Questions - A-Level Maths

WJEC Unit 4 Specimen Q4

11 marks Moderate -0.3

4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects $2 p$ coins that are less than 6.12 grams or greater than 8.12 grams.

The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
\end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
Assume the distribution of the weight of $2 p$ coins is normally distributed. Calculate the proportion of $2 p$ coins that are rejected by this brand of coin counting machine.
A manager suspects that a large batch of $2 p$ coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.
Summary statistics
Mean
Standard
deviation
Minimum
Lower
quartile
Median
Upper
quartile
Maximum
6.89 0.296 6.45 6.63 6.88 7.08 7.48
i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
ii) Assuming the population standard deviation is 0.357 grams, test at the $1 \%$ significance level whether the mean weight of the $2 p$ coins in this batch is less than 7.12 grams.

WJEC Unit 4 Specimen Q5

7 marks Moderate -0.3

5. A hotel owner in Cardiff is interested in what factors hotel guests think are important when staying at a hotel. From a hotel booking website he collects the ratings for 'Cleanliness', 'Location', 'Comfort' and 'Value for money' for a random sample of 17 Cardiff hotels.
(Each rating is the average of all scores awarded by guests who have contributed reviews using a scale from 1 to 10 , where 10 is 'Excellent'.) The scatter graph shows the relationship between 'Value for money' and 'Cleanliness' for the sample of Cardiff hotels. \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-4_693_1033_749_516}

The product moment correlation coefficient for 'Value for money' and 'Cleanliness' for the sample of 17 Cardiff hotels is 0.895 . Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether this correlation is significant. State your conclusion in context.
The hotel owner also wishes to investigate whether 'Value for money' has a significant correlation with 'Cost per night'. He used a statistical analysis package which provided the following output which includes the Pearson correlation coefficient of interest and the corresponding $p$-value.
Value for money Cost per night
Value for money 1
Cost per night
0.047
$( 0.859 )$
1
Comment on the correlation between 'Value for money' and 'Cost per night'.

WJEC Unit 4 Specimen Q6

8 marks Moderate -0.3

An object of mass 4 kg is moving on a horizontal plane under the action of a constant force $4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}$. At time $t = 0 \mathrm {~s}$, its position vector is $7 \mathbf { i } - 26 \mathbf { j }$ with respect to the origin $O$ and its velocity vector is $- \mathbf { i } + 4 \mathbf { j }$.
1. Determine the velocity vector of the object at time $t = 5 \mathrm {~s}$.
2. Calculate the distance of the object from the origin when $t = 2 \mathrm {~s}$.
3. The diagram below shows an object of weight 160 N at a point $C$, supported by two cables $A C$ and $B C$ inclined at angles of $23 ^ { \circ }$ and $40 ^ { \circ }$ to the horizontal respectively. \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}
1. Find the tension in $A C$ and the tension in $B C$.
2. State two modelling assumptions you have made in your solution.
3. The rate of change of a population of a colony of bacteria is proportional to the size of the population $P$, with constant of proportionality $k$. At time $t = 0$ (hours), the size of the population is 10 .
1. Find an expression, in terms of $k$, for $P$ at time $t$.
2. Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
3. A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .
  Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
4. A body is projected at time $t = 0 \mathrm {~s}$ from a point $O$ with speed $V \mathrm {~ms} ^ { - 1 }$ in a direction inclined at an angle of $\theta$ to the horizontal.
1. Write down expressions for the horizontal and vertical components $x \mathrm {~m}$ and $y \mathrm {~m}$ of its displacement from $O$ at time $t \mathrm {~s}$.
2. Show that the range $R \mathrm {~m}$ on a horizontal plane through the point of projection is given by
$$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
Given that the maximum range is 392 m , find, correct to one decimal place,
i) the speed of projection,
ii) the time of flight,
iii) the maximum height attained.

WJEC Further Unit 5 2022 June Q1

5 marks Moderate -0.5

Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.

$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean $\mu$ and variance $0 \cdot 9$, calculate a $90 \%$ confidence interval for $\mu$.

