Questions - A-Level Maths

Pre-U Pre-U 9795/2 Specimen Q3

5 marks Standard +0.8

3 A light spring, of natural length 0.4 m and modulus of elasticity 6.4 N , has one end $A$ attached to the ceiling of a room. A particle of mass $m \mathrm {~kg}$ is attached to the free end of the spring and hangs in equilibrium. The particle is displaced vertically downwards and released from rest. In the subsequent motion the particle does not reach the ceiling and air resistance may be neglected.

Show that the particle oscillates in simple harmonic motion.
Given that the period of the motion is 1.12 s , find
1. the value of $m$, correct to 3 significant figures,
2. the extension of the spring when the particle has a downwards acceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Pre-U Pre-U 9795/2 Specimen Q4

7 marks Challenging +1.2

4 A particle is projected with velocity $V$, at an angle of elevation of $60 ^ { \circ }$ to the horizontal, from a point on a plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. The path of the particle is in a vertical plane through a line of greatest slope. If $R _ { 1 }$ and $R _ { 2 }$ are the respective ranges when the particle is projected up the plane and down the plane, show that $$R _ { 2 } = 2 R _ { 1 }$$

Pre-U Pre-U 9795/2 Specimen Q5

3 marks Standard +0.3

5 When a car of mass 990 kg moves at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a horizontal straight road, the power of its engine is 8.8 kW .

Find the magnitude of the resistance to the motion of the car at this speed.
Assuming that the resistance has magnitude $k v ^ { 2 }$ newtons when the speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the value of the constant $k$. The power of the engine is now increased to 22 kW and remains constant at this value.
Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
Hence show that the car moves about 300 m as its speed increases from $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Pre-U Pre-U 9795/2 Specimen Q6

5 marks Challenging +1.2

6 A simple pendulum consists of a light inextensible string of length 1.5 m with a small bob of mass 0.2 kg at one end. When suspended from a fixed point and hanging at rest under gravity, the bob is given a horizontal speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and it comes instantaneously to rest when the string makes an angle of 0.1 rad with the vertical. At time $t$ seconds after projection the string makes an angle $\theta$ with the vertical.

Show that, neglecting air resistance, $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 40 } { 3 } \{ \cos \theta - \cos ( 0.1 ) \}$$
Find, correct to 2 significant figures,
1. the value of $u$,
2. the tension in the string when $\theta = 0.05 \mathrm { rad }$.
3. By differentiating the above equation for $\left( \frac { \mathrm { d } \theta } { \mathrm { d } t } \right) ^ { 2 }$, or otherwise, show that the motion of the bob can be modelled approximately by simple harmonic motion.
4. Hence find the value of $t$ at which the bob first comes instantaneously to rest.

Pre-U Pre-U 9795/2 Specimen Q7

1 marks Standard +0.8

7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.

Pre-U Pre-U 9795/2 Specimen Q8

5 marks Standard +0.3

8 Specimens of rain were collected at random from the north and south sides of an island and analysed for sulphur content. The results (in suitable units) are given below.

North side	0.12	0.61	0.79	0.16	0.08
South side	1.12	0.27	0.06	0.12	0.24	0.78

Assume that the sulphur contents have normal distributions with population means $\mu _ { N }$ and $\mu _ { S }$ and a common, but unknown, variance.

Calculate a symmetric $95 \%$ confidence interval for the difference in population mean sulphur contents of the rain on the north and south sides of the island, $\mu _ { S } - \mu _ { N }$.
Comment on a claim that the mean sulphur content is the same on both sides of the island.

Pre-U Pre-U 9795/2 Specimen Q9

4 marks Standard +0.8

9 The service time, $X$ minutes, for each customer in a post office is modelled by the probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$ Two customers begin to be served independently at the same instant. The larger of the two service times is $T$ minutes.

By considering the probability that both customers have been served in less than $t$ minutes, show that the cumulative distribution function of $T$ is given by $$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
Find the probability that both customers are served within 10 minutes.
Find the value of the interquartile range of $T$.

