Questions - A-Level Maths

Edexcel PURE 2024 October Q10

Standard +0.8

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-30_563_602_255_735} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region $R$, shown shaded in Figure 5, is bounded by the curve and the $x$-axis.

Show that the area of $R$ is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where $k$ is a constant to be found.
Hence, using algebraic integration, find the exact area of $R$, giving your answer in the form $$p \pi + q$$ where $p$ and $q$ are constants.

Edexcel M2 2024 October Q1

Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

At time $t$ seconds, $t \geqslant 0$, a particle $P$ is moving with velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$, where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point $O$ At time $t = 0 , P$ is at the point with position vector $( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }$.

Find the position vector of $P$ when $t = 3$
Find the magnitude of the acceleration of $P$ when $t = 3$ At time $T$ seconds, $P$ is moving in the direction of the vector $2 \mathbf { i } + \mathbf { j }$
Find the value of $T$

Edexcel M2 2024 October Q2

Standard +0.3

A particle $Q$ of mass 3 kg is moving on a smooth horizontal surface.

Particle $Q$ is moving with velocity $5 \mathbf { i } \mathrm {~ms} ^ { - 1 }$ when it receives a horizontal impulse of magnitude $3 \sqrt { 82 } \mathrm { Ns }$. Immediately after receiving the impulse, the velocity of $Q$ is $( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $x$ and $y$ are positive constants. The kinetic energy gained by $Q$ as a result of receiving the impulse is 138 J .
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $Q$ immediately after receiving the impulse.

Edexcel M2 2024 October Q3

Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-06_275_1143_303_461} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A van of mass 900 kg is moving up a straight road inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 25 }$. The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 240 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 15 kW .

Find the acceleration of the van at the instant when the speed of the van is $12 \mathrm {~ms} ^ { - 1 }$ At the instant when the speed of the van is $14 \mathrm {~ms} ^ { - 1 }$, the trailer is passing the point $A$ on the slope and the towbar breaks. The trailer continues to move up the slope until it comes to rest at the point $B$.
The resistance to the motion of the trailer from non-gravitational forces is still modelled as a constant force of magnitude 240 N .
Use the work-energy principle to find the distance $A B$.

Edexcel M2 2024 October Q4

Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_301_871_319_598} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform lamina $A B C D$ shown in Figure 2 is in the shape of an isosceles trapezium.

$B C$ is parallel to $A D$ and angle $B A D$ is equal to angle $A D C$
$B C = 5 a$ and $A D = 7 a$
the perpendicular distance between $B C$ and $A D$ is $3 a$
the distance of the centre of mass of $A B C D$ from $A D$ is $d$
1. Show that $d = \frac { 17 } { 12 } a$

The uniform lamina $P Q R S$ is a rectangle with $P Q = 5 a$ and $Q R = 9 a$.
The lamina $A B C D$ in Figure 2 is used to cut a hole in $P Q R S$ to form the template shown shaded in Figure 3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_364_876_1567_593} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure}

The template is freely suspended from $P$ and hangs in equilibrium with $P S$ at an angle of $\theta ^ { \circ }$ to the downward vertical.

Find the value of $\theta$

Edexcel M2 2024 October Q5

Standard +0.3

The fixed points $X$ and $Y$ lie on horizontal ground.

At time $t = 0$, a particle $P$ is projected from $X$ with speed $u \mathrm {~ms} ^ { - 1 }$ at angle $\theta$ to the ground. Particle $P$ moves freely under gravity and first hits the ground at $Y$.

Show that $X Y = \frac { u ^ { 2 } \sin 2 \theta } { g }$ The points $A$ and $B$ lie on horizontal ground. A vertical pole $C D$ has length 5 m .
The end $C$ is fixed to the ground between $A$ and $B$, where $A C = 12 \mathrm {~m}$.
At time $t = 0$, a particle $Q$ is projected from $A$ with speed $20 \mathrm {~ms} ^ { - 1 }$ at $60 ^ { \circ }$ to the ground.
Particle $Q$ moves freely under gravity, passes over the pole and first hits the ground at $B$, as shown in Figure 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-14_335_1179_1032_443} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure}
Find the distance $C B$.
Find the height of $Q$ above $D$ at the instant when $Q$ passes over the pole.

