Questions - AQA AS Paper 2

1. Explain why angle $A$ must be opposite the side of length $( m + n )$. 8
(ii) Using the cosine rule, show that $\cos A = \frac { m - 4 n } { 2 ( m - n ) }$
8
You are given that $B C$ is the diameter of a circle, and $A$ lies on the circumference of the circle. The value of $m$ is 8 Calculate the value of $n$.

AQA AS Paper 2 2022 June Q9

6 marks

9 The diagram below shows the graphs of $y = x ^ { 2 } - 4 x - 12$ and $y = x + 2$
\includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-10_933_912_358_566} 9

Write down three inequalities which together describe the shaded region.
9
Find the coordinates of the points $A , B$ and $C$.
9
Find the exact area of the shaded region.
Fully justify your answer.
[0pt] [6 marks]

AQA AS Paper 2 2022 June Q10

10 A bottle of water has a temperature of $6 ^ { \circ } \mathrm { C }$ when it is removed from a refrigerator. It is placed in a room where the temperature is $20 ^ { \circ } \mathrm { C }$
10 minutes later, the temperature of the water is $12 ^ { \circ } \mathrm { C }$
The temperature of the water, $T ^ { \circ } \mathrm { C }$, at time $t$ minutes after it is removed from the refrigerator, may be modelled by the equation $$T = 20 - a \mathrm { e } ^ { - k t }$$ 10

Find the value of $a$. 10
Calculate the value of $k$, giving your answer to two significant figures.
10
Using this model, estimate how long it takes the water to reach a temperature of
$18 ^ { \circ } \mathrm { C }$ after it is taken out of the refrigerator. $18 ^ { \circ } \mathrm { C }$ after it is taken out of the refrigerator. 10
Explain why the model may not be appropriate to predict the temperature of the water three hours after it is taken out of the refrigerator.

AQA AS Paper 2 2022 June Q11

11 Which of the terms below best describes the distribution represented by the boxplot shown in Figure 1? \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-14_154_831_927_584}

\end{figure} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-14_76_1143_1151_450}

\end{figure} Circle your answer.
even
negatively skewed
positively skewed
symmetric

AQA AS Paper 2 2022 June Q12

1 marks

12 Shelly organised an activity weekend for 15 groups of 10 people.
She decided to collect a sample to obtain feedback about the weekend.
To collect the sample Shelly selected two groups at random and then interviewed each member of these two groups. State the name of this sampling method.
Circle your answer.
[0pt] [1 mark] Cluster
Opportunity
Stratified
Systematic
\includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-15_2488_1716_219_153}

AQA AS Paper 2 2022 June Q13

13 Two random samples of 12 NOX emissions (in $\mathrm { g } / \mathrm { km }$ ) were taken from the Large Data Set. One sample was taken from the 2002 data and the other sample from the 2016 data.
The sample data are shown below:

\multirow{2}{*}{2002}	0.031	0.019	0.091	0.025	0.030	0.061
	0.047	0.029	0.059	0.363	0.330	0.376

\multirow{2}{*}{2016}	0.005	0.047	0.053	0.063	0.026	0.013
	0.058	0.012	0.010	0.010	0.008	0.008

The mean and standard deviation of the 2002 sample data are 0.122 and 0.137 respectively. 13

Find the mean and standard deviation of the 2016 sample data giving your answers correct to three decimal places.
13
Siti claims these samples show that, on average, the NOX emissions across all makes of car in all areas of the UK have fallen by over 75\% between 2002 and 2016. 13
1. Show how Siti's claim of 'over 75\%' has been obtained.
  13
(ii) Using your knowledge of the Large Data Set, make two comments on the validity of Siti's claim. Comment 1
\section*{Comment 2}

AQA AS Paper 2 2022 June Q14

2 marks

14 Yingtai visits her local gym regularly. After each visit she chooses one item to eat from the gym's cafe.
This could be an apple, a banana or a piece of cake.
She chooses the item independently each time.
The probability that Yingtai chooses each of these items on any visit is given by: $$\begin{aligned} \mathrm { P } ( \text { Apple } ) & = 0.2
\mathrm { P } ( \text { Banana } ) & = 0.35
\mathrm { P } ( \text { Cake } ) & = 0.45 \end{aligned}$$ For any four randomly selected visits to the gym, find the probability that Yingtai chose: 14

at least one banana.
[0pt] [2 marks]
14
the same item each time.

