Questions - WJEC - A-Level Maths

Number of sixes	0	1	2	3
Frequency	625	384	81	10

Using these results, conduct a goodness of fit test and draw an appropriate conclusion.

WJEC Further Unit 2 2022 June Q6

6. An online survey on the use of social media asked the following question: \begin{displayquote} "Do you use any form of social media?" \end{displayquote} The results for a total of 1953 respondents are shown in the table below.

	Age in years
Use social media	18-29	30-49	50-64	65 or older	Total
Yes	310	412	348	196	1266
No	42	116	196	333	687
Total	352	528	544	529	1953

To test whether there is a relationship between social media use and age, a significance test is carried out at the $5 \%$ level.

State the null and alternative hypotheses.
Show how the expected frequency $228 \cdot 18$ is calculated in the table below.
Expected values Age in years
Use social media 18-29 30-49 50-64 65 or older
Yes $228 \cdot 18$ $342 \cdot 27$ 352.64 342.92
No 123.82 185.73 191.36 186.08
Determine the value of $s$ in the table below.
Chi-squared contributions Age in years
Use social media 18-29 30-49 50-64 65 or older
Yes 29.34 $s$ 0.06 62.94
No 54.07 26-18 0.11 115.99
Complete the significance test, showing all your working.
A student, analysing these data on a spreadsheet, obtains the following output.
\includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-5_202_1271_445_415} Explain why the student must have made an error in calculating the $p$-value.

WJEC Further Unit 2 2022 June Q7

7. Data from a large dataset shows the percentage of children enrolled in secondary education and the percentage of the adult population who are literate. The following graphs show data from 30 randomly selected regions from each of the Arab World, Africa and Asia. In each case, the least squares regression line of '\% Literacy' on '\% Enrolled in Secondary Education' is shown.
\includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_682_1200_584_395} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Africa} \includegraphics[alt={},max width=\textwidth]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_623_1191_1548_397}

\end{figure} \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-7_665_1200_331_434}

Calculate the equation of the least squares regression line of '\% Literacy' ( $y$ ) on '\% Enrolled in Secondary Education' ( $x$ ) for Asia, given the following summary statistics. $$\begin{array} { l l l } \sum x = 2850.836 & \sum y = 2738.656 & S _ { x x } = 88.42142
S _ { y y } = 204.733 & S _ { x y } = 96.60984 & n = 30 \end{array}$$
The Arab World, Africa and Asia each contain a region where $70 \%$ are enrolled in secondary education. The three regression lines are used to estimate the corresponding \% Literacy. Which of these estimates is likely to be the most reliable? Clearly explain your reasoning. \section*{END OF PAPER}

WJEC Further Unit 2 2023 June Q1

The random variable $X$ has mean 17 and variance 64 . The independent random variable $Y$ has mean 10 and variance 16 . Find the value of
1. $\mathrm { E } ( 4 Y - 2 X + 1 )$,
2. $\quad \operatorname { Var } ( 4 Y - 5 X + 3 )$,
3. $\mathrm { E } \left( X ^ { 2 } Y \right)$.
4. For a set of 30 pairs of observations of the variables $x$ and $y$, it is known that $\sum x = 420$ and $\sum y = 240$. The least squares regression line of $y$ on $x$ passes through the point with coordinates $( 19,20 )$.
5. Show that the equation of the regression line of $y$ on $x$ is $y = 2 \cdot 4 x - 25 \cdot 6$ and use it to predict the value of $y$ when $x = 26$.
6. State two reasons why your prediction in part (a) may not be reliable.
7. It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
8. Find the probability that the lifetime of a randomly selected hair dryer is between 1.8 and $2 \cdot 5$ years.
9. Given that $20 \%$ of hair dryers have a lifetime of at least $k$ years, find the value of $k$.
10. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers.
11. State one assumption that you have made in part (c).
12. A continuous random variable $X$ has cumulative distribution function $F$ given by
$$F ( x ) = \begin{cases} 0 & \text { for } x < 0
\frac { 1 } { 4 } x & \text { for } 0 \leqslant x \leqslant 2
\frac { 1 } { 480 } x ^ { 4 } + \frac { 7 } { 15 } & \text { for } 2 < x \leqslant b
1 & \text { for } x > b \end{cases}$$
Show that $b = 4$.
Find $\mathrm { P } ( X \leqslant 2 \cdot 5 )$.
Write down the value of the lower quartile of $X$.
Find the value of the upper quartile of $X$.
Find, correct to three significant figures, the value of $k$ that satisfies the equation $\mathrm { P } ( X > 3 \cdot 5 ) = \mathrm { P } ( X < k )$.

