$ seconds. [3]

WJEC Unit 2 2018 June Q08

7 marks Moderate -0.8

The diagram shows two objects $A$ and $B$, of mass 3 kg and 5 kg respectively, connected by a light inextensible string passing over a light smooth pulley fixed at the end of a smooth horizontal surface. Object $A$ lies on the horizontal surface and object $B$ hangs freely below the pulley. \includegraphics{figure_8} Initially, $B$ is supported so that the objects are at rest with the string just taut. Object $B$ is then released.

Find the magnitude of the acceleration of $A$ and the tension in the string. [6]
State briefly what effect a rough pulley would have on the tension in the string. [1]

WJEC Unit 2 2018 June Q09

6 marks Moderate -0.8

Three forces $\mathbf{L}$, $\mathbf{M}$ and $\mathbf{N}$ are given by $$\mathbf{L} = 2\mathbf{i} + 5\mathbf{j},$$ $$\mathbf{M} = 3\mathbf{i} - 22\mathbf{j},$$ $$\mathbf{N} = 4\mathbf{i} - 23\mathbf{j}.$$ Find the magnitude and direction of the resultant of the three forces. [6]

WJEC Unit 2 2018 June Q10

7 marks Moderate -0.8

A person, of mass 68 kg, stands in a lift which is moving upwards with constant acceleration. The lift is of mass 770 kg and the tension in the lift cable is 8000 N.

Determine the acceleration of the lift, giving your answer correct to two decimal places. [3]
State whether the lift is getting faster, staying at the same speed or slowing down. [1]
Calculate the magnitude of the reaction of the floor of the lift on the person. [3]

WJEC Unit 2 2018 June Q11

12 marks Moderate -0.8

A vehicle moves along a straight horizontal road. Points $A$ and $B$ lie on the road. As the vehicle passes point $A$, it is moving with constant speed 15 ms$^{-1}$. It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point $B$.

Find the distance $AB$. [3]

The vehicle then reverses with a constant acceleration of 2 ms$^{-2}$ for 8 seconds, followed by a constant deceleration of 1·6 ms$^{-2}$, coming to rest at the point $C$, which is between $A$ and $B$.

Calculate the time it takes for the vehicle to reverse from $B$ to $C$. [4]
Sketch a velocity-time graph for the motion of the vehicle. [3]
Determine the distance $AC$. [2]

WJEC Unit 2 2024 June Q1

3 marks Easy -2.0

An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.

What is the intended population in this context? [1]
What type of sampling is this? [1]
How could the sampling process be improved? [1]

WJEC Unit 2 2024 June Q2

10 marks Moderate -0.8

A baker sells 3-5 birthday cakes per hour on average.

State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution. [1]
Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period. [2]
Calculate the probability that, during a randomly selected 3-hour period, the baker sells more than 10 birthday cakes. [3]
The baker sells a birthday cake at 9:30 a.m. Calculate the probability that the baker will sell the next birthday cake before 10:00 a.m. [3]
Select one of the assumptions in part (a) and comment on its reasonableness. [1]

WJEC Unit 2 2024 June Q3

8 marks Moderate -0.3

The following Venn diagram shows the participation of 100 students in three activities, $A$, $B$, and $C$, which represent athletics, baseball and climbing respectively. \includegraphics{figure_3} For these 100 students, participation in athletics and participation in climbing are independent events.

Show that $x = 10$ and find the value of $y$. [5]
Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing. [3]

WJEC Unit 2 2024 June Q4

11 marks Standard +0.3

A company produces sweets of varying colours. The company claims that the proportion of blue sweets is 13·6%. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

State suitable hypotheses for the test. [1]
1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, 5%.
2. Given that there are 6 blue sweets in the sample of 80, complete the significance test. [5]
Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion? [2]
Now suppose that the proportion of blue sweets is 7%. Find the probability of a Type II error. Interpret your answer in context. [3]

WJEC Unit 2 2024 June Q5

8 marks Easy -1.2

In March 2020, the coronavirus pandemic caused major disruption to the lives of individuals across the world. A newspaper published the following graph from the gov.uk website, along with an article which included the following excerpt. "The daily number of vaccines administered continues to fall. In order to get control of the virus, we need the number of people receiving a second dose of the vaccine to keep rocketing. The fear is it will start to drop off soon, which will leave many people still unprotected." \includegraphics{figure_5}

By referring to the graph, explain how the quote could be misleading. [1]

The daily numbers of second dose vaccines, in thousands, over the period April 1st 2021 to May 31st 2021 are shown in the table below.

