Questions - SPS - A-Level Maths

Find the tension in the string, in terms of $m , l , \omega$ and $g$. The particle remains on the surface of the cone.
Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$ [Question 5 Continued]
[0pt] [Question 5 Continued]

SPS SPS FM Statistics 2024 January Q1

1. The continuous random variable $X$ has the distribution $\mathrm { N } ( \mu , 30 )$. The mean of a random sample of 8 observations of $X$ is 53.1. Determine a $95 \%$ confidence interval for $\mu$. You should give the end points of the interval correct to 4 significant figures.

SPS SPS FM Statistics 2024 January Q2

2. At a seaside resort the number $X$ of ice-creams sold and the temperature $Y ^ { \circ } \mathrm { F }$ were recorded on 20 randomly chosen summer days. The data can be summarised as follows. $$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$$

Calculate the equation of the least squares regression line of $y$ on $x$, giving your answer in the form $y = a + b x$.
Explain the significance for the regression line of the quantity $\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }$.
It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as $f ^ { \circ }$ Fahrenheit and $c ^ { \circ }$ Centigrade, the relationship between $f$ and $c$ is $c = \frac { 5 } { 9 } ( f - 32 )$. Find the equation of the new regression line.

SPS SPS FM Statistics 2024 January Q3

3. Eight runners took part in two races. The positions in which the runners finished in the two races are shown in the table.

Runner	A	B	C	D	E	F	G	H
First race	3	1	5	6	2	8	7	4
Second race	4	3	8	7	2	5	6	1

Test at the $5 \%$ significance level whether those runners who do better in one race tend to do better in the other.

SPS SPS FM Statistics 2024 January Q4

4. The manager of a car breakdown service uses the distribution $\operatorname { Po } ( 2.7 )$ to model the number of punctures, $R$, in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.

State a further assumption needed for the Poisson model to be valid.
State the value of the standard deviation of $R$.
Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution $\mathrm { Po } ( 0.8 )$ to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.

SPS SPS FM Statistics 2024 January Q5

5. A company uses two drivers for deliveries.
Driver $A$ charges a fixed rate of $\pounds 80$ per day plus $\pounds 2$ per mile travelled on that day.
Driver $B$ charges a fixed rate of $\pounds 120$ per day plus $\pounds 1.50$ per mile travelled on that day.
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution $\mathrm { N } ( 83,360 )$. Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver $A$ is at least $\pounds 300$ more than the total charge of two randomly chosen days' deliveries by driver $B$.

SPS SPS FM Statistics 2024 January Q6

6. A firm claims that no more than $2 \%$ of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than $2 \%$. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the $5 \%$ significance level, using the null hypothesis $p = 0.02$.

Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
The researcher finds that 18 out of the 600 packets are underweight. A colleague says
" 18 out of 600 is $3 \%$, so there is evidence that the actual proportion of underweight bags is greater than $2 \%$." Criticise this statement.

SPS SPS FM Statistics 2024 January Q7

7. The random variable $X$ was assumed to have a normal distribution with mean $\mu$. Using a random sample of size 128, a significance test was carried out using the following hypotheses.
$\mathrm { H } _ { 0 } : \mu = 30$
$\mathrm { H } _ { 1 } : \mu > 30$
It was found that $\sum x = 3929.6$ and $\sum x ^ { 2 } = 123483.52$. The conclusion of the test was to reject the null hypothesis.

Determine the range of possible values of the significance level of the test.
It was subsequently found that $X$ was not normally distributed. Explain whether this invalidates the conclusion of the test.

SPS SPS FM Statistics 2024 January Q8

8. A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher arranges all 10 books in random order on a shelf.
a) Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.} b) Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.

SPS SPS FM Statistics 2024 January Q9

9. The continuous random variable $X$ has a uniform distribution on the interval $[ - \pi , \pi ]$.
The random variable $Y$ is defined by $Y = \sin X$. Determine the cumulative distribution function of $Y$. END OF TEST

SPS SPS SM Statistics 2024 January Q1

1. At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below.
$n = 205 \quad \sum x = 23042 \quad \sum x ^ { 2 } = 2591716$ Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

Calculate
- The mean mark
- The variance of the marks
- Use your answers to part (a) to write down a possible Normal model for the distribution of marks.

SPS SPS SM Statistics 2024 January Q2

2. The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{0e73f1d0-5532-4995-b39e-759d82c2bd92-04_860_1684_367_130} One of the 150 plants is chosen at random, and its height, $X \mathrm {~cm}$, is noted.

