Questions - WJEC - A-Level Maths

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WJEC Further Unit 6 2023 June Q1

13 marks Challenging +1.2

The diagram shows a uniform rod $A B$, of length 8 m and mass 23 kg , in limiting equilibrium with its end $A$ on rough horizontal ground and point $C$ resting against a smooth fixed cylinder. The rod is inclined at an angle of $30 ^ { \circ }$ to the ground. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is $\frac { 2 } { 3 }$.

Calculate the magnitude of the normal reaction at $C$ and the magnitude of the normal reaction to the ground at $A$.
Find the length $A C$.
Suppose instead that the rod is non-uniform with its centre of mass closer to $A$ than to $B$. Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.

WJEC Further Unit 6 2023 June Q2

7 marks Challenging +1.2

2. You are given that the centre of mass of a uniform solid cone of height $h$ and base radius $r$ is at a height of $\frac { 1 } { 4 } h$ above its base. The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius $3 x$ and height $6 y$, and removing a smaller cone, of base radius $x$, with the same vertex. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575} Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is $\frac { 18 } { 13 } y$.

WJEC Further Unit 6 2023 June Q3

13 marks Standard +0.3

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

Write down the period and amplitude of the motion.
Let $x \mathrm {~m}$ denote the height of water above mean level $t$ hours after 5a.m. Find an expression for $x$ in terms of $t$.
The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
Calculate the rate at which the level of water is falling at 2 p.m.

WJEC Further Unit 6 2023 June Q4

15 marks Challenging +1.2

4. The diagram shows three light rods $A B , B C$ and $C A$ rigidly joined together so that $A B C$ is a right-angled triangle with $A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}$ and $\widehat { A B } = 90 ^ { \circ }$. The rods support a uniform lamina, of density $2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }$, in the shape of a quarter circle $A D E$ with radius 12 cm and centre at the vertex $A$. Three particles are attached to $B C$ : one at $B$, one at $C$ and one at $F$, the midpoint of $B C$. The masses at $C , F$ and $B$ are $50 m \mathrm {~kg} , 30 m \mathrm {~kg}$ and $20 m \mathrm {~kg}$ respectively. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}

Calculate the distance of the centre of mass of the system from
1. $A C$,
2. $A B$.
When the system is freely suspended from a point $P$ on $A C$, it hangs in equilibrium with $A B$ vertical. Write down the length $A P$.
When the system is freely suspended from a point $Q$ on $A D$, it hangs in equilibrium with $Q B$ making an angle of $60 ^ { \circ }$ with the vertical. Find the distance $A Q$.

WJEC Further Unit 6 2023 June Q5

16 marks Challenging +1.2

5. In this question, $\mathbf { i }$ and $\mathbf { j }$ represent unit vectors due east and due north respectively. Two smooth spheres $P$ and $Q$, of equal radii, are moving on a smooth horizontal surface. The mass of $P$ is 2 kg and the mass of $Q$ is 6 kg . The velocity of $P$ is $( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the velocity of $Q$ is $( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. At a particular instant, $Q$ is positioned 12 m east and 48 m south of $P$.

Prove that $P$ and $Q$ will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector $\mathbf { j }$. Immediately after the collision, sphere $Q$ has speed $5 \mathrm {~ms} ^ { - 1 }$.
Determine the coefficient of restitution between the spheres and hence calculate the velocity of $P$ immediately after the collision.
Find the magnitude of the impulse required to stop sphere $P$ after the collision.

WJEC Further Unit 6 2023 June Q6

16 marks Challenging +1.8

6. The diagram on the left shows a train of mass 50 tonnes approaching a buffer at the end of a straight horizontal railway track. The buffer is designed to prevent the train from running off the end of the track. The buffer may be modelled as a light horizontal spring $A B$, as shown in the diagram on the right, which is fixed at the end $A$. The train strikes the buffer so that $P$ makes contact with $B$ at $t = 0$ seconds. While $P$ is in contact with $B$, an additional resistive force of $250000 v \mathrm {~N}$ will oppose the motion of the train, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the train at time $t$ seconds. The spring has natural length 1 m and modulus of elasticity 312500 N . At time $t$ seconds, the compression of the spring is $x$ metres. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-7_358_1506_824_283}

Show that, while $P$ is in contact with $B$, $x$ satisfies the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 20 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 0$$
Given that, when $P$ first makes contact with $B$, the speed of the train is $U \mathrm {~ms} ^ { - 1 }$, find an expression for $x$ in terms of $U$ and $t$.
When the train comes to rest, the compression of the buffer is 0.3 m . Determine the speed of the train when it strikes the buffer.
State which type of damping is described by the motion of $P$. Give a reason for your answer.

WJEC Further Unit 6 Specimen Q1

14 marks Challenging +1.2

A ball of mass 0.4 kg is thrown vertically upwards from a point $O$ with initial speed $17 \mathrm {~ms} ^ { - 1 }$. When the ball is at a height of $x \mathrm {~m}$ above $O$ and its speed is $v \mathrm {~ms} ^ { - 1 }$, the air resistance acting on the ball has magnitude $0.01 v ^ { 2 } \mathrm {~N}$.
1. Show that, as the ball is ascending, $v$ satisfies the differential equation
$$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
Find an expression for $v$ in terms of $x$.
Calculate, correct to two decimal places, the greatest height of the ball.
State, with a reason, whether the speed of the ball when it returns to $O$ is greater than $17 \mathrm {~ms} ^ { - 1 }$, less than $17 \mathrm {~ms} ^ { - 1 }$ or equal to $17 \mathrm {~ms} ^ { - 1 }$.