Edexcel M2 2024 October Q6

Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-18_419_1307_315_379} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A uniform beam $A B$, of weight $5 W$ and length $12 a$, rests with end $A$ on rough horizontal ground.
A package of weight $W$ is attached to the beam at $B$.
The beam rests in equilibrium on a smooth horizontal peg at $C$, with $A C = 9 a$, as shown in Figure 5.
The beam is inclined at an angle $\theta$ to the ground, where $\tan \theta = \frac { 5 } { 12 }$ The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg. The package is modelled as a particle. The normal reaction between the beam and the peg at $C$ has magnitude $k W$ Using the model,

show that $k = \frac { 56 } { 13 }$ The coefficient of friction between $A$ and the ground is $\mu$ Given that the beam is resting in limiting equilibrium,
find the value of $\mu$

Edexcel M2 2024 October Q7

Standard +0.3

A particle $P$ has mass $5 m$ and a particle $Q$ has mass $2 m$.

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
Particle $P$ collides directly with particle $Q$.
Immediately before the collision, the speed of $P$ is $2 u$ and the speed of $Q$ is $3 u$. Immediately after the collision, the speed of $P$ is $x$ and the speed of $Q$ is $y$.
The direction of motion of $Q$ is reversed as a result of the collision.
The coefficient of restitution between $P$ and $Q$ is $e$.

Find the set of values of $e$ for which the direction of motion of $P$ is unchanged as a result of the collision. In the collision, $Q$ receives an impulse of magnitude $\frac { 60 } { 7 } m u$
Show that $e = \frac { 1 } { 5 }$ After the collision, $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$. Particle $Q$ rebounds and there is a second collision between $P$ and $Q$.
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 3 }$
Find, in terms of $m$ and $u$, the magnitude of the impulse received by $Q$ in the second collision between $P$ and $Q$.

Edexcel S2 2024 October Q1

Standard +0.3

During an annual beach-clean, the people doing the clean are asked to conduct a litter survey.
At a particular beach-clean, litter was found at a rate of 4 items per square metre.

Find the probability that, in a randomly selected area of 2 square metres on this beach, exactly 5 items of litter were found. Of the litter found on the beach, 30\% of the items were face masks.
Find the probability that, in a randomly selected area of 5 square metres on this beach, more than 4 face masks were found.
Using a suitable approximation, find the probability that, in a randomly selected area of 20 square metres on this beach, less than 60 items of litter were found that were not face masks.

Edexcel S2 2024 October Q2

Standard +0.3

A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let $X$ be the number of correct answers that Sam achieves.

State the distribution of $X$ Let $M$ be the number of marks that Sam achieves.
1. State the distribution of $M$ in terms of $X$
2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
Given that the probability that more than $n$ but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
find the value of $n$

Edexcel S2 2024 October Q3

Standard +0.3

During Monday afternoons, customers are known to enter a certain shop at a mean rate of 7 customers every 10 minutes.

Suggest a suitable distribution to model the number of customers that enter this shop in a 10-minute interval on Monday afternoons.
State two assumptions necessary for this distribution to be a suitable model of this situation. A new shop manager wants to find out if the rate of customers has changed since they took over.
Write down suitable null and alternative hypotheses that the shop manager should use. The shop manager decides to monitor the number of customers entering the shop in a random 10-minute interval next Monday afternoon.
Using a $3 \%$ level of significance, find the critical region to test whether the rate of customers has changed.
Find the actual significance level of this test based on your critical region from part (d) During the random 10-minute interval that Monday afternoon, 12 customers entered the shop.
Comment on this finding, using the critical region in part (d)

Edexcel S2 2024 October Q4

Standard +0.3

The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$ Given that
Given also that $$4 \times \mathrm { P } ( X < k - 10 ) = \mathrm { P } ( X > k + 20 )$$ (b) find the value of $k$
A piece of wire of length 42 cm is cut into 2 pieces at a random point. Each of the two pieces of the wire is bent to form the outline of a square.
Find the probability that the side length of the larger square minus the side length of the smaller square will be greater than 2 cm .

Edexcel S2 2024 October Q5

Moderate -0.3

The continuous random variable $X$ has a probability density function given by

$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ \frac { 1 } { 4 } & 2 < x \leqslant 3 \\ \frac { 1 } { 4 } ( x - 2 ) & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The cumulative distribution function of $X$ is $\mathrm { F } ( x )$

Show that $\mathrm { F } ( x ) = \frac { 1 } { 4 } \left( 3 x - \frac { x ^ { 2 } } { 2 } \right) - \frac { 5 } { 8 }$ for $1 \leqslant x \leqslant 2$
Find $\mathrm { F } ( x )$ for all values of $x$
Find $\mathrm { P } ( 1.2 < X < 3.1 )$

Edexcel S2 2024 October Q6

Standard +0.3

Two boxes, A and B , each contain a large number of coins.

