Questions - A-Level Maths

Find $\int \left( 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } } \right) \mathrm { d } x$ 6
The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } }$$ The curve passes through the point $( 4,90 )$. Find the equation of the curve.

AQA Paper 3 2024 June Q7

8 marks Moderate -0.8

7 The graphs with equations $$y = 2 + 3 x - 2 x ^ { 2 } \text { and } x + y = 1$$ are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-10_791_721_550_719} The graphs intersect at the points $A$ and $B$
7

On the diagram above, shade and label the region, $R$, that is satisfied by the inequalities $$0 \leq y \leq 2 + 3 x - 2 x ^ { 2 }$$ and $$x + y \geq 1$$ 7
Find the exact coordinates of $A$

AQA Paper 3 2024 June Q8

8 marks Moderate -0.3

8 The temperature $\theta ^ { \circ } \mathrm { C }$ of an oven $t$ minutes after it is switched on can be modelled by the equation $$\theta = 20 \left( 11 - 10 \mathrm { e } ^ { - k t } \right)$$ where $k$ is a positive constant.
Initially the oven is at room temperature.
The maximum temperature of the oven is $T ^ { \circ } \mathrm { C }$
The temperature predicted by the model is shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-12_750_1319_870_424} 8

Find the room temperature.
8
Find the value of $T$
[0pt] [2 marks]
Question 8 continues on the next page 8
The oven reaches a temperature of $86 ^ { \circ } \mathrm { C }$ one minute after it is switched on. 8
1. Find the value of $k$.
  8
(ii) Find the time it takes for the temperature of the oven to be within $1 ^ { \circ } \mathrm { C }$ of its maximum.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-15_2493_1759_173_119} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-16_805_869_459_651}
\end{figure} The centre of the circle is $P$ and the circle intersects the $y$-axis at $Q$ as shown in Figure 1. The equation of the circle is $$x ^ { 2 } + y ^ { 2 } = 12 y - 8 x - 27$$

AQA Paper 3 2024 June Q9

9 marks Standard +0.3

Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where $a , b$ and $k$ are constants to be found.
9
State the coordinates of $P$ 9
Find the $y$-coordinate of $Q$
\section*{Question 9 continues on the next page} 9
The line segment $Q R$ is a tangent to the circle as shown in Figure 2 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-18_885_1180_456_495}
\end{figure} The point $R$ has coordinates $( 9 , - 3 )$.
Find the angle QPR
Give your answer in radians to three significant figures.
It is given that $$f ( x ) = 5 x ^ { 3 } + x$$ Use differentiation from first principles to prove that $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$

AQA Paper 3 2024 June Q11

10 marks Challenging +1.2

11 The curve $C$ with equation $$y = \left( x ^ { 2 } - 8 x \right) \ln x$$ is defined for $x > 0$ and is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-20_862_632_502_767} The shaded region, $R$, lies below the $x$-axis and is bounded by $C$ and the $x$-axis.
Show that the area of $R$ can be written as $$p + q \ln 2$$ where $p$ and $q$ are rational numbers to be found.
[0pt] [10 marks]
\section*{END OF SECTION A}

AQA Paper 3 2024 June Q12

1 marks Easy -3.0

12 A random sample of 84 students was asked how many revision websites they had visited in the past month. The data is summarised in the table below.

Number of websites	Frequency
0	1
1	4
2	18
3	16
4	5
5	37
6	2
7	1

AQA Paper 3 2024 June Q14

5 marks Easy -1.2

14 The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable $\pounds X$ It is given that $$\sum x = 945000 \quad \sum x ^ { 2 } = 2607500000$$ 14

Calculate the mean of $X$. 14
Calculate the standard deviation of $X$.
14
For households in Village B the annual cost of energy in 2021 has mean $\pounds 3100$ and standard deviation £325 Compare the annual cost of energy in 2021 for households in Village A and Village B.

AQA Paper 3 2024 June Q15

9 marks Easy -1.2

15 It is given that $$X \sim \mathrm {~B} ( 48,0.175 )$$ 15

Find the mean of $X$
[0pt] [1 mark] 15
Show that the variance of $X$ is 6.93
[0pt] [1 mark] 15
Find $\mathrm { P } ( X < 10 )$
[0pt] [1 mark] 15
$\quad$ Find $\mathrm { P } ( X \geq 6 )$
[0pt] [2 marks]
15
$\quad$ Find $\mathrm { P } ( 9 \leq X \leq 15 )$
[0pt] [2 marks] L
15
The aeroplanes used on a particular route have 48 seats. The proportion of passengers who use this route to travel for business is known to be 17.5\% Make two comments on whether it would be appropriate to use $X$ to model the number of passengers on an aeroplane who are travelling for business using this route.

