Questions - Edexcel - A-Level Maths

An estate agent asks customers to rank 7 features of a house, $A , B , C , D , E , F$ and $G$, in order of importance. The responses for two randomly selected customers are in the table below.

Rank	1	2	3	4	5	6	7
Customer 1	$A$	$E$	$C$	$F$	$G$	$B$	$D$
Customer 2	$E$	$F$	$C$	$G$	$A$	$D$	$B$

Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses and critical value clearly, test at the $5 \%$ level of significance, whether or not the two customers are generally in agreement.

Edexcel FS2 2024 June Q3

8 marks Standard +0.3

A factory produces bolts. The lengths of the bolts are normally distributed with mean $\mu \mathrm { mm }$ and standard deviation 0.868 mm

A random sample of 15 of these bolts is taken and the mean length is 30.03 mm

Calculate a 90\% confidence interval for $\mu$ A suitable test, at the $10 \%$ level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.
Calculate the critical region for $S ^ { 2 }$ The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a $10 \%$ level of significance to test whether or not there is evidence that
- the mean length of the bolts has changed
- the variance of the length of the bolts has increased
The next month a random sample of 15 bolts is taken.
The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm
With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly.
Give reasons for your answer.

Edexcel FS2 2024 June Q4

9 marks Standard +0.3

The random variable $G$ has a continuous uniform distribution over the interval $[ - 3,15 ]$
1. Calculate $\mathrm { P } ( G > 12 )$
The random variable $H$ has a continuous uniform distribution over the interval [2, w] The random variables $G$ and $H$ are independent and $\mathrm { E } ( H ) = 10$
Show that the probability that $G$ and $H$ are both greater than 12 is $\frac { 1 } { 16 }$ The random variable $A$ is the area on a coordinate grid bounded by $$\begin{aligned} & y = - 3 \\ & y = - 4 | x | + k \end{aligned}$$ where $k$ is a value from the continuous uniform distribution over the interval [5,10]
Calculate the expected value of $A$

Edexcel FS2 2024 June Q5

10 marks Challenging +1.2

A continuous random variable $X$ has probability density function

$$f ( x ) = \left\{ \begin{array} { c l } a x ^ { - 2 } - b x ^ { - 3 } & 2 \leqslant x < \infty \\ 0 & \text { otherwise } \end{array} \right.$$ where $a$ and $b$ are constants. Given that $\mathrm { P } ( X \leqslant 4 ) = \frac { 3 } { 8 }$

use algebraic integration to show that $a = 3$ Show your working clearly.
Find the exact value of the median of $X$

Edexcel FS2 2024 June Q6

12 marks Standard +0.3

A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.

	Set of twin lambs	A	$B$	C	D	$E$	$F$	$G$	$H$
\multirow{2}{*}{Weight gain (kg)}	With food supplement	4.1	5.3	6.0	3.6	5.9	4.2	7.1	6.4
	No food supplement	5.0	4.8	5.2	3.4	5.1	3.9	7.0	6.5

State why a two sample $t$-test is not suitable for use with these data.
Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.
State one assumption, in context, that needs to be made for a paired $t$-test to be valid. For a pair of twin lambs, the random variable $W$ represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
Using the data in the table, calculate a $98 \%$ confidence interval for the mean of $W$ Show your working clearly. The researcher believes that the mean of $W$ is greater than 200 g
Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher's belief.

Edexcel FS2 2024 June Q7

11 marks Challenging +1.2

Two organisations are each asked to carry out a survey to find out the proportion, $p$, of the population that would vote for a particular political party.

The first organisation finds that out of $m$ people, $X$ would vote for this particular political party. The second organisation finds that out of $n$ people, $Y$ would vote for this particular political party. An unbiased estimator, $Q$, of $p$ is proposed where $$Q = k \left( \frac { X } { m } + \frac { Y } { n } \right)$$

Show that $k = \frac { 1 } { 2 }$ A second unbiased estimator, $R$, of $p$ is proposed where $$R = \frac { a X } { m } + \frac { b Y } { n }$$
Show that $a + b = 1$ Given that $m = 100$ and $n = 200$ and that $R$ is a better estimator of $p$ than $Q$
calculate the range of possible values of $a$ Show your working clearly.

Edexcel FS2 2024 June Q8

9 marks Standard +0.3

A company packs chickpeas into small bags and large bags.

The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g A random sample of 3 small bags of chickpeas is taken.

Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g One large bag and one small bag of chickpeas are chosen at random.
Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.

Edexcel FS2 Specimen Q1

13 marks Standard +0.8

The three independent random variables $A , B$ and $C$ each have a continuous uniform distribution over the interval $[ 0,5 ]$.
1. Find the probability that $A , B$ and $C$ are all greater than 3
The random variable $Y$ represents the maximum value of $A , B$ and $C$.
The cumulative distribution function of $Y$ is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
Using algebraic integration, show that $\operatorname { Var } ( Y ) = 0.9375$
Find the mode of $Y$, giving a reason for your answer.
Describe the skewness of the distribution of $Y$. Give a reason for your answer.
Find the value of $k$ such that $\mathrm { P } ( k < Y < 2 k ) = 0.189$

Edexcel FS2 Specimen Q2

9 marks Standard +0.3

A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at 7 positions. The results are shown in the table below.

Position

Distance from

inner bank $\boldsymbol { b } \mathbf { c m }$

Depth $\boldsymbol { s } \mathbf { c m }$

110

120

104

The Spearman's rank correlation coefficient between $b$ and $s$ is $\frac { 6 } { 7 }$

Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a $1 \%$ level of significance.
Without re-calculating the correlation coefficient, explain how the Spearman's rank correlation coefficient would change if
1. the depth for G is 109 instead of 104
2. an extra value H with distance from the inner bank of 800 cm and depth 130 cm is included. The researcher decided to collect extra data and found that there were now many tied ranks.
Describe how you would find the correlation with many tied ranks.

Edexcel FS2 Specimen Q3

7 marks Standard +0.8

A nutritionist studied the levels of cholesterol, $X \mathrm { mg } / \mathrm { cm } ^ { 3 }$, of male students at a large college. She assumed that $X$ was distributed $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$ and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of $\mu$ and $\sigma ^ { 2 }$ as $\hat { \mu }$ and $\hat { \sigma } ^ { 2 }$

A $95 \%$ confidence interval for $\mu$ was found to be $( 1.128,2.232 )$

Show that $\hat { \sigma } ^ { 2 } = 1.79$ (correct to 3 significant figures)
Obtain a $95 \%$ confidence interval for $\sigma ^ { 2 }$

Edexcel FS2 Specimen Q4

13 marks Standard +0.3

The times, $x$ seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	No. of competitors	Sample mean $\overline { \boldsymbol { x } }$	$\sum \boldsymbol { x } ^ { \mathbf { 2 } }$
Girls	8	83.1	55746
Boys	7	88.9	56130

Following the gala, a mother claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

test, at the $10 \%$ level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
Stating your hypotheses clearly, test the mother's claim. Use a $5 \%$ level of significance.

Edexcel FS2 Specimen Q5

13 marks Challenging +1.2

Scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.6,0.6 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution N(4.8, 0.32). The random variables $L$ and $S$ are independent.

A long pole and a short pole are selected at random.

Find the probability that the length of the long pole is more than 4 times the length of the short pole. Show your working clearly. Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
Find the distribution of $T$.
Find $\mathrm { P } ( | L - T | < 0.2 )$

Edexcel FS2 Specimen Q6

12 marks Standard +0.3

A random sample of 10 female pigs was taken. The number of piglets, $x$, born to each female pig and their average weight at birth, $m \mathrm {~kg}$, was recorded. The results were as follows:

Number of piglets, $\boldsymbol { x }$

Average weight at

birth, $\boldsymbol { m } \mathbf { ~ k g }$

(You may use $\mathrm { S } _ { x x } = 82.5$ and $\mathrm { S } _ { m m } = 0.12756$ and $\mathrm { S } _ { x m } = - 2.29$ )

Find the equation of the regression line of $m$ on $x$ in the form $m = a + b x$ as a model for these results.
Show that the residual sum of squares (RSS) is 0.064 to 3 decimal places.
Calculate the residual values.
Write down the outlier.
1. Comment on the validity of ignoring this outlier.
2. Ignoring the outlier, produce another model.
3. Use this model to estimate the average weight at birth if $x = 15$
4. Comment, giving a reason, on the reliability of your estimate.

Edexcel FS2 Specimen Q7

8 marks Standard +0.8

Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, $s \mathrm { mg } / \mathrm { dl }$, in the blood and the corresponding level of the disease protein, $d \mathrm { mg } / \mathrm { dl }$. One of the researchers coded the data for each sample using $x = 10 s$ and $y = 10 ( d - 9 )$ but spilt ink over his work.