In box A

there are only 1 p coins and 2 p coins
the ratio of 1 p coins to 2 p coins is $1 : 3$

In box B

there are only 2 p coins and 5 p coins
the ratio of 2 p coins to 5 p coins is $1 : 4$

One coin is randomly selected from box A and two coins are randomly selected from box B The random variable $T$ represents the total of the values of the three coins selected.

Find the sampling distribution of $T$ The random variable $M$ represents the median of the values of the three coins selected.
Find the sampling distribution of $M$

Edexcel S2 2024 October Q7

Standard +0.3

The continuous random variable $X$ has probability density function given by

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where $a$, $b$ and $c$ are constants. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6e6f7a1a-b577-4f28-a7a9-557b9d325851-24_389_1013_630_529} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the graph of the probability density function $\mathrm { f } ( x )$ The graph consists of two straight line segments of equal length joined at the point where $x = 4$

Show that $a = \frac { 1 } { 16 }$
Hence find
1. the value of $b$
2. the value of $c$
Using algebraic integration, show that $\operatorname { Var } ( X ) = \frac { 8 } { 3 }$
Find, to 2 decimal places, the lower quartile and the upper quartile of $X$ A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where $\mu$ and $\sigma$ are the mean and standard deviation of $X$
Show that the statistician's claim is correct.

Edexcel S3 Q7

Standard +0.8

7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. 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. 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 $\mathrm { P } ( | L - T | < 0.1 )$.
\end{table}
1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows
$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$

(a) Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
(b) find a $99 \%$ confidence interval for the mean wing length of the birds from this group.
2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results \end{table}

A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
1. Explain in detail how the researcher should obtain such a sample.
2. Give one advantage and one disadvantage of
1. quota sampling,
2. systematic sampling.
3. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
Given that the population standard deviation is 0.5 cm ,
1. estimate limits between which $95 \%$ of the heights of the orchids lie,
2. find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
3. Comment on the grower's claim. Give a reason for your answer.
  3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
  Individual $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
  BMI 17.4 21.4 18.9 24.4 19.4 20.1 22.6 18.4 25.8 28.1
  Finishing position 3 5 1 9 6 4 10 2 7 8
1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly and using a one tailed test with a $5 \%$ level of significance, interpret your rank correlation coefficient.
3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
  4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
  5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]
  Number of goals Frequency
  0 40
  1 33
  2 14
  3 8
  4 5
  \captionsetup{labelformat=empty} \caption{Table 1}
  \end{table}
(a) Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
Number of goals Expected Frequency
0 34.994
1 $r$
2 $s$
3 6.752
$\geqslant 4$ 2.221
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} (b) Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places.
(c) Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
  1. Test, using a $5 \%$ level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
  2. State two assumptions you made in carrying out the test in part (a).
  3. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
    119.9
    120.3
    120.1
    120.4
    120.2
  (a) Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of $n$, lies within 0.05 m of its true value.
(b) Find the minimum sample size required.

Edexcel S3 Q8

Moderate -0.3

The random variable $A$ is defined as

$$A = 4 X - 3 Y$$ where $X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)$ and $X$ and $Y$ are independent. Find

$\mathrm { E } ( A )$,
$\operatorname { Var } ( A )$. The random variables $Y _ { 1 } , Y _ { 2 } , Y _ { 3 }$ and $Y _ { 4 }$ are independent and each has the same distribution as $Y$. The random variable $B$ is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
Find $\mathrm { P } ( B > A )$.
advancing learning, changing lives
1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
Test, at the $5 \%$ level of significance, the managing director's claim. State your hypotheses clearly.
2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
find the probability that Philip beats James in the race by more than 2 minutes.
3. A woodwork teacher measures the width, $w \mathrm {~mm}$, of a board. The measured width, $X \mathrm {~mm}$, is normally distributed with mean $w \mathrm {~mm}$ and standard deviation 0.5 mm .

