Questions - AQA - A-Level Maths

Browse by module

All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4

Browse by board

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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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

Sort by: Default | Easiest first | Hardest first

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}

AQA Paper 1 2018 June Q2

1 marks Easy -1.8

2 The graph of $y = 5 ^ { x }$ is transformed by a stretch in the $y$-direction, scale factor 5 State the equation of the transformed graph. Circle your answer.
[0pt] [1 mark] $y = 5 \times 5 ^ { x }$ $y = 5 ^ { \frac { x } { 5 } }$ $y = \frac { 1 } { 5 } \times 5 ^ { x }$ $y = 5 ^ { 5 x }$

AQA Paper 1 2018 June Q3

1 marks Easy -1.8

3 A periodic sequence is defined by $U _ { n } = \sin \left( \frac { n \pi } { 2 } \right)$ State the period of this sequence. Circle your answer. $82 \pi \quad 4 \quad \pi$

AQA Paper 1 2018 June Q4

3 marks Moderate -0.8

4 The function f is defined by $\mathrm { f } ( x ) = \mathrm { e } ^ { x - 4 } , x \in \mathbb { R }$ Find $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.

AQA Paper 1 2018 June Q5

6 marks Standard +0.3

5 A curve is defined by the parametric equations $$\begin{aligned} & x = 4 \times 2 ^ { - t } + 3 \\ & y = 3 \times 2 ^ { t } - 5 \end{aligned}$$ 5

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 } \times 2 ^ { 2 t }$ 5
Find the Cartesian equation of the curve in the form $x y + a x + b y = c$, where $a , b$ and $c$ are integers.

AQA Paper 1 2018 June Q6

12 marks Standard +0.8

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $\frac { 1 } { \sqrt { 4 + x } }$ 6
Hence, find the first three terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 3 } } }$ 6 (d) (i) Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.
[0pt] [2 marks]
6 (d) (ii) Edward goes on to use the expansion from part (b) to find an approximation for $\int _ { - 2 } ^ { 0 } \frac { 1 } { \sqrt { 4 - x ^ { 3 } } } \mathrm {~d} x$ Explain why Edward's approximation is invalid.

AQA Paper 1 2018 June Q7

8 marks Moderate -0.3

7 Three points $A , B$ and $C$ have coordinates $A ( 8,17 ) , B ( 15,10 )$ and $C ( - 2 , - 7 )$ 7

Show that angle $A B C$ is a right angle.
7
$\quad A , B$ and $C$ lie on a circle.
7 (b) (i) Explain why $A C$ is a diameter of the circle.
7 (b) (ii) Determine whether the point $D ( - 8 , - 2 )$ lies inside the circle, on the circle or outside the circle. Fully justify your answer.

AQA Paper 1 2018 June Q8

8 marks Standard +0.3

8 The diagram shows a sector of a circle $O A B$. $C$ is the midpoint of $O B$.
Angle $A O B$ is $\theta$ radians. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-10_700_963_536_534} 8

Given that the area of the triangle $O A C$ is equal to one quarter of the area of the sector $O A B$, show that $\theta = 2 \sin \theta$ 8
Use the Newton-Raphson method with $\theta _ { 1 } = \pi$, to find $\theta _ { 3 }$ as an approximation for $\theta$. Give your answer correct to five decimal places.
8
Given that $\theta = 1.89549$ to five decimal places, find an estimate for the percentage error in the approximation found in part (b).
Turn over for the next question

AQA Paper 1 2018 June Q9

9 marks Standard +0.8

9 An arithmetic sequence has first term $a$ and common difference $d$. The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. 9

Show that $4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }$ 9
Given that the sixth term of the sequence is 25 , find the smallest possible value of $a$.

AQA Paper 1 2018 June Q10

8 marks Moderate -0.3

10 A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using $$m = m _ { 0 } \mathrm { e } ^ { - k t }$$ where $m _ { 0 }$ milligrams is the initial mass of caffeine in the body and $m$ milligrams is the mass of caffeine in the body after $t$ hours. On average, it takes 5.7 hours for the mass of caffeine in the body to halve.
One cup of strong coffee contains 200 mg of caffeine.
10

The scientist drinks two strong cups of coffee at 8 am. Use the model to estimate the mass of caffeine in the scientist's body at midday.
10
The scientist wants the mass of caffeine in her body to stay below 480 mg
10 (b)
Use the model to find the earliest time
coffee.
Give your answer to the nearest minute
□

AQA Paper 1 2018 June Q11

10 marks Standard +0.3

11 The daily world production of oil can be modelled using $$V = 10 + 100 \left( \frac { t } { 30 } \right) ^ { 3 } - 50 \left( \frac { t } { 30 } \right) ^ { 4 }$$ where $V$ is volume of oil in millions of barrels, and $t$ is time in years since 1 January 1980. 11

1. The model is used to predict the time, $T$, when oil production will fall to zero.
  Show that $T$ satisfies the equation $$T = \sqrt [ 3 ] { 60 T ^ { 2 } + \frac { 162000 } { T } }$$ 11
  1. (ii) Use the iterative formula $T _ { n + 1 } = \sqrt [ 3 ] { 60 T _ { n } { } ^ { 2 } + \frac { 162000 } { T _ { n } } }$, with $T _ { 0 } = 38$, to find the values of $T _ { 1 } , T _ { 2 }$, and $T _ { 3 }$, giving your answers to three decimal places.
    11
2. (iii) Explain the relevance of using $T _ { 0 } = 38$ 11
3. From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as $$V = 4.5 \times 1.063 ^ { t }$$ Use the models to show that the country's use of oil and the world production of oil will be equal during the year 2029.
  [0pt] [4 marks] $12 \quad \mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2$

AQA Paper 1 2018 June Q12

10 marks Standard +0.3

Prove that ( $2 x + 1$ ) is a factor of $\mathrm { p } ( x )$ 12
Factorise $\mathrm { p } ( x )$ completely.
12
Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

AQA Paper 1 2018 June Q13

10 marks Standard +0.3

13 A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models the logo on an $x - y$ plane as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-20_492_492_511_776} Use calculus to find the maximum area of the rectangle.
Fully justify your answer.

