Questions AS Paper 2 - A-Level Maths

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AQA AS Paper 2 2022 June Q4

4 marks Moderate -0.8

4 The equation $9 x ^ { 2 } + 4 x + p ^ { 2 } = 0$ has no real solutions for $x$. Find the set of possible values of $p$.
Fully justify your answer.
[0pt] [4 marks]

AQA AS Paper 2 2022 June Q5

5 marks Moderate -0.8

5 Kaya is investigating the function $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 12 x + 45$$ Kaya makes two statements.
Statement 1: $\mathrm { f } ( 3 ) = 0$ Statement 2: this shows that ( $x + 3$ ) must be a factor of $\mathrm { f } ( x )$.
5

State, with a reason, whether each of Kaya's statements is correct. Statement 1: $\_\_\_\_$ Statement 2: $\_\_\_\_$ 5
Fully factorise f (x).

AQA AS Paper 2 2022 June Q6

6 marks Moderate -0.8

6 An on-line science website states:
'To find a dog's equivalent human age in years, multiply the natural logarithm of the dog's age in years by 16 then add 31.' 6

Calculate the equivalent age to the nearest human year of a dog aged 5 years. 6
A dog's equivalent age in human years is 40 years. Find the dog's actual age to the nearest month.
6
Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs.

AQA AS Paper 2 2022 June Q7

4 marks Moderate -0.8

7 The expression $$\frac { 3 - \sqrt { } n } { 2 + \sqrt { } n }$$ can be written in the form $a + b \sqrt { } n$, where $a$ and $b$ and $n$ are rational but $\sqrt { } n$ is irrational. Find expressions for $a$ and $b$ in terms of $n$.

AQA AS Paper 2 2022 June Q8

7 marks Standard +0.8

8 Triangle $A B C$ has sides of length $( m - n ) , m$ and $( m + n )$ where $0 < 2 n < m$ Angle $A$ is the largest angle in the triangle.
8

1. Explain why angle $A$ must be opposite the side of length $( m + n )$. 8
  1. (ii) Using the cosine rule, show that $\cos A = \frac { m - 4 n } { 2 ( m - n ) }$ 8
2. 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

12 marks Standard +0.3

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

8 marks Moderate -0.3

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.

\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 (b) (i) Show how Siti's claim of 'over 75\%' has been obtained.
13 (b) (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

7 marks Moderate -0.8

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

5 marks Standard +0.3

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

8 marks Standard +0.3

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. (i) Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.
  16
2. (ii) Find the mean number of heads obtained from 7 tosses of the coin.
  16
3. 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}

Edexcel AS Paper 2 2018 June Q1

3 marks Moderate -0.8

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

4 marks Moderate -0.3

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

7 marks Moderate -0.3

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

8 marks Moderate -0.8

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

8 marks Moderate -0.3

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

4 marks Moderate -0.8

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

7 marks Moderate -0.3

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

10 marks Standard +0.3

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 marks Standard +0.3

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

4 marks Easy -1.8

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

5 marks Moderate -0.8

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

5 marks Easy -1.2

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.

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

Questions AS Paper 2 (315 questions)

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AQA AS Paper 2 2022 June Q4

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Edexcel AS Paper 2 2018 June Q1

Edexcel AS Paper 2 2018 June Q2

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