Pre-U Pre-U 9795/2 Specimen Q10

4 marks Standard +0.3

10

$X , Y$ and $Z$ are independent random variables having Poisson distributions with means $\lambda , \mu$ and $\lambda + \mu$ respectively. Find $\mathrm { P } ( X = 0$ and $Y = 2 ) , \mathrm { P } ( X = 1$ and $Y = 1 )$ and $\mathrm { P } ( X = 2$ and $Y = 0 )$. Hence verify that $\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )$.
In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
1. during a particular month both the male absence rate and the female absence rate are equal to 3,
2. during a particular month the total of the male and female absence rates is equal to 6,
3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .

Pre-U Pre-U 9795/2 Specimen Q11

11 marks Standard +0.3

11 The time, $T$ years, before a particular type of washing machine breaks down may be taken to have probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$ where $a$ and $b$ are positive constants. It may be assumed that, if $n$ is a positive integer, $$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$

Records show that the mean of $T$ is 1.5 . Show that $b = \frac { 4 } { 3 }$ and find the value of $a$.
Find $\operatorname { Var } ( T )$.
Calculate $\mathrm { P } ( T < 1.5 )$. State, giving a reason, whether this value indicates that the median of $T$ is smaller than the mean of $T$ or greater than the mean of $T$.

Pre-U Pre-U 9795/2 Specimen Q12

8 marks Standard +0.8

12 A game is played in which the number of points scored, $X$, has the probability distribution given in the following table.

$x$	- 1	1	3
$\mathrm { P } ( X = x )$	$\frac { 16 } { 25 }$	$\frac { 8 } { 25 }$	$\frac { 1 } { 25 }$

Write down the probability generating function of $X$.
Use this generating function to find the mean and variance of $X$.
The game is played 4 times (independently) and the total number of points scored is denoted by $Y$. Show that the probability generating function of $Y$ can be written in the form $$\frac { \left( a + t ^ { 2 } \right) ^ { 8 } } { b t ^ { 4 } }$$ where $a$ and $b$ are constants.
Hence find $\mathrm { P } ( Y < 0 )$.

WJEC Unit 1 2018 June Q1

Moderate -0.8

Showing all your working, simplify a) $\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }$,
b) $\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }$.

WJEC Unit 1 2018 June Q2

Standard +0.3

The points $A$ and $B$ have coordinates $( - 1,10 )$ and $( 5,1 )$ respectively. The straight line $L$ has equation $2 x - 3 y + 6 = 0$. a) The line $L$ intersects the line $A B$ at the point $C$. Find the coordinates of $C$.
b) Determine the ratio in which the line $L$ divides the line $A B$.
c) The line $L$ crosses the $x$-axis at the point $D$. Find the coordinates of $D$.
d) i) Show that $L$ is perpendicular to $A B$.
ii) Calculate the area of the triangle $A C D$.

WJEC Unit 1 2018 June Q3

Moderate -0.8

Solve the following equation for values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$. $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$

0

4

a) Given that $y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 8$. b) Find $\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x$.

0

5

The diagram below shows a sketch of $y = f ( x )$. \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of $y = 4 + f ( x )$, clearly indicating any asymptotes.
b) Sketch the graph of $y = f ( x - 3 )$, clearly indicating any asymptotes.
□
0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve $C$ with equation $y = 14 + 5 x - x ^ { 2 }$ and line $L$ with equation $y = x + 2$. The line intersects the curve at the points $A$ and $B$.
a) Find the coordinates of $A$ and $B$.
b) Calculate the area enclosed by $L$ and $C$.

0

7

Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

WJEC Unit 1 2018 June Q8

Moderate -0.8

Given that $( x - 2 )$ and $( x + 2 )$ are factors of the polynomial $2 x ^ { 3 } + p x ^ { 2 } + q x - 12$, a) find the values of $p$ and $q$,
b) determine the other factor of the polynomial.

WJEC Unit 1 2018 June Q9

5 marks Moderate -0.3

The triangle $A B C$ is such that $A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}$ and $A \widehat { B C } = 32 ^ { \circ }$. Find two possible values for the area of the triangle $A B C$.

1

0

a) Use the binomial theorem to expand $( a + \sqrt { b } ) ^ { 4 }$.
b) Hence, deduce an expression in terms of $a$ and $b$ for $( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }$.