AQA Paper 3 2024 June Q16

4 marks Moderate -0.8

16 A medical student believes that, in adults, there is a negative correlation between the amount of nicotine in their blood stream and their energy level. The student collected data from a random sample of 50 adults. The correlation coefficient between the amount of nicotine in their blood stream and their energy level was - 0.45 Carry out a hypothesis test at the $2.5 \%$ significance level to determine if this sample provides evidence to support the student's belief. For $n = 50$, the critical value for a one-tailed test at the $2.5 \%$ level for the population correlation coefficient is 0.2787

AQA Paper 3 2024 June Q17

15 marks Moderate -0.3

17 In 2019, the lengths of new-born babies at a clinic can be modelled by a normal distribution with mean 50 cm and standard deviation 4 cm . 17

This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-29_375_531_644_817} \captionsetup{labelformat=empty} \caption{Length (cm)}
\end{figure} 17
State the probability that the length of a new-born baby is less than 50 cm .
17
Find the probability that the length of a new-born baby is more than 56 cm .
17
Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm .
17
Determine the length exceeded by 95\% of all new-born babies at the clinic.
17
In 2020, the lengths of 40 new-born babies at the clinic were selected at random.
The total length of the 40 new-born babies was 2060 cm .
Carry out a hypothesis test at the $10 \%$ significance level to investigate whether the mean length of a new-born baby at the clinic in 2020 has increased compared to 2019. You may assume that the length of a new-born baby is still normally distributed with standard deviation 4 cm .

AQA Paper 3 2024 June Q18

7 marks Moderate -0.8

(ii)
[0pt] [2 marks]
\end{tabular}}
\hline \end{tabular} \end{center}

AQA Paper 3 2024 June Q19

9 marks Standard +0.3

19 It is known that 80\% of all diesel cars registered in 2017 had carbon monoxide (CO) emissions less than $0.3 \mathrm {~g} / \mathrm { km }$. Talat decides to investigate whether the proportion of diesel cars registered in 2022 with CO emissions less than $0.3 \mathrm {~g} / \mathrm { km }$ has changed. Talat will carry out a hypothesis test at the 10\% significance level on a random sample of 25 diesel cars registered in 2022. 19

1. State suitable null and alternative hypotheses for Talat's test. 19
(ii) Using a 10\% level of significance, find the critical region for Talat's test.
19
(iii) In his random sample, Talat finds 18 cars with CO emissions less than $0.3 \mathrm {~g} / \mathrm { km }$. State Talat's conclusion in context. 19
Talat now wants to use his random sample of 25 diesel cars, registered in 2022, to investigate whether the proportion of diesel cars in England with CO emissions more than $0.5 \mathrm {~g} / \mathrm { km }$ has changed from the proportion given by the Large Data Set. Using your knowledge of the Large Data Set, give two reasons why it is not possible for Talat to do this.
[0pt] [2 marks]

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.

Previous
1
2
3
...
1193
1194
1195
1196
1197
...
1232
1233
Next

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 (30808 questions)

Browse by module

Browse by board

AQA Paper 3 2024 June Q4

AQA Paper 3 2024 June Q5

AQA Paper 3 2024 June Q6

AQA Paper 3 2024 June Q7

AQA Paper 3 2024 June Q8

AQA Paper 3 2024 June Q9

AQA Paper 3 2024 June Q11

AQA Paper 3 2024 June Q12

AQA Paper 3 2024 June Q14

AQA Paper 3 2024 June Q15

AQA Paper 3 2024 June Q16

AQA Paper 3 2024 June Q17

AQA Paper 3 2024 June Q18

AQA Paper 3 2024 June Q19

Edexcel AS Paper 2 2018 June Q1

Edexcel AS Paper 2 2018 June Q2

Edexcel AS Paper 2 2018 June Q3

Edexcel AS Paper 2 2018 June Q4

Edexcel AS Paper 2 2018 June Q5

Edexcel AS Paper 2 2018 June Q6

Edexcel AS Paper 2 2018 June Q7

Edexcel AS Paper 2 2018 June Q8

Edexcel AS Paper 2 2018 June Q9

Edexcel AS Paper 2 Specimen Q1

Edexcel AS Paper 2 Specimen Q2