Show that $\mathrm { P } ( 20 < X < 30 ) = 0.147$, correct to 3 significant figures. Sam suggests that the distribution of $X$ can be well modelled by the distribution $\mathrm { N } ( 40,100 )$.
1. Give a brief justification for the use of the normal distribution in this context.
2. Give a brief justification for the choice of the parameter values 40 and 100 .
Use Sam's model to find $\mathrm { P } ( 20 < X < 30 )$. Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, $m$ and $s$, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution $\mathrm { N } \left( m , s ^ { 2 } \right)$ as her model.
Use Nina's model to find $\mathrm { P } ( 20 < X < 30 )$.

Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.

By considering the different ranges of values of $X$ given in the table, discuss how well the two models fit the original distribution. Table for (e)(i):

$x$	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
Probability obtained from Nina's model, $\mathrm { N } \left( m , s ^ { 2 } \right)$	0.030		0.153	0.188			0.130	0.023

SPS SPS SM Statistics 2024 January Q3

3. Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
Mean
Standard
deviation
Median
Lower
quartile
Upper
quartile
21.0 4.20 20.5 18.0 22.9
Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the $5 \%$ significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.

SPS SPS SM Statistics 2024 January Q4

4. The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.

Geography code	Local authority	Work mainly at or from home	Underground, metro, light rail or tram	Bus, minibus or coach	Driving a car or van	Passenger in a car or van	On foot
E06000047	County Durham	0.74\%	0.05\%	-1.50\%	4.58\%	-2.99\%	-0.97\%
E06000005	Darlington	0.26\%	-0.01\%	-3.25\%	3.06\%	-1.28\%	0.99\%
E08000020	Gateshead	-0.01\%	-0.01\%	-2.28\%	4.62\%	-2.35\%	-0.18\%
E06000001	Hartlepool	0.03\%	-0.04\%	-1.62\%	4.80\%	-2.38\%	-0.26\%
E06000002	Middlesbrough	-0.34\%	-0.01\%	-2.32\%	2.19\%	-1.33\%	0.67\%
E08000021	Newcastle upon Tyne	0.10\%	-0.23\%	-0.67\%	-0.48\%	-1.51\%	1.75\%
E08000022	North Tyneside	0.05\%	0.54\%	-1.18\%	3.30\%	-2.21\%	-0.60\%
E06000048	Northumberland	1.39\%	-0.08\%	-0.95\%	3.50\%	-2.37\%	-1.44\%
E06000003	Redcar and Cleveland	-0.02\%	-0.01\%	-2.09\%	4.20\%	-2.06\%	-0.49\%
E08000023	South Tyneside	-0.36\%	2.03\%	-3.05\%	4.50\%	-2.41\%	-0.51\%
E06000004	Stockton-on-Tees	0.14\%	0.03\%	-2.02\%	3.52\%	-2.01\%	-0.15\%
E08000024	Sunderland	0.17\%	1.48\%	-3.11\%	4.89\%	-2.21\%	-0.52\%

\section*{Increase in percentage of employees travelling to work by various methods} The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough

In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
- The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
- The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
- Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011.
- In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011.

SPS SPS SM Statistics 2024 January Q5

5. Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.

	male	female
black	1	3
yellow	2	1
chocolate	1	1

Four puppies are chosen at random to train as guide dogs.
(b) Determine the probability that at least 3 black puppies are chosen.
(c) Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
(d) Explain whether the 2 events
'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. A firm claims that no more than $2 \%$ of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than $2 \%$. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the $5 \%$ significance level, using the null hypothesis $p = 0.02$.
(a) Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
(b) The researcher finds that 18 out of the 600 packets are underweight. A colleague says
" 18 out of 600 is $3 \%$, so there is evidence that the actual proportion of underweight bags is greater than $2 \%$." Criticise this statement.

SPS SPS SM Statistics 2024 January Q7

7. The probability distribution of a random variable $X$ is modelled as follows.
$\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 ,
0 & \text { otherwise, } \end{cases}$
where $k$ is a constant.

Show that $k = \frac { 12 } { 25 }$.
Show in a table the values of $X$ and their probabilities.
The values of three independent observations of $X$ are denoted by $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$. Find $\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)$. In a game, a player notes the values of successive independent observations of $X$ and keeps a running total. The aim of the game is to reach a total of exactly 7 .
Determine the probability that a total of exactly 7 is first reached on the 5th observation. END OF TEST

SPS SPS SM Pure 2024 February Q1

1. Find $\int \left( 2 x ^ { 4 } - x \sqrt { x } \right) \mathrm { d } x$.

SPS SPS SM Pure 2024 February Q2

2. The coefficient of $x ^ { 8 }$ in the expansion of $( 2 x + k ) ^ { 12 }$, where $k$ is a positive integer, is 79200000.
Determine the value of $k$.

