Questions - AQA Paper 3 - 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 Paper 3 2020 June Q16

4 marks Moderate -0.8

An educational expert found that the correlation coefficient between the hours of revision and the scores achieved by 25 students in their A-level exams was 0.379 Her data came from a bivariate normal distribution. Carry out a hypothesis test at the 1\% significance level to determine if there is a positive correlation between the hours of revision and the scores achieved by students in their A-level exams. The critical value of the correlation coefficient is 0.4622 [4 marks]

AQA Paper 3 2020 June Q17

8 marks Moderate -0.8

The lifetime of Zaple smartphone batteries, $X$ hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours.

1. Find P($X \neq 8$) [1 mark]
2. Find P($6 < X < 10$) [1 mark]
Determine the lifetime exceeded by 90\% of Zaple smartphone batteries. [2 marks]
A different smartphone, Kaphone, has its battery's lifetime, $Y$ hours, modelled by a normal distribution with mean 7 hours and standard deviation $\sigma$. 25\% of randomly selected Kaphone batteries last less than 5 hours. Find the value of $\sigma$, correct to three significant figures. [4 marks]

AQA Paper 3 2020 June Q18

14 marks Standard +0.3

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.

Type of defect	Colour	Fabric	Sewing	Sizing
Probability	0.25	0.30	0.40	0.05

Shirts with defects are packed in boxes of 30 at random.

Find the probability that:
1. a box contains exactly 5 shirts with a colour defect [2 marks]
2. a box contains fewer than 15 shirts with a sewing defect [2 marks]
3. a box contains at least 20 shirts which do not have a fabric defect. [3 marks]
Tiana wants to investigate the proportion, $p$, of defective shirts with a fabric defect. She wishes to test the hypotheses H$_0$: $p = 0.3$ H$_1$: $p < 0.3$ She takes a random sample of 60 shirts with a defect and finds that $x$ of them have a fabric defect.
1. Using a 5\% level of significance, find the critical region for $x$. [5 marks]
2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

Show that the first three terms, in descending powers of $x$, of the expansion of $$(2x - 3)^{10}$$ are given by $$1024x^{10} + px^9 + qx^8$$ where $p$ and $q$ are integers to be found. [3 marks]
Find the constant term in the expansion of $$\left(2x - \frac{3}{x}\right)^{10}$$ [2 marks]

AQA Paper 3 2021 June Q5

13 marks Moderate -0.8

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}

1. Find the area of this flowerbed. [2 marks]
2. Find the cost of the edging strip required for this flowerbed. [3 marks]
A flowerbed is to be made with an area of 20 m²
1. Show that the cost, £$C$, of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where $r$ is the radius measured in metres. [3 marks]
2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when $r \approx 4.5$ Fully justify your answer. [5 marks]

AQA Paper 3 2021 June Q6

4 marks Standard +0.3

Given that $x > 0$ and $x \neq 25$, fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]

AQA Paper 3 2021 June Q7

10 marks Moderate -0.8

A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the $n$th minute after the rain stops, the volume of water that drips into the bucket is $W_n$ millilitres.

Find $W_2$ [1 mark]
Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of $A$. [2 marks]
Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d). [2 marks]

AQA Paper 3 2021 June Q8

6 marks Standard +0.3

Given that $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \cos x \, dx = a\pi + b$$ find the exact value of $a$ and the exact value of $b$. Fully justify your answer. [6 marks]

AQA Paper 3 2021 June Q9

9 marks Standard +0.3

A function f is defined for all real values of $x$ as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when $x = 0$ and $x = -\frac{15}{4}$

1. Find $f''(x)$ [2 marks]
2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
State the range of values of $x$ for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
A second function g is defined for all real values of $x$ as $$g(x) = x^4 - 5x^3$$
1. State the single transformation which maps f onto g. [1 mark]
2. State the range of values of $x$ for which g is an increasing function. [1 mark]

AQA Paper 3 2021 June Q10

1 marks Easy -2.5

Anke has collected data from 30 similar-sized cars to investigate any correlation between the age of the car and the current market value. She calculates the correlation coefficient. Which of the following statements best describes her answer of $-1.2$? Tick ($\checkmark$) one box. [1 mark] Definitely incorrect Probably incorrect Probably correct Definitely correct