14
apple twice and cake twice

AQA AS Paper 2 2022 June Q15

15 The discrete random variable $X$ is modelled by the probability distribution defined by: $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c x & x = 1,2
k x ^ { 2 } & x = 3,4
0 & \text { otherwise } \end{array} \right.$$ where $k$ and $c$ are constants.
15

State, in terms of $k$, the probability that $X = 3$
15
Given that $\mathrm { P } ( X \geq 3 ) = 3 \times \mathrm { P } ( X \leq 2 )$
Find the exact value of $k$ and the exact value of $c$.
\includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-21_2488_1716_219_153}

AQA AS Paper 2 2022 June Q16

16 It is believed that a coin is biased so that the probability of obtaining a head when the coin is tossed is 0.7 16

Assume that the probability of obtaining a head when the coin is tossed is indeed 0.7
16
1. Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.
  16
(ii) Find the mean number of heads obtained from 7 tosses of the coin.
16
Harry believes that the probability of obtaining a head for this coin is actually greater than 0.7 To test this belief he tosses the coin 35 times and obtains 28 heads. Carry out a hypothesis test at the $10 \%$ significance level to investigate Harry's belief.
\includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-24_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-28_2498_1722_213_147}

AQA AS Paper 2 2023 June Q1

1 Simplify $\log _ { 2 } 8 ^ { a }$
Circle your answer.
$a ^ { 3 }$
$2 a$
3a
$8 a$

AQA AS Paper 2 2023 June Q2

1 marks

2 It is given that $\sin \theta = \frac { 4 } { 5 }$ and $90 ^ { \circ } < \theta < 180 ^ { \circ }$
Find the value of $\cos \theta$ Circle your answer.
[0pt] [1 mark]
$- \frac { 3 } { 4 }$
$- \frac { 3 } { 5 }$
$\frac { 3 } { 5 }$
$\frac { 3 } { 4 }$

AQA AS Paper 2 2023 June Q3

Find $\int \left( 2 x ^ { 3 } + \frac { 8 } { x ^ { 2 } } \right) \mathrm { d } x$ 3
A curve has gradient function $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + \frac { 8 } { x ^ { 2 } }$
The $x$-intercept of the curve is at the point $( 2,0 )$
Find the equation of the curve.
Fully justify your answer.
$4 \quad$ Find the exact solution of the equation $\ln ( x + 1 ) + \ln ( x - 1 ) = \ln 15 - 2 \ln 7$

AQA AS Paper 2 2023 June Q5

5 It is given that $\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ and $\cos 15 ^ { \circ } = \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$
Use these two expressions to show that $\tan 15 ^ { \circ } = 2 - \sqrt { 3 }$
Fully justify your answer.

AQA AS Paper 2 2023 June Q6

6 A curve has equation $$y = 2 x ^ { 2 } + p x + 1$$ A line has equation $$y = 5 x - 2$$ Find the set of values of $p$ for which the line intersects the curve at two distinct points.
Give your answer in exact form.
$7 \quad$ The curve $C$ has equation $y = \mathrm { f } ( x )$
$C$ has a maximum point at $P$ with coordinates ( $a , 2 b$ ) as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-07_483_643_456_788}

AQA AS Paper 2 2023 June Q7

$\quad C$ is mapped by a single transformation onto curve $C _ { 1 }$ with equation $y = \mathrm { f } ( x + 2 )$ State the coordinates of the maximum point on curve $C _ { 1 }$
7
$\quad C$ is mapped by a single transformation onto curve $C _ { 2 }$ with equation $y = 4 \mathrm { f } ( x )$ State the coordinates of the maximum point on curve $C _ { 2 }$
7
$\quad C$ is mapped by a stretch in the $x$-direction onto curve $C _ { 3 }$ with equation $y = \mathrm { f } ( 3 x )$ State the scale factor of the stretch.

