Questions - AQA - A-Level Maths

Sketch the two possible positions of the circle.
\includegraphics[max width=\textwidth, alt={}, center]{8d9ace4b-0c15-48bd-9b0d-302f57ea9759-08_892_1244_742_376}
\multirow{3}{*}{}
Show that $\tan ^ { 2 } 15 ^ { \circ }$ can be written in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.
Fully justify your answer.
[0pt] [3 marks]

AQA AS Paper 2 2018 June Q9

3 marks Moderate -0.5

9 It is given that $\cos 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ and $\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

AQA AS Paper 2 2018 June Q10

5 marks Standard +0.3

10 In the binomial expansion of $( 1 + x ) ^ { n }$, where $n \geq 4$, the coefficient of $x ^ { 4 }$ is $1 \frac { 1 } { 2 }$ times the sum of the coefficients of $x ^ { 2 }$ and $x ^ { 3 }$ Find the value of $n$.

AQA AS Paper 2 2018 June Q11

9 marks Standard +0.8

11 Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. bends a rectangle of steel to make an open cylinder and welds the joint. She She bends this cylinder to the circumference of a circular base then welds this cylinder to the circumference of a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{8d9ace4b-0c15-48bd-9b0d-302f57ea9759-12_552_524_497_753} The planter must have a capacity of $8000 \mathrm {~cm} ^ { 3 }$ Welding is time consuming, so Rakti wants the total length of weld to be a minimum.
Calculate the radius, $r$, and height, $h$, of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy.

AQA AS Paper 2 2018 June Q12

8 marks Moderate -0.3

12 Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease $\mathrm { A } , n _ { \mathrm { A } }$, can be modelled by the formula $$n _ { \mathrm { A } } = a \mathrm { e } ^ { 0.1 t }$$ where $t$ is the time in years after 1 January 2017.
The number of trees affected by disease $\mathrm { B } , n _ { \mathrm { B } }$, can be modelled by the formula $$n _ { \mathrm { B } } = b \mathrm { e } ^ { 0.2 t }$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease.
On 1 January 2018 a total of 331 trees were affected by a fungal disease.
12

Show that $b = 90$, to the nearest integer, and find the value of $a$.

12
Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020.
[1 mark]
12
Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A.
12
Comment on the long-term accuracy of the model.

AQA AS Paper 2 2018 June Q13

1 marks Easy -1.8

13 The table below shows the probability distribution for a discrete random variable $X$.

$\boldsymbol { x }$	0	1	2	3	4 or more
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.35	0.25	$k$	0.14	0.1

Find the value of $k$. Circle your answer.
0.140 .160 .1801

AQA AS Paper 2 2018 June Q14

1 marks Easy -1.8

14 Given that $\sum x = 364 , \sum x ^ { 2 } = 19412 , n = 10$, find $\sigma$, the standard deviation of $X$. Circle your answer.
[0pt] [1 mark]
24.844 .1616 .21941 .2

AQA AS Paper 2 2018 June Q15

6 marks Moderate -0.8

15 Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.
15

Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. 15
Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions.
15
State two assumptions that are necessary for the distribution you have used in part (a) to be valid.

AQA AS Paper 2 2018 June Q16

4 marks Easy -1.8

16 Kevin is the Principal of a college. He wishes to investigate types of transport used by students to travel to college.
There are 3200 students in the college and Kevin decides to survey 60 of them.
Describe how he could obtain a simple random sample of size 60 from the 3200 students.
The table below is an extract from the Large Data Set, showing the purchased quantities of fats and oils for the South East of England in 2014.

Other vegetable and salad oils

Kim claims that more olive oil was purchased in the South East than soft margarine.
Explain why Kim may be incorrect.

AQA AS Paper 2 2018 June Q18

6 marks Easy -1.3

18 Jennie is a piano teacher who teaches nine pupils. She records how many hours per week they practice the piano along with their most recent practical exam score.

Student	Practice (hours per week)	Practical exam score (out of 100)
Donovan	50	64
Vazquez	6	71
Higgins	3	55
Begum	2.5	47
Collins	1	80
Coldbridge	4	61
Nedbalek	4.5	65
Carter	8	83
White	11	92

She plots a scatter diagram of this data, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{8d9ace4b-0c15-48bd-9b0d-302f57ea9759-20_862_1516_1434_262} 18

Identify two possible outliers by name, giving a possible explanation for the position on the scatter diagram of each outlier. First outlier $\_\_\_\_$
Possible reason $\_\_\_\_$
Second outlier $\_\_\_\_$
Possible reason $\_\_\_\_$
18
Jennie discards the two outliers.
18
1. Describe the correlation shown by the scatter diagram for the remaining points.
  18
(ii) Interpret this correlation in the context of the question.
In the past, he has found that $70 \%$ of all seeds successfully germinate and grow into cucumber plants. He decides to try out a new brand of seed.
The producer of this brand claims that these seeds are more likely to successfully germinate than other brands of seeds. Martin sows 20 of this new brand of seed and 18 successfully germinate.
Carry out a hypothesis test at the $5 \%$ level of significance to investigate the producer's claim.

