Questions AS Paper 2 - A-Level Maths

Using David's model: 10
1. state the population of rabbits on the island on 1 January 2016; 10
(ii) predict the population of rabbits on 1 January 2021. 10
Use David's model to find the value of $t$ when $R = 150$, giving your answer to three significant figures.
[0pt] [2 marks] 10
Give one reason why David's model may not be appropriate.
[0pt] [1 mark] 10
On the same island, the population of crickets, $C$, can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where $t$ is the time in years after 1 January 2016.
Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
[0pt] [3 marks]

AQA AS Paper 2 Specimen Q11

9 marks

11 The circle with equation $( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5$ has centre $C$. 11

1. Write down the radius of the circle. 11
(ii) Write down the coordinates of $C$.
[0pt] [1 mark] 11
The point $P ( 5 , - 1 )$ lies on the circle.
Find the equation of the tangent to the circle at $P$, giving your answer in the form $y = m x + c$
[0pt] [4 marks] 11
The point $Q ( 3,3 )$ lies outside the circle and the point $T$ lies on the circle such that $Q T$ is a tangent to the circle. Find the length of $Q T$.
[0pt] [4 marks]

AQA AS Paper 2 Specimen Q12

3 marks

Given that $n$ is an even number, prove that $9 n ^ { 2 } + 6 n$ has a factor of 12
[0pt] [3 marks]
12
Determine if $9 n ^ { 2 } + 6 n$ has a factor of 12 for any integer $n$.
END OF SECTION A

AQA AS Paper 2 Specimen Q13

13 The number of pots of yoghurt, $X$, consumed per week by adults in Milton is a discrete random variable with probability distribution given by

$\boldsymbol { x }$	0	1	2	3	4	5	6	7 or more
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.30	0.10	0.05	0.07	0.03	0.16	0.09	0.20

Find $\mathrm { P } ( 3 \leq X < 6 )$ Circle the correct answer. $0.26 \quad 0.31 \quad 0.35 \quad 0.40$

AQA AS Paper 2 Specimen Q14

3 marks

14 In the Large Data Set, the emissions of carbon dioxide are measured in what units? Circle your answer.
[0pt] [1 mark]
mg/litre
g/litre
g/km
mg/km A school took 225 children on a trip to a theme park.
After the trip the children had to write about their favourite ride at the park from a choice of three. The table shows the number of children who wrote about each ride.

\multirow{2}{*}{}		Ride written about
		The Drop	The Beanstalk	The Giant	Total
\multirow{3}{*}{Year group}	Year 7	24	45	23	92
	Year 8	36	17	22	75
	Year 9	20	13	25	58
	Total	80	75	70	225

Three children were randomly selected from those who went on the trip.
Calculate the probability that one wrote about 'The Drop', one wrote about ‘The Beanstalk’ and one wrote about The Giant’.
[0pt] [2 marks]

AQA AS Paper 2 Specimen Q16

2 marks

16 The boxplot below represents the time spent in hours by students revising for a history exam.
\includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-18_373_753_427_778} 16

Use the information in the boxplot to state the value of a measure of central tendency of the revision times, stating clearly which measure you are using.
[0pt] [1 mark] 16
Use the information in the boxplot to explain why the distribution of revision times is negatively skewed.
[0pt] [1 mark]

AQA AS Paper 2 Specimen Q17

6 marks

17 The table below is an extract from the Large Data Set.

Make	Region	Engine size	Mass	CO2	CO
VAUXHALL	South West	1398	1163	118	0.463
VOLKSWAGEN	London	999	1055	106	0.407
VAUXHALL	South West	1248	1225	85	0.141
BMW	South West	2979	1635	194	0.139
TOYOTA	South West	1995	1650	123	0.274
BMW	South West	2979	0	244	0.447
FORD	South West	1596	0	165	0.518
TOYOTA	South West	1299	1050	144
VAUXHALL	London	1398	1361	140	0.695
FORD	North West	4951	1799	299	0.621

1. Calculate the standard deviation of the engine sizes in the table.
  [0pt] [1 mark] 17
(ii) The mean of the engine sizes is 2084
Any value more than 2 standard deviations from the mean can be identified as an outlier. Using this definition of an outlier, show that the sample of engine sizes has exactly one outlier. Fully justify your answer.
[0pt] [3 marks] 17
Rajan calculates the mean of the masses of the cars in this extract and states that it is 1094 kg. Use your knowledge of the Large Data Set to suggest what error Rajan is likely to have made in his calculation.
[0pt] [1 mark] 17
Rajan claims there is an error in the data recorded in the table for one of the Toyotas from the South West, because there is no value for its carbon monoxide emissions. Use your knowledge of the Large Data Set to comment on Rajan's claim.
[0pt] [1 mark]