WJEC Further Unit 2 2023 June Q5

5. (a) Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.
(b) A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.

Tractor	Farmer's rank	PTO (horsepower)	Price ( $\boldsymbol { \pounds } \mathbf { 1 0 0 0 s }$ )
A	1	77.5	80
B	6	87.9	45
C	5	$53 \cdot 0$	47
D	4	$41 \cdot 0$	53
E	2	$112 \cdot 0$	60
F	3	$90 \cdot 0$	61

Spearman's rank correlation coefficient between the farmer's ranks and the price is 0.9429 .

Test at the $5 \%$ significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm.
Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO.
How should the tractor salesman interpret the results in (i) and (ii)?

WJEC Further Unit 2 2023 June Q6

6. A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that $10 \%$ of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution $B ( 20,0 \cdot 1 )$. The numbers of boats that were not taken for 110 randomly selected days are given below.

Number of boats not taken

5 or

Frequency

State suitable hypotheses to carry out a goodness of fit test.
Here is part of the table for a $\chi ^ { 2 }$ goodness of fit test on the data.
Number of boats not taken 0 1 2 3 4 5 or more
Observed 10 35 29 25 8 3
Expected $f$ 29.72 $g$ 20.91 9.88 $4 \cdot 75$
1. Calculate the values of $f$ and $g$.
2. By completing the test, give the conclusion the manager should reach. The cost of hiring a boat is $\pounds 15$. Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give $\pounds 20$ (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
2. Hence calculate the expected daily net income for the company following the manager's decision.
Is the manager justified in her decision? Give a reason for your answer.

WJEC Further Unit 2 2024 June Q1

6 marks

Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.
1. (i) Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
  (ii) Justify any distribution you have used in answering (a)(i).
2. On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
  1. find the expected time he catches his first fish,
  2. calculate the probability that he will not catch a fish by 3 pm .
3. On average, only $2 \%$ of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
  [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
4. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
PLEASE DO NOT WRITE ON THIS PAGE

WJEC Further Unit 2 2024 June Q2

2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, $X$ minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function. $$F ( x ) = \begin{cases} 0 & x < - 2
\frac { x + 2 } { 5 } & - 2 \leqslant x < 1
\frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2
1 & x > 2 \end{cases}$$

Calculate the upper quartile for the 'allowed time'.
Find $f ( x )$, the probability density function, for all values of $x$.
1. Calculate the mean 'allowed time'.
2. Interpret your answer in context.

WJEC Further Unit 2 2024 June Q3

A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean $2 \cdot 2$.

Carry out a chi-squared goodness of fit test, using a $10 \%$ significance level.
A chi-squared goodness of fit test for the Poisson distribution with mean $2 \cdot 5$ is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.

WJEC Further Unit 2 2024 June Q4

4. An author poses the following question: Does using cash for transactions affect people's financial behaviour?
She collects data on 'Cash transactions as a \% of all transactions' and 'Household debt as a $\%$ of net disposable income' from a random sample of 25 countries. The table below shows the data she collected. There are missing values, $p$ and $q$, for Malta and Denmark respectively.