AQA AS Paper 2 2018 June Q19

7 marks Moderate -0.8

19 Martin grows cucumbers from seed.

AQA AS Paper 2 2019 June Q1

1 marks Easy -1.8

1 Find the gradient of the curve $y = \mathrm { e } ^ { - 3 x }$ at the point where it crosses the $y$-axis. Circle your answer.
$\begin{array} { l l l } - 3 & - 1 & 1 \end{array}$

AQA AS Paper 2 2019 June Q2

1 marks Easy -1.3

2 Find the centre of the circle $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12$
Tick ( $\checkmark$ ) one box.
(-2, -3) □
(-2, 3) □
$( 2 , - 3 )$ □
$( 2,3 )$ □

AQA AS Paper 2 2019 June Q3

2 marks Moderate -0.8

3 It is given that $\sin \theta = - 0.1$ and $180 ^ { \circ } < \theta < 270 ^ { \circ }$ Find the exact value of $\cos \theta$

AQA AS Paper 2 2019 June Q4

4 marks Moderate -0.8

4 Show that, for $x > 0$ $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$

AQA AS Paper 2 2019 June Q5

4 marks Moderate -0.3

5 A triangular prism has a cross section $A B C$ as shown in the diagram below. Angle $A B C = 25 ^ { \circ }$
Angle $A C B = 30 ^ { \circ }$
$B C = 40$ millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.

AQA AS Paper 2 2019 June Q6

5 marks Moderate -0.3

6 A curve has equation $y = \frac { 2 } { x \sqrt { x } }$
\includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-05_508_549_420_744} The region enclosed between the curve, the $x$-axis and the lines $x = 1$ and $x = a$ has area 3 units. Given that $a > 1$, find the value of $a$.
Fully justify your answer.

AQA AS Paper 2 2019 June Q7

6 marks Moderate -0.3

7 The points $A ( a , 3 )$ and $B ( 10,6 )$ lie on a circle.
$A B$ is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where $c , d$ and $e$ are rational numbers. Find the values of $a , c , d$ and $e$.

AQA AS Paper 2 2019 June Q8

10 marks Standard +0.3

8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point $R ( 2,3 )$
The gradient of the curve at $R$ is 8
8

Find the value of $p$ and the value of $q$.
8
Calculate the area enclosed between the normal to the curve at $R$ and the coordinate 8
axes.
$9 \quad$ A curve $C$ has equation $y = \mathrm { f } ( x )$ where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$

AQA AS Paper 2 2019 June Q9

10 marks Moderate -0.3

Find the exact coordinates of the turning points of $C$.
Determine the nature of each turning point.
Fully justify your answer.
9
State the coordinates of the turning points of the curve $$y = \mathrm { f } ( x + 1 ) - 4$$

AQA AS Paper 2 2019 June Q10

10 marks Moderate -0.3

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is $0 ^ { \circ } \mathrm { C }$. She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below.

Time (minutes)	10	20
Temperature (degrees Celsius)	30	12

Zena models the relationship between $\theta$, the temperature of the water in ${ } ^ { \circ } \mathrm { C }$, and $t$, the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where $A$ and $k$ are constants. 10

Using $t = 0$, explain how the value of $A$ relates to the experiment. 10
Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
Using Zena's results, calculate the values of $A$ and $k$.
10
Zena states that the temperature of the water will be less than $1 ^ { \circ } \mathrm { C }$ after 45 minutes. Determine whether the model supports this statement.
10
Explain why Zena's model is unlikely to accurately give the value of $\theta$ after 45 minutes.

AQA AS Paper 2 2019 June Q11

1 marks Easy -2.0

11 A survey is undertaken to find out the most popular political party in London.
The first 1100 available people from London are surveyed.
Identify the name of this type of sampling.
Circle your answer.
simple random
opportunity
stratified
quota

AQA AS Paper 2 2019 June Q12

1 marks Easy -1.8

12 Manny is studying the price and number of pages of a random sample of books.
He calculates the value of the product moment correlation coefficient between the price and number of pages in each book as 1.05 Which of the following best describes the value 1.05 ?
Tick ( $\checkmark$ ) one box.
definitely correct □
probably correct □
probably incorrect □
definitely incorrect □
\includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-15_2488_1716_219_153}

AQA AS Paper 2 2019 June Q13

6 marks Easy -1.2

13 Denzel wants to buy a car with a propulsion type other than petrol or diesel.
He takes a sample, from the Large Data Set, of the CO2 emissions, in $\mathrm { g } / \mathrm { km }$, of cars with one particular propulsion type. The sample is as follows $$\begin{array} { l l l l l l l l } 82 & 13 & 96 & 49 & 96 & 92 & 70 & 81 \end{array}$$ 13

Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.
13
Calculate the mean of the sample.
13
Calculate the standard deviation of the sample.
13
Denzel claims that the value 13 is an outlier. 13
1. Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
  13
(ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.

AQA AS Paper 2 2019 June Q14

4 marks Easy -1.2

14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where $c$ is a constant.
14

Show that $c = \frac { 1 } { 10 }$
14
Calculate $\mathrm { P } ( X \geq 1 )$

12	Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020.
	[1 mark]

\(\boldsymbol { x }\)	0	1	2	3	4 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.35	0.25	\(k\)	0.14	0.1

	\multirow{3}{*}{}

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