WJEC Further Unit 6 Specimen Q2

15 marks Challenging +1.2

2. (a) Prove that the centre of mass of a uniform solid cone of height $h$ and base radius $b$ is at a height of $\frac { 1 } { 4 } h$ above its base.
(b) A uniform solid cone $C _ { 1 }$ has height 3 m and base radius 2 m . A smaller cone $C _ { 2 }$ of height 2 m and base radius 1 m is contained symmetrically inside $C _ { 1 }$. The bases of $C _ { 1 }$ and $C _ { 2 }$ have a common centre and the axis of $C _ { 2 }$ is part of the axis of $C _ { 1 }$. If $C _ { 2 }$ is removed from $C _ { 1 }$, show that the centre of mass of the remaining solid is at a distance of $\frac { 11 } { 5 } \mathrm {~m}$ from the vertex of $C _ { 1 }$.
(c) The remaining solid is suspended from a string which is attached to a point on the outer curved surface at a distance of $\frac { 1 } { 3 } \sqrt { 13 } \mathrm {~m}$ from the vertex of $C _ { 1 }$. Given that the axis of symmetry is inclined at an angle of $\alpha$ to the vertical, find $\tan \alpha$.

WJEC Further Unit 6 Specimen Q3

10 marks Standard +0.8

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of $20 \mathrm {~ms} ^ { - 1 }$. At time $t \mathrm {~s}$ the body experiences a resistance of magnitude $( 0.2 + 0.03 v ) \mathrm { N }$ where $v \mathrm {~ms} ^ { - 1 }$ is its speed.

Show that $v$ satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
Find an expression for $t$ in terms of $v$.
Calculate, to the nearest second, the time taken for the body to come to rest.

WJEC Further Unit 6 Specimen Q4

11 marks Challenging +1.2

4. The diagram shows a uniform lamina consisting of a rectangular section $G P Q E$ with a semi-circular section EFG of radius 4 cm . Quadrants $A P B$ and $C Q D$ each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

Write down the distance of the centre of mass of the lamina $A B C D E F G$ from $A G$.
Determine the distance of the centre of mass of the lamina $A B C D E F G$ from $B C$.
The lamina $A B C D E F G$ is suspended freely from the point $E$ and hangs in equilibrium. Calculate the angle EG makes with the vertical.

WJEC Further Unit 6 Specimen Q5

13 marks Standard +0.3

5. A particle $A$, of mass $m \mathrm {~kg}$, has position vector $11 \mathbf { i } + 6 \mathbf { j }$ and a velocity $2 \mathbf { i } + 7 \mathbf { j }$. At the same moment, second particle $B$, of mass $2 m \mathrm {~kg}$, has position vector $7 \mathbf { i } + 10 \mathbf { j }$ and a velocity $5 \mathbf { i } + 4 \mathbf { j }$.

If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
Determine the impulse exerted by $A$ on $B$.
Calculate the loss of kinetic energy caused by the collision.

WJEC Further Unit 6 Specimen Q6

17 marks Standard +0.3

6. The diagram shows a playground ride consisting of a seat $P$, of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m . \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity $\lambda$.

Find the value of $\lambda$. The seat $P$ is now pushed vertically downwards a further 0.05 m and is then released from rest.
Show that $P$ makes Simple Harmonic oscillations of period $\frac { \pi } { 7 }$ and write down the amplitude of the motion.
Find the maximum speed of $P$.
Calculate the speed of $P$ when it is at a distance 0.03 m from the equilibrium position.
Find the distance of $P$ from the equilibrium position 1.6 s after it is released.[3]
State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.

WJEC Unit 4 Specimen Q1

6 marks Moderate -0.3

It is known that $4 \%$ of a population suffer from a certain disease. When a diagnostic test is applied to a person with the disease, it gives a positive response with probability 0.98 . When the test is applied to a person who does not have the disease, it gives a positive response with probability 0.01 .
1. Using a tree diagram, or otherwise, show that the probability of a person who does not have the disease giving a negative response is 0.9504 .
The test is applied to a randomly selected member of the population.
Find the probability that a positive response is obtained.
Given that a positive response is obtained, find the probability that the person has the disease.

WJEC Unit 4 Specimen Q2

9 marks Challenging +1.2

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability $p$ of hitting the target. Successive shots are independent.

Determine the probability that Jeff wins the game
i) with his first shot,
ii) with his second shot.
Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
Find the range of values of $p$ for which Mary is more likely to win the game than Jeff.

WJEC Unit 4 Specimen Q3

7 marks Standard +0.8

3. A string of length 60 cm is cut a random point.

Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than $100 \mathrm {~cm} ^ { 2 }$.

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}
4. Find the tension in $A C$ and the tension in $B C$.
5. State two modelling assumptions you have made in your solution.
6. 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 .
7. Find an expression, in terms of $k$, for $P$ at time $t$.
8. Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
9. 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.
10. 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.
11. Write down expressions for the horizontal and vertical components $x \mathrm {~m}$ and $y \mathrm {~m}$ of its displacement from $O$ at time $t \mathrm {~s}$.
12. 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 $\alpha$.

Questions — WJEC (504 questions)

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WJEC Further Unit 6 2023 June Q1

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WJEC Further Unit 6 2023 June Q6

WJEC Further Unit 6 Specimen Q1

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WJEC Further Unit 6 Specimen Q6

WJEC Unit 4 Specimen Q1

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WJEC Unit 4 Specimen Q3

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WJEC Unit 4 Specimen Q5

WJEC Unit 4 Specimen Q6

WJEC Further Unit 5 2022 June Q1

WJEC Further Unit 5 2022 June Q2

WJEC Further Unit 5 2022 June Q3

WJEC Further Unit 5 2022 June Q4

WJEC Further Unit 5 2022 June Q5

WJEC Further Unit 5 2022 June Q6

WJEC Further Unit 5 2022 June Q7