The following summary statistics and unfinished scatter diagram are the only remaining information. $$\sum d ^ { 2 } = 1081.74 \quad \mathrm {~S} _ { d s } = 59.524$$ and $$\sum y = 64 \quad \mathrm {~S} _ { x x } = 2658.9$$ $d \mathrm { mg } / \mathrm { dl }$
\includegraphics[max width=\textwidth, alt={}, center]{e777c787-0d39-4d84-a0f9-fc4a6712184f-22_983_1534_840_303}

Use the formula for $\mathrm { S } _ { x x }$ to show that $\mathrm { S } _ { s s } = 26.589$
Find the value of the product moment correlation coefficient between $s$ and $d$.
With reference to the unfinished scatter diagram, comment on your result in part (b).

Edexcel FM1 2019 June Q1

8 marks Standard +0.8

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a871044a-17c5-440d-8d8f-886939603dd4-02_307_889_244_589} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the plan of part of a smooth horizontal floor, where $W _ { 1 }$ and $W _ { 2 }$ are two fixed parallel vertical walls. The walls are 3 metres apart. A particle lies at rest at a point $O$ on the floor between the two walls, where the point $O$ is $d$ metres, $0 < d \leqslant 3$, from $W _ { 1 }$ At time $t = 0$, the particle is projected from $O$ towards $W _ { 1 }$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is $\frac { 2 } { 3 }$
The particle returns to $O$ at time $t = T$ seconds, having bounced off each wall once.

Show that $T = \frac { 45 - 5 d } { 4 u }$ The value of $u$ is fixed, the particle still hits each wall once but the value of $d$ can now vary.
Find the least possible value of $T$, giving your answer in terms of $u$. You must give a reason for your answer.

Edexcel FM1 2019 June Q2

11 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a871044a-17c5-440d-8d8f-886939603dd4-06_524_638_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 represents the plan view of part of a horizontal floor, where $A B$ and $B C$ are fixed vertical walls with $A B$ perpendicular to $B C$. A small ball is projected along the floor towards $A B$ with speed $6 \mathrm {~ms} ^ { - 1 }$ on a path that makes an angle $\alpha$ with $A B$, where $\tan \alpha = \frac { 4 } { 3 }$. The ball hits $A B$ and then hits $B C$.
Immediately after hitting $A B$, the ball is moving at an angle $\beta$ to $A B$, where $\tan \beta = \frac { 1 } { 3 }$
The coefficient of restitution between the ball and $A B$ is $e$.
The coefficient of restitution between the ball and $B C$ is $\frac { 1 } { 2 }$
By modelling the ball as a particle and the floor and walls as being smooth,

show that the value of $e = \frac { 1 } { 4 }$
find the speed of the ball immediately after it hits $B C$.
Suggest two ways in which the model could be refined to make it more realistic.

Edexcel FM1 2019 June Q3

9 marks Challenging +1.2

A particle $P$, of mass 0.5 kg , is moving with velocity ( $4 \mathbf { i } + 4 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse I of magnitude 2.5 Ns.

As a result of the impulse, the direction of motion of $P$ is deflected through an angle of $45 ^ { \circ }$ Given that $\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )$ Ns, find all the possible pairs of values of $\lambda$ and $\mu$.

Edexcel FM1 2019 June Q4

12 marks Standard +0.8

A car of mass 600 kg pulls a trailer of mass 150 kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude 200 N . At the instant when the speed of the car is $v \mathrm {~ms} ^ { - 1 }$, the resistance to the motion of the car is modelled as a force of magnitude $( 200 + \lambda v ) \mathrm { N }$, where $\lambda$ is a constant.

When the engine of the car is working at a constant rate of 15 kW , the car is moving at a constant speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Show that $\lambda = 8$ Later on, the car is pulling the trailer up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 15 }$
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 200 N at all times. At the instant when the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude $( 200 + 8 v ) \mathrm { N }$. The engine of the car is again working at a constant rate of 15 kW .
When $v = 10$, the towbar breaks. The trailer comes to instantaneous rest after moving a distance $d$ metres up the road from the point where the towbar broke.
Find the acceleration of the car immediately after the towbar breaks.
Use the work-energy principle to find the value of $d$.