1

a) The vectors $\mathbf { u }$ and $\mathbf { v }$ are defined by $\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }$ and $\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }$. Determine the range of values for $\mu$ such that $\mu | \mathbf { v } | > | \mathbf { u } |$.
b) The point $A$ has position vector $11 \mathbf { i } - 4 \mathbf { j }$ and the point $B$ has position vector $21 \mathbf { i } + \mathbf { j }$. Determine the position vector of the point $C$, which lies between $A$ and $B$, such that $A C : C B$ is $2 : 3$.

1

2

Find the values of $m$ for which the equation $4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )$ has real roots. [5]

WJEC Unit 1 2018 June Q13

Moderate -0.8

A curve $C$ has equation $y = x ^ { 3 } - 3 x ^ { 2 }$. a) Find the stationary points of $C$ and determine their nature.
b) Draw a sketch of $C$, clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether $\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x$ is positive or
negative, giving a reason for your answer.

1

4

In each of the two statements below, $c$ and $d$ are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\ & \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true.

1

5

The value of a car, $\pounds V$, may be modelled as a continuous variable. At time $t$ years, the value of the car is given by $V = A \mathrm { e } ^ { k t }$, where $A$ and $k$ are constants. When the car is new, it is worth $\pounds 30000$. When the car is two years old, it is worth $\pounds 20000$. Determine the value of the car when it is six years old, giving your answer correct to the nearest $\pounds 100$.

1

6

The curve $C$ has equation $y = 7 + 13 x - 2 x ^ { 2 }$. The point $P$ lies on $C$ and is such that the tangent to $C$ at $P$ has equation $y = x + c$, where $c$ is a constant. Find the coordinates of $P$ and the value of $c$.

1

7

a) Solve $2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2$.
b) Solve $3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }$.
c) Express $4 ^ { x } - 10 \times 2 ^ { x }$ in terms of $y$, where $y = 2 ^ { x }$. Hence solve the equation $4 ^ { x } - 10 \times 2 ^ { x } = - 16$.

WJEC Unit 1 2018 June Q18

Moderate -0.5

The coordinates of three points $A , B , C$ are $( 4,6 ) , ( - 3,5 )$ and $( 5 , - 1 )$ respectively. a) Show that $B \widehat { A C }$ is a right angle.
b) A circle passes through all three points $A , B , C$. Determine the equation of the circle.

WJEC Unit 2 2022 June Q2

Standard +0.3

The probability distribution for $X$, the lifetime of a light bulb, in hours, is given below.

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

a) Suppose that a random sample of 40 light bulbs is tested, and a histogram is drawn of their lifetimes. Calculate the expected height of the bar for the interval $262 \leqslant x < 265$.
b) Now suppose that the last two intervals are changed to $265 \leqslant x < 268$ and $268 \leqslant x < 300$. Explain why it is not possible to tell what will happen to the expected heights of the last two bars.
c) Celyn collects a different random sample of 40 light bulbs to test. She draws a histogram of their lifetimes and finds that it is different to the histogram referred to in part (a). Should Celyn be concerned that the two histograms are different?

WJEC Unit 2 2022 June Q3

Moderate -0.3

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.

0

4

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.

0

6

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}

$\mathbf { 0 }$

7

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.

\(x\)	- 1	1	3
\(\mathrm { P } ( X = x )\)	\(\frac { 16 } { 25 }\)	\(\frac { 8 } { 25 }\)	\(\frac { 1 } { 25 }\)

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

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Pre-U Pre-U 9795/2 Specimen Q3

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WJEC Unit 1 2018 June Q1

WJEC Unit 1 2018 June Q2

WJEC Unit 1 2018 June Q3

WJEC Unit 1 2018 June Q8

WJEC Unit 1 2018 June Q9

WJEC Unit 1 2018 June Q13

WJEC Unit 1 2018 June Q18

WJEC Unit 2 2022 June Q2

WJEC Unit 2 2022 June Q3

WJEC Unit 2 2022 June Q8

WJEC Unit 2 2022 June Q11

WJEC Unit 3 2019 June Q1

WJEC Unit 3 2019 June Q2

WJEC Unit 3 2019 June Q3

WJEC Unit 3 2019 June Q6

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