AQA AS Paper 2 2023 June Q8

8 Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4 .
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-09_2488_1716_219_153}

AQA AS Paper 2 2023 June Q9

1 marks

9 A craft artist is producing items and selling them in a local market. The selling price, $\pounds P$, of an item is inversely proportional to the number of items produced, $n$ 9

When $n = 10 , P = 24$
Show that $P = \frac { 240 } { n }$ 9
The production cost, $\pounds C$, of an item is inversely proportional to the square of the number of items produced, $n$ When $n = 10 , C = 60$ Find the set of values of $n$ for which $P > C$
9
Explain the significance to the craft artist of the range of values found in part (b).
[0pt] [1 mark]

AQA AS Paper 2 2023 June Q10

10 A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure $x \mathrm {~cm}$ and $y \mathrm {~cm}$ and the sides of the triangle measure $x \mathrm {~cm}$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-12_405_492_630_863} 10

1. You are given that $\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$
  Explain why the area of the triangle is $\frac { \sqrt { 3 } } { 4 } x ^ { 2 }$ 10
(ii) Show that the area enclosed by the wire, $A \mathrm {~cm} ^ { 2 }$, can be expressed by the formula $$A = 33 x - \frac { 1 } { 4 } ( 6 - \sqrt { 3 } ) x ^ { 2 }$$ 10
Use calculus to find the value of $x$ for which the wire encloses the maximum area. Give your answer in the form $p + q \sqrt { 3 }$, where $p$ and $q$ are integers. Fully justify your answer.
$L _ { 1 }$ is a tangent to the circle $C$ at the point $P ( 6,5 )$
The line $L _ { 2 }$ has equation $y = x + 3$
$L _ { 2 }$ is a tangent to the circle $C$ at the point $Q ( 0,3 )$
The lines $L _ { 1 }$ and $L _ { 2 }$ and the circle $C$ are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-14_702_714_726_753}

AQA AS Paper 2 2023 June Q11

4 marks

Show that the equation of the radius of the circle through $P$ is $y = 7 x - 37$

11
Find the equation of $C$ Do not write outside the box
[4 marks]

AQA AS Paper 2 2023 June Q12

12 The mass of a bag of nuts produced by a company is known to have a mean of 40 grams and a standard deviation of 3 grams. The company produces five different flavours of nuts.
The bags of nuts are packed in large boxes.
Given the information above, identify the continuous variable from the options below.
Tick ( $\checkmark$ ) one box. The flavours of the bags of nuts The known standard deviation of the mass of a bag of nuts
□ The mass of an individual bag of nuts
□ The number of bags of nuts in a large box
□
The number of bags of nuts in a large box □

AQA AS Paper 2 2023 June Q13

13 The table below shows the frequencies for a set of data from a continuous variable $X$

$\boldsymbol { x }$	Frequency
$11 < x \leq 21$	7
$21 < x \leq 24$	9
$24 < x \leq 42$	36
$42 < x \leq 50$	18

A histogram is drawn to represent this data.
Find the frequency density of the bar in the histogram representing the class $24 < x \leq 42$ Circle your answer. 2183670

AQA AS Paper 2 2023 June Q14

1 marks

14 The manager of a factory wants to introduce a bonus scheme. The factory has 65 employees who work in production and 28 employees who work in the office. The manager decides to collect the opinions of a sample of these 93 employees.
14

Explain how the manager could collect a simple random sample of 20 employees.
14
The manager collected a simple random sample of 20 employees.
The manager noticed that all 20 of the employees in the sample worked in production and therefore the sample was not representative. State a different method of sampling that would give a representative sample.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-19_2488_1716_219_153}