AQA Paper 1 2018 June Q14

7 marks Standard +0.3

14 Some students are trying to prove an identity for $\sin ( A + B )$. They start by drawing two right-angled triangles $O D E$ and $O E F$, as shown. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle $D O E = A$ and angle $E O F = B$.
In triangle OFR,
Line $1 \quad \sin ( A + B ) = \frac { R F } { O F }$ Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 $=$ $\_\_\_\_$ $+ \cos A \sin B$ 14

Explain why $\frac { P F } { E F } \times \frac { E F } { O F }$ in Line 4 leads to $\cos A \sin B$ in Line 5
14
Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = $+ \cos A \sin B$ 14
Explain why the argument used in part (a) only proves the identity when $A$ and $B$ are acute angles. 14
Another student claims that by replacing $B$ with $- B$ in the identity for $\sin ( A + B )$ it is possible to find an identity for $\sin ( A - B )$. Assuming the identity for $\sin ( A + B )$ is correct for all values of $A$ and $B$, prove a similar result for $\sin ( A - B )$.

AQA Paper 1 2018 June Q15

6 marks Moderate -0.5

15 A curve has equation $y = x ^ { 3 } - 48 x$ The point $A$ on the curve has $x$ coordinate - 4
The point $B$ on the curve has $x$ coordinate $- 4 + h$ 15

Show that the gradient of the line $A B$ is $h ^ { 2 } - 12 h$ 15
Explain how the result of part (a) can be used to show that $A$ is a stationary point on the curve. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-25_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-26_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-27_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-28_2498_1721_213_150}

AQA Paper 1 2020 June Q1

2 marks Easy -1.2

1 The first three terms, in ascending powers of $x$, of the binomial expansion of $( 9 + 2 x ) ^ { \frac { 1 } { 2 } }$ are given by $$( 9 + 2 x ) ^ { \frac { 1 } { 2 } } \approx a + \frac { x } { 3 } - \frac { x ^ { 2 } } { 54 }$$ where $a$ is a constant. 1

State the range of values of $x$ for which this expansion is valid.
Circle your answer. $| x | < \frac { 2 } { 9 }$ $| x | < \frac { 2 } { 3 }$ $| x | < 1$ $| x | < \frac { 9 } { 2 }$ 1
Find the value of $a$.
Circle your answer.
[0pt] [1 mark]
1239

AQA Paper 1 2020 June Q2

1 marks Easy -1.8

2 A student is searching for a solution to the equation $\mathrm { f } ( x ) = 0$ He correctly evaluates $$f ( - 1 ) = - 1 \text { and } f ( 1 ) = 1$$ and concludes that there must be a root between - 1 and 1 due to the change of sign.
Select the function $\mathrm { f } ( x )$ for which the conclusion is incorrect.
Circle your answer. $$\mathrm { f } ( x ) = \frac { 1 } { x } \quad \mathrm { f } ( x ) = x \quad \mathrm { f } ( x ) = x ^ { 3 } \quad \mathrm { f } ( x ) = \frac { 2 x + 1 } { x + 2 }$$

AQA Paper 1 2020 June Q3

1 marks Easy -1.2

3 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius 2 \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-03_374_455_1187_790} The angle $A O B$ is $\theta$ radians and the perimeter of the sector is 6
Find the value of $\theta$ Circle your answer.
[0pt] [1 mark]
1 $\sqrt { 3 }$ 2
3

AQA Paper 1 2020 June Q4

5 marks Easy -1.2

Sketch the graph of \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-04_933_1093_349_475} 4
Solve the inequality $$4 - | 2 x - 6 | > 2$$

Questions — AQA (3620 questions)

Browse by module

Browse by board

AQA AS Paper 2 2022 June Q10

AQA AS Paper 2 2022 June Q11

AQA AS Paper 2 2022 June Q12

AQA AS Paper 2 2022 June Q13

AQA AS Paper 2 2022 June Q14

AQA AS Paper 2 2022 June Q15

AQA AS Paper 2 2022 June Q16

AQA Paper 1 2018 June Q2

AQA Paper 1 2018 June Q3

AQA Paper 1 2018 June Q4

AQA Paper 1 2018 June Q5

AQA Paper 1 2018 June Q6

AQA Paper 1 2018 June Q7

AQA Paper 1 2018 June Q8

AQA Paper 1 2018 June Q9

AQA Paper 1 2018 June Q10

AQA Paper 1 2018 June Q11

AQA Paper 1 2018 June Q12

AQA Paper 1 2018 June Q13

AQA Paper 1 2018 June Q14

AQA Paper 1 2018 June Q15

AQA Paper 1 2020 June Q1

AQA Paper 1 2020 June Q2

AQA Paper 1 2020 June Q3

AQA Paper 1 2020 June Q4