Country	Cash transactions as a \% of all transactions $\boldsymbol { x }$	Household debt as a \% of net disposable income $\boldsymbol { y }$	Country	Cash transactions as a \% of all transactions $\boldsymbol { x }$	Household debt as a \% of net disposable income $\boldsymbol { y }$
Malta	92	$p$	France	68	120
Mexico	90	-14	Luxembourg	64	177
Greece	88	107	Belgium	63	113
Spain	87	110	Finland	54	137
Italy	86	87	Estonia	48	82
Austria	85	91	The Netherlands	45	247
Portugal	81	131	UK	42	147
Slovenia	80	56	Australia	37	214
Germany	80	95	USA	32	109
Ireland	79	154	Sweden	20	187
Slovakia	78	74	South Korea	14	182
Lithuania	75	46	Denmark	$q$	261
Latvia	71	43

The summary statistics and scatter diagram below are for the other 23 countries. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Household debt versus Cash transactions} \includegraphics[alt={},max width=\textwidth]{1538fa56-5b61-40ec-bb02-cf1ed9da5eb0-13_664_1296_511_379}

\end{figure} $$\begin{gathered} \sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435
\sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043 \end{gathered}$$

Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
Calculate the equation of the least squares regression line of Household debt as a \% of net disposable income $( y )$ on Cash transactions as a \% of all transactions ( $x$ ). The regression line $x$ on $y$ is given below. $$x = - 0 \cdot 24 y + 91 \cdot 92$$
By selecting the appropriate regression line in each case, estimate the values of $p$ and $q$ in the table.
Comment on the reliability of your answers in part (c).
Interpret the negative value of $y$ for Mexico.

WJEC Further Unit 2 2024 June Q5

5. Lily is interested in the relationship between the way in which students learned Welsh and their attitude towards the Welsh language. Students were categorised as having learned Welsh in one of three ways:

from one Welsh-speaking parent/carer at home,
from two Welsh-speaking parents/carers at home,
at school only, for those with no Welsh-speaking parents/carers at home.

The students were asked to rate their attitude towards the Welsh language from 'Very negative' to 'Very positive'. The following data for a random sample of 253 students were collected as part of a project.

	Learned Welsh
Attitude	From two parents/carers	From one parent/carer	At school only	Total
Very negative	2	14	30	46
Slightly negative	4	20	21	45
Neutral	12	17	8	37
Slightly positive	21	19	11	51
Very positive	25	21	28	74
Total	64	91	98	253

Lily intends to carry out a chi-squared test for independence at the $5 \%$ level. She produces the following tables which are incomplete.

Expected Frequencies	Learned Welsh
Attitude	From two parents/carers	From one parent/carer	At school only
Very negative	11.64	16.55	17.82
Slightly negative	11.38	16.19	17.43
Neutral	9.36	13.31	14.33
Slightly positive	12.90	18.34	19.75
Very positive	F	26.62	28.66

Chi-Squared Contributions	Learned Welsh
Attitude	From two parents/carers	From one parent/carer	At school only
Very negative	7.98	0.39	8.33
Slightly negative	$4 \cdot 79$	0.90	0.73
Neutral	$0 \cdot 74$	1.02	G
Slightly positive	5.08	0.02	3.88
Very positive	2.11	1.19	0.02
Total	20.70	3.52	H

Calculate the values of $F , G$ and $H$.
Carry out Lily's chi-squared test for independence at the $5 \%$ level.
By referring to the figures in the tables on pages 16 and 17, give two comments on the relationship between the way students learned Welsh and their attitude towards the Welsh language.

WJEC Further Unit 2 2024 June Q6

6. Penelope makes 8 cakes per week. Each cake costs $\pounds 20$ to make and sells for $\pounds 60$. She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is $0 \cdot 3$. She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is $\pounds 206$. Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Further Unit 2 Specimen Q1

The random variable $X$ has mean14 and standard deviation 5. The independent random variable $Y$ has mean 12 and standard deviation 3. The random variable $W$ is given by $W = X Y$. Find the value of
1. $\quad \mathrm { E } ( W )$,
2. $\quad \operatorname { Var } ( W )$.
3. The queueing times, $T$ minutes, of customers at a local Post Office are modelled by the probability density function
$$\begin{array} { l l } f ( t ) = \frac { 1 } { 2500 } t \left( 100 - t ^ { 2 } \right) & \text { for } 0 \leq t \leq 10
f ( t ) = 0 & \text { otherwise. } \end{array}$$
Determine the mean queueing time.
1. Find the cumulative distribution function, $F ( t )$, of $T$.
2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
3. Find the median queueing time.