AQA AS Paper 2 Specimen Q18

4 marks

18 Neesha wants to open an Indian restaurant in her town.
Her cousin, Ranji, has an Indian restaurant in a neighbouring town. To help Neesha plan her menu, she wants to investigate the dishes chosen by a sample of Ranji's customers. Ranji has a list of the 750 customers who dined at his restaurant during the past month and the dish that each customer chose. Describe how Neesha could obtain a simple random sample of size 50 from Ranji's customers.
[0pt] [4 marks]

AQA AS Paper 2 Specimen Q19

9 marks

19 Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size $n$, can be modelled by the binomial distribution $\mathrm { B } ( n , 0.20 )$. 19

There are 10 students in Ellie's statistics class.
Using the distributional model suggested by Ellie, find the probability that, of the students in her class: 19
1. two or fewer eat at least five portions of fruit and vegetables every day;
  [0pt] [1 mark] 19
(ii) at least one but fewer than four eat at least five portions of fruit and vegetables every day;
[0pt] [2 marks] 19
Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day. 19
1. Name the sampling method used by Declan. 19
(ii) Describe one weakness of this sampling method.
19
(iii) Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the $5 \%$ significance level to investigate whether Declan's belief is supported by this evidence.
[0pt] [6 marks]

Edexcel AS Paper 2 2018 June Q1

A company is introducing a job evaluation scheme. Points ( $x$ ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( $\pounds y$ ) will then be allocated to each job according to the number of points awarded.

Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was $y = 4.5 x - 47$

Describe the correlation between points and pay.
Give an interpretation of the gradient of this regression line.
Explain why this model might not be appropriate for all jobs in the company.

Edexcel AS Paper 2 2018 June Q2

A factory buys $10 \%$ of its components from supplier $A , 30 \%$ from supplier $B$ and the rest from supplier $C$. It is known that $6 \%$ of the components it buys are faulty.

Of the components bought from supplier $A , 9 \%$ are faulty and of the components bought from supplier $B , 3 \%$ are faulty.

Find the percentage of components bought from supplier $C$ that are faulty. A component is selected at random.
Explain why the event "the component was bought from supplier $B$ " is not statistically independent from the event "the component is faulty".

Edexcel AS Paper 2 2018 June Q3

Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is $\frac { 1 } { 3 }$
Naasir and his friends play the game 15 times.
1. Find the probability that Naasir wins
  1. exactly 2 games,
  2. more than 5 games.
Naasir claims he has a method to help him win more than $\frac { 1 } { 3 }$ of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
Stating your hypotheses clearly, test Naasir's claim at the $5 \%$ level of significance.

Edexcel AS Paper 2 2018 June Q4

Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.

\begin{table}[h]

Windspeed	$\mathrm { n } / \mathrm { a }$	6	7	8	9	11	12	13	14	16
Frequency	13	2	3	2	2	3	1	2	1	2

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table}

Calculate the mean for these data.
Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]
Month $A$ $B$ $C$ $D$ $E$
Mean 7.58 8.26 8.57 8.57 11.57
Standard Deviation 2.93 3.89 3.46 3.87 4.64
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
1. State the meaning of the * symbol on some of the box plots.
2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked $Y$. \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}

Edexcel AS Paper 2 2018 June Q5

5. A biased spinner can only land on one of the numbers $1,2,3$ or 4 . The random variable $X$ represents the number that the spinner lands on after a single spin and $\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )$ for $r = 1,2$ Given that $\mathrm { P } ( X = 2 ) = 0.35$

find the complete probability distribution of $X$. Ambroh spins the spinner 60 times.
Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable $Y = \frac { 12 } { X }$
Find $\mathrm { P } ( Y - X \leqslant 4 )$

Edexcel AS Paper 2 2018 June Q6

A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The ball hits the ground $T$ seconds after leaving the man's hand.
Using the model, find the value of $T$.

Edexcel AS Paper 2 2018 June Q7

A train travels along a straight horizontal track between two stations, $A$ and $B$.

In a model of the motion, the train starts from rest at $A$ and moves with constant acceleration $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, coming to rest at $B$.