WJEC Further Unit 5 2022 June Q2

15 marks Challenging +1.2

2. Geraint is a beekeeper. The amounts of honey, $X \mathrm {~kg}$, that he collects annually, from each hive are modelled by the normal distribution $\mathrm { N } \left( 15,5 ^ { 2 } \right)$. At location $A$, Geraint has three hives and at location $B$ he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.

1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location $A$.
2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location $A$.
Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
Find the probability that Geraint collects at least twice as much honey from location B as from location A.

WJEC Further Unit 5 2022 June Q3

8 marks Standard +0.3

3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.

Student	Area	Stress Level
Heledd	North	67
Mair	North	55
Hywel	South	26
Gwyn	South	70
Liam	South	36
Marcin	South	57
Gosia	South	32
Kestutas	North	64
Erica	North	60
Tomos	North	22

State one reason why a Mann-Whitney test is appropriate.
Conduct a Mann-Whitney test at a significance level as close to $5 \%$ as possible. State your conclusion clearly.
How could this investigation be improved?

WJEC Further Unit 5 2022 June Q4

12 marks Standard +0.3

4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.

Calculate an approximate $95 \%$ confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
Give two reasons why the interval is approximate.
Suppose that a $99 \%$ confidence interval is required, and that the width of the interval is to be no greater than $0 \cdot 04$. Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.

WJEC Further Unit 5 2022 June Q5

13 marks Standard +0.3

5. A laboratory carrying out screening for a certain blood disorder claims that the average time taken for test results to be returned is 38 hours. A reporter for a national newspaper suspects that the results take longer, on average, to be returned than claimed by the laboratory. The reporter finds the time, $x$ hours, for 50 randomly selected results, in order to conduct a hypothesis test. The following summary statistics were obtained. $$\sum x = 2163 \quad \sum x ^ { 2 } = 98508$$

Calculate the $p$-value for the reporter's hypothesis test, and complete the test using a $5 \%$ level of significance. Hence write a headline for the reporter to use.
Explain the relevance or otherwise of the Central Limit Theorem to your answer in part (a).
Briefly explain why a random sample is preferable to taking a batch of 50 consecutive results.
On another occasion, the reporter took a different random sample of 10 results.
1. State, with a reason, what type of hypothesis test the reporter should use on this occasion.
2. State one assumption required to carry out this test.

WJEC Further Unit 5 2022 June Q6

8 marks Standard +0.8

6. A zoologist knows that the median body length of adults in a species of fire-bellied toads is 4.2 cm . The zoologist thinks he has discovered a new subspecies of fire-bellied toads. If there is sufficient evidence to suggest the median body length differs from 4.2 cm , he will continue his studies to confirm whether he has discovered a new subspecies. Otherwise, he will abandon his studies on fire-bellied toads. The lengths of 10 randomly selected adult toads from the group being investigated are given below. $$\begin{array} { l l l l l l l l l l } 5 \cdot 0 & 3 \cdot 2 & 4 \cdot 9 & 4 \cdot 0 & 3 \cdot 3 & 4 \cdot 2 & 6 \cdot 1 & 4 \cdot 3 & 4 \cdot 8 & 5 \cdot 9 \end{array}$$ Carry out a suitable Wilcoxon signed rank test at a significance level as close to $1 \%$ as possible and give your conclusion in context.

WJEC Further Unit 5 2022 June Q7

19 marks Challenging +1.2

7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral $A B C D$, where $\widehat { B A D } = \alpha , \widehat { B C D } = \beta$ and $\alpha + \beta = 180 ^ { \circ }$. These angles are measured.
The random variables $X$ and $Y$ denote the measured values, in degrees, of $\widehat { B A D }$ and $\widehat { B C D }$ respectively. You are given that $X$ and $Y$ are independently normally distributed with standard deviation $\sigma$ and means $\alpha$ and $\beta$ respectively.