WJEC Further Unit 2 Specimen Q3

3. A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.

Student	A	B	C	D	E	F	G	H
History mark	73	59	83	49	57	82	67	60
Geography mark	55	51	58	59	44	66	49	67

Calculate Spearman's rank correlation coefficient for this data set.
Hence determine whether or not, at the $5 \%$ significance level, there is evidence of a positive association between marks in History and marks in Geography.
Explain why it might not have been appropriate to use Pearson's product moment correlation coefficient to test association using this data set.

WJEC Further Unit 2 Specimen Q4

4. A year 12 student wishes to study at a Welsh university. For a randomly chosen year between 2000 and 2017 she collected data for seven universities in Wales from the Complete University Guide website. The data are for the variables:

'Entry standards' - the average UCAS tariff score of new undergraduate students;
'Student satisfaction' - a measure of student views of the teaching quality at the university taken from the National Student Survey (maximum 5);
'Graduate prospects' - a measure of the employability of a university's first degree graduates (maximum 100);
'Research quality' - a measure of the quality of the research undertaken in the university (maximum 4).
1. Pearson's product-moment correlation coefficients, for each pairing of the four variables, are shown in the table below.
  Discuss the correlation between graduate prospects and the other three variables.

Variable	Entry standards	Student satisfaction	Graduate prospects	Research quality
Entry standards	1
Student satisfaction	-0.030	1
Graduate prospects	0.772	0.236	1
Research quality	0.866	0.066	0.827	1

Calculate the equation of the least squares regression line to predict 'Entry standards'( $y )$ from 'Research quality'( $x$ ), given the summary statistics: $$\sum x = 22.24 , \sum y = 2522 , S _ { x x } = 1.0542 , S _ { y y } = 20193.5 , S _ { x y } = 122.72 .$$

The data for one of the Welsh universities are missing. This university has a research quality of 3.00 . Use your equation to predict the entry standard for this university.

WJEC Further Unit 2 Specimen Q5

5. The manager of a hockey team studies last season's results and puts forward the theory that the number of goals scored per match by her team can be modelled by a Poisson distribution with mean 2.0. The number of goals scored during the season are summarised below.

Goals scored	0	1	2	3	4 or more
Frequency	6	11	15	10	8

State suitable hypotheses to carry out a goodness of fit test.
Carry out a $\chi ^ { 2 }$ goodness of fit test on this data set, using a $5 \%$ level of significance and draw a conclusion in context.

WJEC Further Unit 2 Specimen Q6

6. Customers arrive at a shop such that the number of arrivals in a time interval of $t$ minutes follows a Poisson distribution with mean $0.5 t$.

Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m.
A customer arrives at exactly 11 a.m.
1. Let the next customer arrive at $T$ minutes past 11 a.m. Show that $$P ( T > t ) = \mathrm { e } ^ { - 0.5 t }$$
2. Hence find the probability density function, $f ( t )$, of $T$.
3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers.

WJEC Further Unit 2 Specimen Q7

7. The Pew Research Center's Internet Project offers scholars access to raw data sets from their research. One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked: Do you know your child's password for any of [his/her] social media accounts?
Responses to this question were received from 493 families. The table below provides a summary of their responses.

	Age (years)			Total
Parent know password	13	14	15
Yes	76	75	67	218
No	66	103	106	275
Total	142	178	173	493

A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
1. Clearly state the null and alternative hypotheses.
2. Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above. Expected values:
  Age (years)
  Parent knows
  password
  $\mathbf { 1 3 }$ $\mathbf { 1 4 }$ $\mathbf { 1 5 }$
  Yes 62.79 78.71 76.50
  No 99.29 96.50
3. Obtain the two chi-squared contributions that are missing from the table below. Chi-squared contributions:
  Age (years)
  Parent knows
  password
  $\mathbf { 1 3 }$ $\mathbf { 1 4 }$ $\mathbf { 1 5 }$
  Yes 0.175 1.180
  No 2.203 0.935
  The following output was obtained from the statistical package that was used to undertake the analysis: $$\text { Pearson chi-squared } ( 2 ) = 7.409 \quad p \text {-value } = 0.0305$$
4. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
5. Interpret the output obtained from the statistical test in terms of the initial hypotheses.
Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a).