Find the probability that $X$ is within 0.6 mm of $w$. The same board is measured 16 times and the results are recorded.
Find the probability that the mean of these results is within 0.3 mm of $w$. Given that the mean of these 16 measurements is 35.6 mm ,
find a $98 \%$ confidence interval for $w$.
1. 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 seven positions. The results are shown in the table below.
advancing learning, changing lives \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than $5 \%$.
(5)
3. The table below shows the population and the number of council employees for different towns and villages. \end{table} A nswers without working may not gain full credit. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{ $0 - 3$ & 8
\hline $3 - 5$ & 12
\hline $5 - 6$ & 13
\hline $6 - 8$ & 9
\hline $8 - 12$ & 8
\hline \captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
1. Show that an estimate of $\bar { X } = 5.49$ and an estimate of $S _ { X } ^ { 2 } = 6.88$ The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
  Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
  Waiting Time $\mathrm { x } < 3$ $3 - 5$ $5 - 6$ $6 - 8$ $\mathrm { x } > 8$
  Expected Frequency 8.56 12.73 7.56 a b
  \captionsetup{labelformat=empty} \caption{Table 2}
  \end{table}
2. Find the value of a and the value of b .
3. Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.
  \section*{Q uestion 4 continued}
  1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
  One large and 3 small bottles of Blumen are chosen at random.
1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
  \section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is $0.5 \mathrm {~kg} ^ { 2 }$ and the variance of the yield for the new variety of plant is $0.75 \mathrm {~kg} ^ { 2 }$. A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
1. Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
  7. Lambs are born in a shed on M ill Farm. The birth weights, $x \mathrm {~kg}$, of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
(a) Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
Calculate a $95 \%$ confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
\section*{Q uestion 7 continued} \end{figure}

Pre-U Pre-U 9795/1 2010 June Q1

4 marks Standard +0.8

1 The equation $x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0$ has roots $\alpha , \beta$ and $\gamma$. Use the substitution $y = 1 + x ^ { 2 }$ to find an equation, with integer coefficients, whose roots are $1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }$ and $1 + \gamma ^ { 2 }$.

Pre-U Pre-U 9795/1 2010 June Q2

5 marks Standard +0.3

2 Use the method of differences to express $\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }$ in terms of $n$, and hence deduce the sum of the infinite series $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$

Pre-U Pre-U 9795/1 2010 June Q3

4 marks Moderate -0.8

3 The points $A ( 1,3 ) , B ( 4,36 )$ and $C ( 9,151 )$ lie on the curve with equation $y = p + q x + r x ^ { 2 }$.

Using this information, write down three simultaneous equations in $p , q$ and $r$.
Re-write this system of equations in the matrix form $\mathbf { C x } = \mathbf { a }$, where $\mathbf { C }$ is a $3 \times 3$ matrix, $\mathbf { x }$ is an unknown vector, and $\mathbf { a }$ is a fixed vector.
By finding $\mathbf { C } ^ { - 1 }$, determine the values of $p , q$ and $r$.

Pre-U Pre-U 9795/1 2010 June Q4

5 marks Standard +0.3

4

Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
Solve the equation $5 \cosh x + 3 \sinh x = 12$, giving your answers in the form $\ln ( p \pm q \sqrt { 2 } )$ for rational numbers $p$ and $q$ to be determined.

Pre-U Pre-U 9795/1 2010 June Q5

8 marks Standard +0.8

5 A curve has equation $y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }$ for $x \neq - 3$.

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 1$ at all points on the curve.
Sketch the curve, justifying all significant features.

Pre-U Pre-U 9795/1 2010 June Q6

8 marks Challenging +1.2

6

The set $S$ consists of all $2 \times 2$ matrices of the form $\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)$, where $n \in \mathbb { Z }$.
1. Show that $S$, under the operation of matrix multiplication, forms a group $G$. [You may assume that matrix multiplication is associative.]
2. State, giving a reason, whether $G$ is abelian.
3. The group $H$ is the set $\mathbb { Z }$ together with the operation of addition. Explain why $G$ is isomorphic to $H$.
4. The plane transformation $T$ is given by the matrix $\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)$, where $n$ is a non-zero integer. Describe $T$ fully.

Pre-U Pre-U 9795/1 2010 June Q7

9 marks Challenging +1.2

7 A curve $C$ has polar equation $r = 2 + \cos \theta$ for $- \pi < \theta \leqslant \pi$.

The point $P$ on $C$ corresponds to $\theta = \alpha$, and the point $Q$ on $C$ is such that $P O Q$ is a straight line, where $O$ is the pole. Show that the length $P Q$ is independent of $\alpha$.
Find, in an exact form, the area of the region enclosed by $C$.
Show that $\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)$ is a cartesian equation for $C$. Identify the coordinates of the point which is included in this cartesian equation but is not on $C$.

Pre-U Pre-U 9795/1 2010 June Q8

10 marks Challenging +1.8

8 For the differential equation $t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 6 - 4 t ^ { 2 } \right) x = 0$, use the substitution $x = t ^ { 2 } u$ to find a differential equation involving $t$ and $u$ only. Hence solve the above differential equation, given that $x = \mathrm { e } - 1$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 4 \mathrm { e }$ when $t = 1$.

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Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221

Waiting Time	\(\mathrm { x } < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(\mathrm { x } > 8\)
Expected Frequency	8.56	12.73	7.56	a	b