Questions — AQA AS Paper 2 (137 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

AQA AS Paper 2 Specimen Q7

7 Solve the equation $$\sin \theta \tan \theta + 2 \sin \theta = 3 \cos \theta \quad \text { where } \cos \theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
Fully justify your answer.
[0pt] [5 marks]

AQA AS Paper 2 Specimen Q8

8 Prove that the function \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 15 x - 1\) is an increasing function.
[0pt] [6 marks]

AQA AS Paper 2 Specimen Q9

9 A curve has equation \(y = \mathrm { e } ^ { 2 x }\)
Find the coordinates of the point on the curve where the gradient of the curve is \(\frac { 1 } { 2 }\) Give your answer in an exact form.
[0pt] [5 marks]
David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.

AQA AS Paper 2 Specimen Q10

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

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

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

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

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

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

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

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]

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