SPS SPS SM Pure 2024 February Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-06_709_1052_287_552} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$
The table below shows corresponding values of $x$ and $y$ for this curve between $x = 0.5$ and $x = 0.9$ The values of $y$ are given to 4 significant figures.

$x$	0.5	0.6	0.7	0.8	0.9
$y$	1.632	1.711	1.786	1.859	1.930

Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for $$\int _ { 0.5 } ^ { 0.9 } \mathrm { f } ( x ) \mathrm { d } x$$ Give your answer to 3 significant figures.
Using your answer to part (a), deduce an estimate for $$\int _ { 0.5 } ^ { 0.9 } ( 3 \mathrm { f } ( x ) + 2 ) \mathrm { d } x$$

SPS SPS SM Pure 2024 February Q4

4. Relative to a fixed origin $O$,
the point $A$ has position vector $( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )$,
the point $B$ has position vector $( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$,
and the point $C$ has position vector $( a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k } )$, where $a$ is a constant and $a < 0$
$D$ is the point such that $\overrightarrow { A B } = \overrightarrow { B D }$.

Find the position vector of $D$.
(2) Given $| \overrightarrow { A C } | = 4$
find the value of $a$.
(3)

SPS SPS SM Pure 2024 February Q5

5. The diagram shows the graph of $y = 1.5 + \sin ^ { 2 } x$ for $0 \leqslant x \leqslant 2 \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-10_513_1266_349_210}

Show that the equation of the graph can be written in the form $y = a - b \cos 2 x$ where $a$ and $b$ are constants to be determined.
Write down the period of the function $1.5 + \sin ^ { 2 } x$.
Determine the $x$-coordinates of the points of intersection of the graph of $y = 1.5 + \sin ^ { 2 } x$ with the graph of $y = 1 + \cos 2 x$ in the interval $0 \leqslant x \leqslant 2 \pi$.

SPS SPS SM Pure 2024 February Q6

6. Curve $C$ has equation $$y = \left( x ^ { 2 } - 5 x + 8 \right) \mathrm { e } ^ { x ^ { 2 } } \quad x \in \mathbb { R }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( 2 x ^ { 3 } - 10 x ^ { 2 } + 18 x - 5 \right) \mathrm { e } ^ { x ^ { 2 } }$$ Given that
- $C$ has only one stationary point
- the stationary point has $x$ coordinate $\alpha$
- $\frac { \mathrm { d } y } { \mathrm {~d} x } \approx - 0.5$ at $x = 0.3$
- $\frac { \mathrm { d } y } { \mathrm {~d} x } \approx 0.9$ at $x = 0.4$
- explain why $0.3 < \alpha < 0.4$
- Show that $\alpha$ is a solution of the equation
$$x = \frac { 5 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 9 \right) }$$
Using the iteration formula $$x _ { n + 1 } = \frac { 5 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 9 \right) } \quad \text { with } x _ { 1 } = 0.3$$ find
1. the value of $x _ { 3 }$ to 4 decimal places,
2. the value of $\alpha$ to 4 decimal places.

SPS SPS SM Pure 2024 February Q7

7. The function f is defined by $$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where $k$ is a positive constant.

Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where $\mathrm { g } ( x )$ is a function to be found. Given that the curve with equation $y = \mathrm { f } ( x )$ has at least one stationary point,
find the range of possible values of $k$.

SPS SPS SM Pure 2024 February Q8

4 marks

Evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$
Find the smallest positive exact value of $\theta$, in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$ [4 marks]

SPS SPS SM Pure 2024 February Q9

Express $2 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ Give the exact value of $R$ and the value of $\alpha$ in radians to 3 decimal places. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-18_456_1150_488_584} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows the cross-section of a water wheel.
The wheel is free to rotate about a fixed axis through the point $C$.
The point $P$ is at the end of one of the paddles of the wheel, as shown in Figure 6.
The water level is assumed to be horizontal and of constant height.
The vertical height, $H$ metres, of $P$ above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where $t$ is the time in seconds after the wheel starts rotating.
Using the model, find
1. the maximum height of $P$ above the water level,
2. the value of $t$ when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, $P$ is below the water level for a total of $T$ seconds. According to the model,
find the value of $T$ giving your answer to 3 significant figures.
(Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
Explain how the equation of the model should be refined to take this into account.

\(x\)	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
Probability obtained from Nina's model, \(\mathrm { N } \left( m , s ^ { 2 } \right)\)	0.030		0.153	0.188			0.130	0.023

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