For this model of the motion of the train between $A$ and $B$,
1. state the value of the constant velocity of the train,
2. state the time for which the train is decelerating,
3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
Using the model, find the total time taken by the train to travel from $A$ to $B$.
Suggest one improvement that could be made to the model of the motion of the train from $A$ to $B$ in order to make the model more realistic.

Edexcel AS Paper 2 2018 June Q8

A particle, $P$, moves along the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the displacement, $x$ metres, of $P$ from the origin $O$, is given by $x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)$
1. Find the times when $P$ is instantaneously at rest.
2. Find the total distance travelled by $P$ in the time interval $0 \leqslant t \leqslant 2$
3. Show that $P$ will never move along the negative $x$-axis.

Edexcel AS Paper 2 2018 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two small balls, $P$ and $Q$, have masses $2 m$ and $k m$ respectively, where $k < 2$.
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, $P$ moves downwards with an acceleration of magnitude $\frac { 5 g } { 7 }$ The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

find, in terms of $m$ and $g$, the tension in the string,
explain why the acceleration of $Q$ also has magnitude $\frac { 5 g } { 7 }$
find the value of $k$.
Identify one limitation of the model that will affect the accuracy of your answer to part (c).

Edexcel AS Paper 2 Specimen Q1

Sara is investigating the variation in daily maximum gust, $t \mathrm { kn }$, for Camborne in June and July 1987.

She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.

State the sampling technique Sara used.
From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
Calculate the standard deviation.

Edexcel AS Paper 2 Specimen Q2

The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway.
\includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}

Delay (minutes)	Number of motorists
4-6	6
7-8
9	17
10-12	45
13-15	9
16-20

Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.

Edexcel AS Paper 2 Specimen Q3

The Venn diagram shows the probabilities for students at a college taking part in various sports.
$A$ represents the event that a student takes part in Athletics.
$T$ represents the event that a student takes part in Tennis.
$C$ represents the event that a student takes part in Cricket.
$p$ and $q$ are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}

The probability that a student selected at random takes part in Athletics or Tennis is 0.75

Find the value of $p$.
State, giving a reason, whether or not the events $A$ and $T$ are statistically independent. Show your working clearly.
Find the probability that a student selected at random does not take part in Athletics or Cricket.

Edexcel AS Paper 2 Specimen Q4

Sara was studying the relationship between rainfall, $r \mathrm {~mm}$, and humidity, $h \%$, in the UK. She takes a random sample of 11 days from May 1987 for Leuchars from the large data set.

She obtained the following results.

$h$	93	86	95	97	86	94	97	97	87	97	86
$r$	1.1	0.3	3.7	20.6	0	0	2.4	1.1	0.1	0.9	0.1

Sara examined the rainfall figures and found $$Q _ { 1 } = 0.1 \quad Q _ { 2 } = 0.9 \quad Q _ { 3 } = 2.4$$ A value that is more than 1.5 times the interquartile range (IQR) above $Q _ { 3 }$ is called an outlier.

Show that $r = 20.6$ is an outlier.
Give a reason why Sara might:
1. include
2. exclude
  this day's reading. Sara decided to exclude this day's reading and drew the following scatter diagram for the remaining 10 days' values of $r$ and $h$.
  \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-08_988_1081_1555_420}
Give an interpretation of the correlation between rainfall and humidity. The equation of the regression line of $r$ on $h$ for these 10 days is $r = - 12.8 + 0.15 h$
Give an interpretation of the gradient of this regression line.
1. Comment on the suitability of Sara's sampling method for this study.
2. Suggest how Sara could make better use of the large data set for her study.

Edexcel AS Paper 2 Specimen Q5

5. (a) The discrete random variable $X \sim \mathrm {~B} ( 40,0.27 )$ $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that $30 \%$ of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a $10 \%$ level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.

\(\boldsymbol { x }\)	0	1	2	3	4	5	6	7 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.30	0.10	0.05	0.07	0.03	0.16	0.09	0.20

Windspeed	\(\mathrm { n } / \mathrm { a }\)	6	7	8	9	11	12	13	14	16
Frequency	13	2	3	2	2	3	1	2	1	2

Month	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
Mean	7.58	8.26	8.57	8.57	11.57
Standard Deviation	2.93	3.89	3.46	3.87	4.64

\(h\)	93	86	95	97	86	94	97	97	87	97	86
\(r\)	1.1	0.3	3.7	20.6	0	0	2.4	1.1	0.1	0.9	0.1

Questions AS Paper 2 (308 questions)

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