Calculate, correct to two decimal places, the probability that $X + Y$ will differ from $180 ^ { \circ }$ by less than $\sigma$.
Show that $T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )$ is an unbiased estimator for $\alpha$ and verify that it is a better estimator than $X$ for $\alpha$.
Now consider $T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)$.
1. Show that $T _ { 2 }$ is an unbiased estimator for $\alpha$ for all values of $\lambda$.
2. Find $\operatorname { Var } \left( T _ { 2 } \right)$ in terms of $\lambda$ and $\sigma$.
3. Hence determine the value of $\lambda$ which gives the best unbiased estimator for $\alpha$.

OCR FM1 AS 2021 June Q2

11 marks Standard +0.8

2 \includegraphics[max width=\textwidth, alt={}, center]{60f72141-4a99-4907-93b1-adb0cd66948e-2_211_1276_1427_365} Three particles $A , B$ and $C$ are free to move in the same straight line on a large smooth horizontal surface. Their masses are $1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively (see diagram). The coefficient of restitution in collisions between any two of them is $\frac { 3 } { 4 }$. Initially, $B$ and $C$ are at rest and $A$ is moving with a velocity of $4.0 \mathrm {~ms} ^ { - 1 }$ towards $B$.
a) Show that immediately after the collision between $A$ and $B$ the speed of $B$ is $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
b) Find the velocity of $A$ immediately after this collision. $B$ subsequently collides with $C$.
c) Find, in terms of $m$, the velocity of $B$ after its collision with $C$.
d) Given that the direction of motion of $B$ is reversed by the collision with $C$, find the range of possible values of $m$. The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now $30 \mathrm {~ms} ^ { - 1 }$. The resistance to motion of the trailer may also be assumed constant.
(b) Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is $15 \mathrm {~ms} ^ { - 1 }$ and their acceleration is $0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
(c) (i) Find the power of the engine of the car at this instant.
(ii) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of $10 \mathrm {~ms} ^ { - 1 }$. When the speed is $10 \mathrm {~ms} ^ { - 1 }$ or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
(d) Without carrying out any further calculations,

explain whether the time taken to attain a speed of $20 \mathrm {~m} ^ { - 1 }$ would be predicted to be lower, the same or higher using the refined model compared with the original model,
explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

OCR FM1 AS 2021 June Q2

8 marks Standard +0.3

2 A particle moves in a straight line with constant acceleration. Its initial and final velocities are $u$ and $v$ respectively and at time $t$ its displacement from its starting position is $s$. An equation connecting these quantities is $s = k \left( u ^ { \alpha } + v ^ { \beta } \right) t ^ { \gamma }$, where $k$ is a dimensionless constant.

Use dimensional analysis to find the values of $\alpha , \beta$ and $\gamma$.
By considering the case where the acceleration is zero, determine the value of $k$.

OCR FM1 AS 2021 June Q3

10 marks Standard +0.8

3
Two particles $A$ and $B$ are connected by a light inextensible string. Particle $A$ has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre $O$. The string passes through a small smooth hole at $O$. Particle $B$ moves in a horizontal circle in such a way that it is always vertically below $A$. The angle that the portion of the string below the table makes with the downwards vertical through $O$ is $\theta$, where $\cos \theta = \frac { 4 } { 5 }$ (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{75f629e7-969d-43ae-8222-031875ae54ae-02_453_696_1571_552}

Find the time taken for the particles to perform a complete revolution.
Find the mass of $B$.

OCR FM1 AS 2021 June Q1

6 marks Standard +0.3

1 \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-02_494_390_251_255} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point $O$. A small bead $B$ is threaded onto the wire. $B$ is held with $O B$ vertical and is then projected horizontally with an initial speed of $8.4 \mathrm {~ms} ^ { - 1 }$ (see diagram).

Find the speed of $B$ at the instant when $O B$ makes an angle of 0.8 radians with the downward vertical through $O$.
Determine whether $B$ has sufficient energy to reach the point on the wire vertically above $O$.