Edexcel FM1 2019 June Q5

11 marks Standard +0.8

A particle $P$ of mass $3 m$ and a particle $Q$ of mass $2 m$ are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of $P$ is $u$ and the speed of $Q$ is $2 u$.
Immediately after the collision $P$ and $Q$ are moving in opposite directions.
The coefficient of restitution between $P$ and $Q$ is $e$.

Find the range of possible values of $e$, justifying your answer. Given that $Q$ loses 75\% of its kinetic energy as a result of the collision,
find the value of $e$.

Edexcel FM1 2019 June Q6

12 marks Standard +0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere $A$ has mass 0.2 kg and another smooth uniform sphere $B$, with the same radius as $A$, has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of $A$ is $( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $B$ is $( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf { i }$
The coefficient of restitution between the spheres is $\frac { 3 } { 7 }$

Find the velocity of $A$ immediately after the collision.
Find the magnitude of the impulse received by $A$ in the collision.
Find, to the nearest degree, the size of the angle through which the direction of motion of $A$ is deflected as a result of the collision.

Edexcel FM1 2019 June Q7

12 marks Standard +0.8

A particle $P$, of mass $m$, is attached to one end of a light elastic spring of natural length $a$ and modulus of elasticity kmg.

The other end of the spring is attached to a fixed point $O$ on a ceiling.
The point $A$ is vertically below $O$ such that $O A = 3 a$
The point $B$ is vertically below $O$ such that $O B = \frac { 1 } { 2 } a$
The particle is held at rest at $A$, then released and first comes to instantaneous rest at the point $B$.

Show that $k = \frac { 4 } { 3 }$
Find, in terms of $g$, the acceleration of $P$ immediately after it is released from rest at $A$.
Find, in terms of $g$ and $a$, the maximum speed attained by $P$ as it moves from $A$ to $B$.

Edexcel FM1 2020 June Q1

7 marks Moderate -0.5

A particle $P$ of mass 0.5 kg is moving with velocity ( $4 \mathbf { i } + 3 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse $\mathbf { J }$ Ns. Immediately after receiving the impulse, $P$ is moving with velocity $( - \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. Find the magnitude of $\mathbf { J }$.
The angle between the direction of the impulse and the direction of motion of $P$ immediately before receiving the impulse is $\alpha ^ { \circ }$
Find the value of $\alpha$

Edexcel FM1 2020 June Q2

9 marks Standard +0.3

A truck of mass 1200 kg is moving along a straight horizontal road.

At the instant when the speed of the truck is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the truck is modelled as a force of magnitude $( 900 + 9 v ) \mathrm { N }$. The engine of the truck is working at a constant rate of 25 kW .

Find the deceleration of the truck at the instant when $v = 25$ Later on, the truck is moving up a straight road that is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 20 }$ At the instant when the speed of the truck is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the truck from non-gravitational forces is modelled as a force of magnitude ( $900 + 9 v$ ) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the value of $V$.

Edexcel FM1 2020 June Q3

14 marks Standard +0.8

Two particles, $A$ and $B$, have masses $3 m$ and $4 m$ respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of $A$ is $2 u$ and the speed of $B$ is $u$.

The coefficient of restitution between $A$ and $B$ is $e$.

Show that the direction of motion of each of the particles is unchanged by the collision.
(8) After the collision with $A$, particle $B$ collides directly with a third particle, $C$, of mass $2 m$, which is at rest on the surface. The coefficient of restitution between $B$ and $C$ is also $e$.
Show that there will be a second collision between $A$ and $B$.

Edexcel FM1 2020 June Q4

9 marks Standard +0.3

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.]

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{361d263e-0ee1-47e9-8fc2-0f127f1c2d7e-12_588_633_301_724} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where $A B$ represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is $( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Immediately after the impact the velocity of the ball is $( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.
The coefficient of restitution between the ball and the wall is $e$.

Show that $A B$ is parallel to $( 2 \mathbf { i } + 3 \mathbf { j } )$.
Find the value of $e$.

Rank	1	2	3	4	5	6	7
Customer 1	\(A\)	\(E\)	\(C\)	\(F\)	\(G\)	\(B\)	\(D\)
Customer 2	\(E\)	\(F\)	\(C\)	\(G\)	\(A\)	\(D\)	\(B\)

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	No. of competitors	Sample mean \(\overline { \boldsymbol { x } }\)	\(\sum \boldsymbol { x } ^ { \mathbf { 2 } }\)
Girls	8	83.1	55746
Boys	7	88.9	56130

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