WJEC Further Unit 3 2018 June Q1

10 marks

\begin{enumerate} \item Two objects, $A$ of mass 18 kg and $B$ of mass 7 kg , are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of $4 \mathrm {~ms} ^ { - 1 }$ and in the same direction. Object $B$ collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of $3 \mathrm {~ms} ^ { - 1 }$. Subsequently, the two objects $A$ and $B$ collide directly. The coefficient of restitution between the two objects is $\frac { 5 } { 7 }$.

Find the coefficient of restitution between $B$ and the wall.
Determine the speed of $A$ and the speed of $B$ immediately after the two objects collide.
Calculate the impulse exerted by $A$ on $B$ due to the collision and clearly state its units.
Find the loss in energy due to the collision between $A$ and $B$.
State the direction of motion of $A$ relative to the wall after the collision with $B$. \item A car of mass 750 kg is moving on a slope inclined at an angle $\theta$ to the horizontal, where $\sin \theta = 0 \cdot 1$. When the car's engine is working at a constant power $P \mathrm {~W}$, the car can travel at maximum speeds of $14 \mathrm {~ms} ^ { - 1 }$ up the slope and $28 \mathrm {~ms} ^ { - 1 }$ down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of $P$ and determine the resistance to the motion of the car when its speed is $10 \cdot 5 \mathrm {~ms} ^ { - 1 }$.
[0pt] [10] \item A light elastic string of natural length 1.5 m and modulus of elasticity 490 N has one end attached to a fixed point $A$ and the other end attached to a particle $P$ of mass 30 kg . Initially, $P$ is held at rest vertically below $A$ such that the distance $A P$ is 0.6 m . It is then allowed to fall vertically.

WJEC Further Unit 3 2018 June Q5

5. A particle $P$, of mass $m \mathrm {~kg}$, is attached to one end of a light inextensible string of length $l \mathrm {~m}$. The other end of the string is attached to a fixed point $O$. Initially, $P$ is held at rest with the string just taut and making an angle of $60 ^ { \circ }$ with the downward vertical. It is then given a velocity $u \mathrm {~ms} ^ { - 1 }$ perpendicular to the string in a downward direction.

1. When the string makes an angle $\theta$ with the downward vertical, the velocity of the particle is $v$ and the tension in the string is $T$. Find an expression for $T$ in terms of $m , l , u ^ { 2 }$ and $\theta$.
2. Given that $P$ describes complete circles in the subsequent motion, show that $u ^ { 2 } > 4 l g$.
Given that now $u ^ { 2 } = 3 l g$, find the position of the string when circular motion ceases. Briefly describe the motion of $P$ after circular motion has ceased.
The string is replaced by a light rigid rod. Given that $P$ describes complete circles in the subsequent motion, show that $u ^ { 2 } > k l g$, where $k$ is to be determined.

WJEC Further Unit 3 2018 June Q6

6. A vehicle of mass 1200 kg is moving with a constant speed of $40 \mathrm {~ms} ^ { - 1 }$ around a horizontal circular path which is on a test track banked at an angle of $60 ^ { \circ }$ to the horizontal. There is no tendency to sideslip at this speed. The vehicle is modelled as a particle.

Calculate the normal reaction of the track on the vehicle.
Determine
1. the radius of the circular path,
2. the angular speed of the vehicle and clearly state its units.
What further assumption have you made in your solution to (b)? Briefly explain what effect this assumption has on the radius of the circular path.

WJEC Further Unit 3 2019 June Q1

The diagram shows a spring of natural length 0.15 m enclosed in a smooth horizontal tube. One end of the spring $A$ is fixed and the other end $B$ is compressed against a ball of mass $0 \cdot 1 \mathrm {~kg}$.
\includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-2_241_714_639_632}

Initially, the ball is held in equilibrium by a force of 21 N so that the compressed length of the spring is $\frac { 2 } { 5 }$ of its natural length.

Calculate the modulus of elasticity of the spring.
The ball is released by removing the force. Determine the speed of the ball when the spring reaches its natural length. Give your answer correct to two significant figures.