OCR FM1 AS 2021 June Q2

11 marks Moderate -0.3

2 A student is studying the speed of sound, $u$, in a gas under different conditions.
He assumes that $u$ depends on the pressure, $p$, of the gas, the density, $\rho$, of the gas and the wavelength, $\lambda$, of the sound in the relationship $u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }$, where $k$ is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)

Use the fact that density is mass per unit volume to find $[ \rho ]$.
Given that the units of $p$ are $\mathrm { Nm } ^ { - 2 }$, determine the values of $\alpha , \beta$ and $\gamma$.
Comment on what the value of $\gamma$ means about how fast sounds of different wavelengths travel through the gas. The student carries out two experiments, $A$ and $B$, to measure $u$. Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of $u$ in experiment $B$ is double the value in experiment $A$.
By what factor has the density of the gas in experiment $A$ been multiplied to give the density of the gas in experiment $B$ ? Particles $A$ of mass $2 m$ and $B$ of mass $m$ are on a smooth horizontal floor. $A$ is moving with speed $u$ directly towards a vertical wall, and $B$ is at rest between $A$ and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-03_211_795_285_244} $A$ collides directly with $B$. The coefficient of restitution in this collision is $\frac { 1 } { 2 }$. $B$ then collides with the wall, rebounds, and collides with $A$ for a second time.
1. Show that the speed of $B$ after its second collision with $A$ is $\frac { 1 } { 2 } u$. The first collision between $A$ and $B$ occurs at a distance $d$ from the wall. The second collision between $A$ and $B$ occurs at a distance $\frac { 1 } { 5 } d$ from the wall.
2. Find the coefficient of restitution for the collision between $B$ and the wall.

OCR FM1 AS 2021 June Q1

7 marks Standard +0.3

1 A particle $A$ of mass 3.6 kg is attached by a light inextensible string to a particle $B$ of mass 2.4 kg . $A$ and $B$ are initially at rest, with the string slack, on a smooth horizontal surface. $A$ is projected directly away from $B$ with a speed of $7.2 \mathrm {~ms} ^ { - 1 }$.

Calculate the speed of $A$ after the string becomes taut.
Find the impulse exerted on $A$ at the instant that the string becomes taut.
Find the loss in kinetic energy as a result of the string becoming taut.

OCR FM1 AS 2021 June Q2

11 marks Standard +0.3

2 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight horizontal road using maximum power, its acceleration is $3.3 \mathrm {~ms} ^ { - 2 }$. In an initial model of the motion of the car it is assumed that the resistance to motion is constant.

Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be $21.6 \mathrm {~ms} ^ { - 1 }$.
Explain what this value means about the models used in parts (a) and (b). \includegraphics[max width=\textwidth, alt={}, center]{aa25b8a6-9a5a-4de2-9534-18db8a175c34-03_583_378_169_255} As shown in the diagram, $A B$ is a long thin rod which is fixed vertically with $A$ above $B$. One end of a light inextensible string of length 1 m is attached to $A$ and the other end is attached to a particle $P$ of mass $m _ { 1 } \mathrm {~kg}$. One end of another light inextensible string of length 1 m is also attached to $P$. Its other end is attached to a small smooth ring $R$, of mass $m _ { 2 } \mathrm {~kg}$, which is free to move on $A B$. Initially, $P$ moves in a horizontal circle of radius 0.6 m with constant angular velocity $\omega \mathrm { rads } ^ { - 1 }$. The magnitude of the tension in string $A P$ is denoted by $T _ { 1 } \mathrm {~N}$ while that in string $P R$ is denoted by $T _ { 2 } \mathrm {~N}$.
1. By considering forces on $R$, express $T _ { 2 }$ in terms of $m _ { 2 }$.
2. Show that
  1. $T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)$,
  2. $\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }$.
  3. Deduce that, in the case where $m _ { 1 }$ is much bigger than $m _ { 2 } , \omega \approx 3.5$. In a different case, where $m _ { 1 } = 2.5$ and $m _ { 2 } = 2.8 , P$ slows down. Eventually the system comes to rest with $P$ and $R$ hanging in equilibrium.
3. Find the total energy lost by $P$ and $R$ as the angular velocity of $P$ changes from the initial value of $\omega \mathrm { rads } ^ { - 1 }$ to zero.