Daily number	Midpoint	Frequency	Percentage
of 2nd dose	$x$	$f$
vaccines
(1000s)
$0 \leqslant v < 100$	50	2	3·3
$100 \leqslant v < 200$	150	8	13·1
$200 \leqslant v < 300$	250	10	16·4
$300 \leqslant v < 400$	350	13	21·3
$400 \leqslant v < 500$	450	26	42·6
$500 \leqslant v < 600$	550	2	3·3
Total		61	100

1. Calculate estimates of the mean and standard deviation for the daily number of second dose vaccines given over this period. You may use $\sum x^2 f = 8272500$. [4]
2. Comment on the skewness of these data. [1]
Give a possible reason for the pattern observed in this graph. [1]
State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect. [1]

WJEC Unit 2 2024 June Q6

4 marks Easy -1.2

A ship $S$ is moving with constant velocity $(4\mathbf{i} - 7\mathbf{j})\text{ms}^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north respectively. Find the speed and direction of $S$, giving the direction as a three-figure bearing, correct to the nearest degree. [4]

WJEC Unit 2 2024 June Q7

4 marks Moderate -0.8

The diagram below shows a forklift truck being used to raise two boxes, $P$ and $Q$, vertically. Box $Q$ rests on horizontal forks and box $P$ rests on top of box $Q$. Box $P$ has mass 25 kg and box $Q$ has mass 55 kg. \includegraphics{figure_7}

When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box $Q$ is 820 N. Calculate the magnitude of the acceleration. [3]
Calculate the reaction of box $Q$ on box $P$ when they are moving vertically upwards with constant speed. [1]

WJEC Unit 2 2024 June Q8

7 marks Moderate -0.3

A particle, of mass 4 kg, moves in a straight line under the action of a single force $F$ N, whose magnitude at time $t$ seconds is given by $$F = 12\sqrt{t} - 32 \quad \text{for} \quad t \geqslant 0.$$

Find the acceleration of the particle when $t = 9$. [2]
Given that the particle has velocity $-1\text{ms}^{-1}$ when $t = 4$, find an expression for the velocity of the particle at $t$ s. [3]
Determine whether the speed of the particle is increasing or decreasing when $t = 9$. [2]

WJEC Unit 2 2024 June Q9

9 marks Moderate -0.3

The diagram below shows an object $A$, of mass $2m$ kg, lying on a horizontal table. It is connected to another object $B$, of mass $m$ kg, by a light inextensible string, which passes over a smooth pulley $P$, fixed at the edge of the table. Initially, object $A$ is held at rest so that object $B$ hangs freely with the string taut. \includegraphics{figure_9} Object $A$ is then released.

When object $B$ has moved downwards a vertical distance of 0·4 m, its speed is 1·2 ms$^{-1}$. Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of $B$ is 1·8 ms$^{-2}$. [2]
During the motion, object $A$ experiences a constant resistive force of 22 N. Find the value of $m$ and hence determine the tension in the string. [6]
What assumption did the word 'inextensible' in the description of the string enable you to make in your solution? [1]

WJEC Unit 2 2024 June Q10

11 marks Moderate -0.8

A car, starting from rest at a point $A$, travels along a straight horizontal road towards a point $B$. The distance between points $A$ and $B$ is 1·9 km. Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of 26 ms$^{-1}$. The car continues at 26 ms$^{-1}$ for 1 minute, before decelerating at a constant rate of 0·75 ms$^{-2}$ until it passes the point $B$.

Show that the car travels 156 m while it is accelerating. [2]
1. Work out the distance travelled by the car while travelling at a constant speed. [1]
2. Hence calculate the length of time for which the car is decelerating until it passes the point $B$. [5]
Sketch a displacement-time graph for the motion of the car between $A$ and $B$. [3]

\(X\)	\(256 \leqslant x < 259\)	\(259 \leqslant x < 262\)	\(262 \leqslant x < 265\)	\(265 \leqslant x < 267\)	\(267 \leqslant x < 300\)
Probability	0.05	0.25	0.45	0.20	0.05

\(h ( \mathrm {~cm} )\)	\(98 \cdot 6\)	\(112 \cdot 1\)	\(97 \cdot 9\)	\(110 \cdot 2\)	\(89 \cdot 2\)	\(111 \cdot 7\)	\(87 \cdot 0\)	\(94 \cdot 7\)	\(100 \cdot 4\)	\(104 \cdot 0\)	\(88 \cdot 4\)
\(t ( \mathrm {~cm} )\)	\(48 \cdot 3\)	\(87 \cdot 2\)	\(55 \cdot 2\)	\(68 \cdot 0\)	\(48 \cdot 5\)	\(63 \cdot 2\)	\(49 \cdot 5\)	\(55 \cdot 7\)	\(59 \cdot 1\)	\(64 \cdot 0\)	\(52 \cdot 4\)

Questions Unit 2 (35 questions)

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WJEC Unit 2 2022 June Q2

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