WJEC Further Unit 3 2019 June Q2

2. A particle of mass 0.5 kg is moving under the action of a single force $\mathbf { F N }$ so that its velocity $\mathrm { v } \mathrm { ms } ^ { - 1 }$ at time $t$ seconds is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 8 t \mathbf { j } + 2 \mathrm { e } ^ { - t } \mathbf { k }$$

Find an expression for the acceleration of the particle at time $t \mathrm {~s}$.
Determine an expression for F.v at time $t \mathrm {~s}$.
Find the kinetic energy of the particle at time $t \mathrm {~s}$.
Describe the relationship between the kinetic energy of a particle and the rate of working of the force acting on the particle. Verify this relationship using your answers to part (b) and part (c).

WJEC Further Unit 3 2019 June Q3

3. The position vectors $\mathbf { r } _ { A }$ and $\mathbf { r } _ { B }$, in kilometres, of two small aeroplanes $A$ and $B$ relative to a fixed point $O$ are given by $$\begin{aligned} & \mathbf { r } _ { A } = ( 60 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) + ( 168 \mathbf { i } + 132 \mathbf { j } ) t
& \mathbf { r } _ { B } = ( 62 \mathbf { i } + 3 \mathbf { k } ) + ( 160 \mathbf { i } + p \mathbf { j } + q \mathbf { k } ) t \end{aligned}$$ where $t$ denotes the time in hours after 9:00 a.m. and $p , q$ are constants.
The aeroplanes $A$ and $B$ are on course to collide.

Show that $p = 140$ and $q = 4$.
Find an expression for the square of the distance between $A$ and $B$ at time $t$ hours after 9:00 a.m.
Both aeroplanes have systems that activate an alarm if they come within 600 m of each other. Determine the time when the alarms are first activated.

WJEC Further Unit 3 2019 June Q4

4. A car of mass 1200 kg has an engine that is capable of producing a maximum power of 80 kW . When in motion, the car experiences a constant resistive force of 2000 N .

Calculate the maximum possible speed of the car when travelling on a straight horizontal road.
The car travels up a slope inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 20 }$. If the car's engine is working at $80 \%$ capacity, calculate the acceleration of the car at the instant when its speed is $20 \mathrm {~ms} ^ { - 1 }$.
Explain why the assumption of a constant resistive force may be unrealistic.

Expected values	Age in years
Use social media	18-29	30-49	50-64	65 or older
Yes	\(228 \cdot 18\)	\(342 \cdot 27\)	352.64	342.92
No	123.82	185.73	191.36	186.08

Number of boats not taken	0	1	2	3	4	5 or more
Observed	10	35	29	25	8	3
Expected	\(f\)	29.72	\(g\)	20.91	9.88	\(4 \cdot 75\)

Chi-squared contributions	Age in years
Use social media	18-29	30-49	50-64	65 or older
Yes	29.34	\(s\)	0.06	62.94
No	54.07	26-18	0.11	115.99

Questions — WJEC (325 questions)

Browse by module

Browse by board

WJEC Further Unit 2 2022 June Q5

WJEC Further Unit 2 2022 June Q6

WJEC Further Unit 2 2022 June Q7

WJEC Further Unit 2 2023 June Q1

WJEC Further Unit 2 2023 June Q5

WJEC Further Unit 2 2023 June Q6

WJEC Further Unit 2 2024 June Q1

WJEC Further Unit 2 2024 June Q2

WJEC Further Unit 2 2024 June Q3

WJEC Further Unit 2 2024 June Q4

WJEC Further Unit 2 2024 June Q5

WJEC Further Unit 2 2024 June Q6

WJEC Further Unit 2 Specimen Q1

WJEC Further Unit 2 Specimen Q3

WJEC Further Unit 2 Specimen Q4

WJEC Further Unit 2 Specimen Q5

WJEC Further Unit 2 Specimen Q6

WJEC Further Unit 2 Specimen Q7

WJEC Further Unit 3 2018 June Q1

WJEC Further Unit 3 2018 June Q5

WJEC Further Unit 3 2018 June Q6

WJEC Further Unit 3 2019 June Q1

WJEC Further Unit 3 2019 June Q2

WJEC Further Unit 3 2019 June Q3

WJEC Further Unit 3 2019 June Q4