OCR FM1 AS 2021 June Q4

14 marks Standard +0.3

4
2.4
&

B1 for each of two correct statements about the models.

If commenting on the accuracy of (a), must emphasise that (a) is very inaccurate or at least quite inaccurate

Do not allow e.g.

- model (a) is not very effective

- Neither model is accurate

- (a) and (b) are not very accurate

Clear comparison between the accuracy of the two models (must emphasise that (b) is fairly accurate or considerably more accurate than (a)), or other suitable distinct second comment

Do not allow e.g.

- model (b) is more accurate than model (a)

- (b) is not accurate

Do not allow statement claiming that resistance is proportional to speed, or to speed ${ } ^ { 2 }$

Suitable comments for (a):

- is very inaccurate

- predicted speed is nearly three times the actual value

- constant resistance is not a suitable model

- both models underestimate the resistance (as top speed is lower than expected)

For the linear model (b)

- is fairly accurate (but probably underestimates the resistance at higher speeds)

- resistance is not proportional to speed but is a much better model than constant resistance

3

(a)

$T _ { 2 } \cos \theta = m _ { 2 } g$ $T _ { 2 } = \frac { m _ { 2 } \times 9.8 } { 0.8 } = 12.25 m _ { 2 }$

M1

A1

[2]

1.1a

1.1

Resolving $T _ { 2 }$ vertically and balancing forces on $R$

Do not allow extra forces present

Allow use of g, e.g. $\frac { 5 } { 4 } g m _ { 2 }$

In this solution $\theta$ is the angle between $R P$ and $R A$ Sin may be seen instead if $\theta$ is measured horizontally.

Do not allow incomplete expressions e.g. $\frac { m _ { 2 } g } { \sin 53.13 }$

3

(b)

(i)

\(\begin{aligned}

T _ { 2 } \cos \theta + m _ { 1 } g = T _ { 1 } \cos \theta

T _ { 1 } = T _ { 2 } + \frac { 9.8 m _ { 1 } } { 0.8 } =

\qquad 12.25 m _ { 2 } + 12.25 m _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right) \end{aligned}\)

M1

A1

[2]

3.1b

2.1

Vertical forces on $P$; 3 terms including resolving of $T _ { 1 }$; allow sign error

AG Dividing by $\cos \theta ( = 0.8 )$, substituting their $T _ { 2 }$ and rearranging

Allow 12.25 instead of $\frac { 49 } { 4 }$

Or $T _ { 1 } \cos \theta = m _ { 1 } g + m _ { 2 } g$ (equation for the system as a whole)

At least one intermediate step must be seen

3

(b)

(ii)

\(\begin{aligned}

T _ { 1 } \sin \theta + T _ { 2 } \sin \theta = m _ { 1 } a

12.25 \left( m _ { 1 } + m _ { 2 } \right) \times 0.6 + 12.25 m _ { 2 } \times 0.6 = m _ { 1 } \times 0.6 \omega ^ { 2 }

\omega ^ { 2 } = \frac { 7.35 m _ { 1 } + 14.7 m _ { 2 } } { 0.6 m _ { 1 } } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } } \end{aligned}\)

M1

A1

[3]

3.1b

1.1

2.1

NII horizontally for $P ; 3$ terms including resolving of tensions; allow sign error

Substituting for $T _ { 1 }$, their $T _ { 2 } , \sin \theta$ and $\alpha$

AG Must see an intermediate step

Could see $a$ or $0.6 \omega ^ { 2 }$ or $\frac { v ^ { 2 } } { 0.6 }$ or $\omega ^ { 2 } r$ or $\frac { v ^ { 2 } } { r } \sin \theta = 0.6$

must be $a = 0.6 \omega ^ { 2 }$

3

(c)

\(\begin{aligned}

\text { E.g } m _ { 1 } \gg m _ { 2 } \Rightarrow \frac { 2 m _ { 2 } } { m _ { 1 } } \approx 0 \text { or } \frac { 49 m _ { 2 } } { 4 m _ { 1 } } \approx 0

\omega \approx \sqrt { \frac { 49 m } { 4 m } } = 3.5 \end{aligned}\)

M1 A1

[2]

1.1

Allow argument such as if $m _ { 1 } \gg m _ { 2 }$ then $m _ { 1 } + 2 m _ { 2 } \approx m _ { 1 }$

AG $m$ may be missing

SC1 for result following argument that $m _ { 2 }$ is negligible (by comparison with $m _ { 1 }$ ) without justification, or using trial values of $m _ { 1 }$ and $m _ { 2 }$ with $m _ { 1 } \gg m _ { 2 }$.

Do not allow the assumption that $m _ { 2 } = 0$

If using trial values, $m _ { 1 }$ must be at least $70 \times m _ { 2 }$ to give $\omega = 3.5$ to 1 dp .

3

\multirow{3}{*}{(d)}

$v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }$

Final energy $= 2.5 \times g \times 1$ $\text { Initial } \mathrm { KE } = \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }$

Initial PE $= 2.5 \times g \times 1.2 + 2.8 \times g \times 0.4$

Energy loss $= 17.8605 + 40.376 - 24.5 = 33.7365$

M1

B1

M1

A1

1.2

1.1

3.2a

Use of $v = r \omega$ with values for $m _ { 1 }$ and $m _ { 2 }$

(Assuming zero PE level at 2 m below $A$; other values possible)

Do not allow use of $\omega = 3.5$

oe with different zero PE level awrt 33.7

( $v = 3.78 , v ^ { 2 } = 14.2884$ )

NB $\omega = 6.3$ (24.5)

(17.8605)

(40.376)

Alternate method $v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }$

Initial KE $= \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }$

$\triangle P E$ for $m _ { 1 } = \pm 2.5 \times 9.8 \times ( 0.8 - 1 )$

$\triangle P E$ for $m _ { 2 } = \pm 2.8 \times 9.8 ( 1.6 - 2 )$

Energy loss $= 17.8605 + 4.9 + 10.976$

M1

A1

Use of $v = r \omega$ with values for $m _ { 1 }$ and $m _ { 2 }$

Or $- \triangle P E$ $= 2.5 \times 9.8 \times 0.2 + 2.8 \times 9.8 \times 0.4$

awrt 33.7

( $v = 3.78 , v ^ { 2 } = 14.2884$ ) $\mathrm { NB } \omega = 6.3$

(17.8605)

$( \pm 4.9 )$

$( \pm 10.976 )$

$( \pm 15.876 )$

Or 15.876 + 17.8605

[5]

OCR FM1 AS 2021 June Q1

7 marks Moderate -0.8

1 A particle $P$ of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of $2.4 \mathrm {~ms} ^ { - 1 }$ when it strikes a vertical wall directly. It rebounds at a speed of $1.6 \mathrm {~ms} ^ { - 1 }$.

Find the coefficient of restitution between $P$ and the wall.
Determine the impulse applied to $P$ by the wall, stating its direction.
Find the loss of kinetic energy of $P$ as a result of the collision.
State, with a reason, whether the collision is perfectly elastic.

OCR FM1 AS 2021 June Q2

14 marks Moderate -0.3

2 A particle $P$ of mass 2.4 kg is moving in a straight line $O A$ on a horizontal plane. $P$ is acted on by a force of magnitude 30 N in the direction of motion. The distance $O A$ is 10 m . \begin{enumerate}[label=(\alph*)] \item Find the work done by this force as $P$ moves from $O$ to $A$. The motion of $P$ is resisted by a constant force of magnitude $R \mathrm {~N}$. The velocity of $P$ increases from $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $O$ to $18 \mathrm {~ms} ^ { - 1 }$ at $A$. \item Find the value of $R$. \item Find the average power used in overcoming the resistance force on $P$ as it moves from $O$ to $A$. When $P$ reaches $A$ it collides directly with a particle $Q$ of mass 1.6 kg which was at rest at $A$ before the collision. The impulse exerted on $Q$ by $P$ as a result of the collision is 17.28 Ns . \item

Find the speed of $Q$ after the collision.
Hence show that the collision is inelastic. It is required to model the motion of a car of mass $m \mathrm {~kg}$ travelling at a constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ around a circular portion of banked track. The track is banked at $30 ^ { \circ }$ (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
For a particular portion of banked track, $r = 24$.
(b) Find the value of $v$ as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
(c) Explain

OCR FP1 AS 2021 June Q2

12 marks Standard +0.3

2
The position vector of point $A$ is $\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }$.
The line $l$ passes through $A$ and is perpendicular to $\mathbf { a }$.

Determine the shortest distance between the origin, $O$, and $l$. $l$ is also perpendicular to the vector $\mathbf { b }$ where $\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }$.
Find a vector which is perpendicular to both $\mathbf { a }$ and $\mathbf { b }$.
Write down an equation of $l$ in vector form. $P$ is a point on $l$ such that $P A = 2 O A$.
Find angle $P O A$ giving your answer to 3 significant figures. $C$ is a point whose position vector, $\mathbf { c }$, is given by $\mathbf { c } = p \mathbf { a }$ for some constant $p$. The line $m$ passes through $C$ and has equation $\mathbf { r } = \mathbf { c } + \mu \mathbf { b }$. The point with position vector $9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }$ lies on $m$.
Find the value of $p$. \section*{In this question you must show detailed reasoning.} You are given that $\alpha , \beta$ and $\gamma$ are the roots of the equation $5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$.
1. Find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.
2. Find a cubic equation whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$ giving your answer in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d = 0$ where $a , b , c$ and $d$ are integers.

OCR FP1 AS 2021 June Q4

6 marks Standard +0.3

4 In this question you must show detailed reasoning. $\mathbf { M }$ is the matrix $\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)$.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)$, for any positive integer $n$.

OCR FP1 AS 2021 June Q1

4 marks Easy -1.3

1 Matrices $\mathbf { P }$ and $\mathbf { Q }$ are given by $\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)$ and $\mathbf { Q } = ( ( 1 + k ) - 1 )$ where $k$ is a constant.
Exactly one of statements A and B is true.
Statement A: $\quad \mathbf { P }$ and $\mathbf { Q }$ (in that order) are conformable for multiplication.
Statement B: $\quad \mathbf { Q }$ and $\mathbf { P }$ (in that order) are conformable for multiplication.

State, with a reason, which one of A and B is true.
Find either $\mathbf { P Q }$ or $\mathbf { Q P }$ in terms of $k$.

OCR FP1 AS 2021 June Q2

14 marks Standard +0.3

2 In this question you must show detailed reasoning. You are given that $\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185$ and that $2 + \mathrm { i }$ is a root of the equation $\mathrm { f } ( z ) = 0$.

Express $\mathrm { f } ( z )$ as the product of two quadratic factors with integer coefficients.
Solve $\mathrm { f } ( z ) = 0$. Two loci on an Argand diagram are defined by $C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}$ and $C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}$ where $r _ { 1 } > r _ { 2 }$. You are given that two of the points representing the roots of $\mathrm { f } ( z ) = 0$ are on $C _ { 1 }$ and two are on $C _ { 2 } \cdot R$ is the region on the Argand diagram between $C _ { 1 }$ and $C _ { 2 }$.
Find the exact area of $R$.
$\omega$ is the sum of all the roots of $\mathrm { f } ( z ) = 0$. Determine whether or not the point on the Argand diagram which represents $\omega$ lies in $R$.

OCR FP1 AS 2021 June Q3

5 marks Standard +0.8

3 A transformation T is represented by the matrix $\mathbf { T }$ where $\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)$. A quadrilateral $Q$, whose area is 12 units, is transformed by T to $Q ^ { \prime }$. Find the smallest possible value of the area of $Q ^ { \prime }$.

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