AQA Paper 3 2021 June Q11

1 marks Easy -1.2

The random variable $X$ is such that $X \sim B(n, p)$ The mean value of $X$ is 225 The variance of $X$ is 144 Find $p$. Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8

AQA Paper 3 2021 June Q12

3 marks Easy -1.8

An electoral register contains 8000 names. A researcher decides to select a systematic sample of 100 names from the register. Explain how the researcher should select such a sample. [3 marks]

AQA Paper 3 2021 June Q13

6 marks Moderate -0.8

The table below is an extract from the Large Data Set.

Propulsion Type	Region	Engine Size	Mass	CO₂	Particulate Emissions
2	London	1896	1533	154	0.04
2	North West	1896	1423	146	0.029
2	North West	1896	1353	138	0.025
2	South West	1998	1547	159	0.026
2	London	1896	1388	138	0.025
2	South West	1896	1214	130	0.011
2	South West	1896	1480	146	0.029
2	South West	1896	1413	146	0.024
2	South West	2496	1695	192	0.034
2	South West	1422	1251	122	0.025
2	South West	1995	2075	175	0.034
2	London	1896	1285	140	0.036
2	North West	1896	0	146

1. Calculate the mean and standard deviation of CO₂ emissions in the table. [2 marks]
2. Any value more than 2 standard deviations from the mean can be identified as an outlier. Determine, using this definition of an outlier, if there are any outliers in this sample of CO₂ emissions. Fully justify your answer. [2 marks]
Maria claims that the last line in the table must contain two errors. Use your knowledge of the Large Data Set to comment on Maria's claim. [2 marks]

AQA Paper 3 2021 June Q14

7 marks Standard +0.3

$A$ and $B$ are two events such that $$P(A \cap B) = 0.1$$ $$P(A' \cap B') = 0.2$$ $$P(B) = 2P(A)$$

Find $P(A)$ [4 marks]
Find $P(B|A)$ [2 marks]
Determine if $A$ and $B$ are independent events. [1 mark]

AQA Paper 3 2021 June Q15

7 marks Standard +0.3

A team game involves solving puzzles to escape from a room. Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes. After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed. To test this claim, a random sample of 100 teams is selected. The total time to solve the puzzles and escape for the 100 teams is 6780 minutes. Assuming that the times are normally distributed, test at the 2% level the claim that the mean time has changed. [7 marks]

AQA Paper 3 2021 June Q16

4 marks Moderate -0.3

The discrete random variable $X$ has the probability function $$P(X = x) = \begin{cases} c(7 - 2x) & x = 0, 1, 2, 3 \\ k & x = 4 \\ 0 & \text{otherwise} \end{cases}$$ where $c$ and $k$ are constants.

Show that $16c + k = 1$ [2 marks]
Given that $P(X \geq 3) = \frac{5}{8}$ find the value of $c$ and the value of $k$. [2 marks]

AQA Paper 3 2021 June Q17

11 marks Standard +0.3

James is playing a mathematical game on his computer. The probability that he wins is 0.6 As part of an online tournament, James plays the game 10 times. Let $Y$ be the number of games that James wins.

State two assumptions, in context, for $Y$ to be modelled as $B(10, 0.6)$ [2 marks]
Find $P(Y = 4)$ [1 mark]
Find $P(Y \geq 4)$ [2 marks]
After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them. Test at a 5% significance level whether James's claim is correct. [6 marks]

AQA Paper 3 2021 June Q18

10 marks Moderate -0.3

A factory produces jars of jam and jars of marmalade.

The weight, $X$ grams, of jam in a jar can be modelled as a normal variable with mean 372 and a standard deviation of 3.5
1. Find the probability that the weight of jam in a jar is equal to 372 grams. [1 mark]
2. Find the probability that the weight of jam in a jar is greater than 368 grams. [2 marks]
The weight, $Y$ grams, of marmalade in a jar can be modelled as a normal variable with mean $\mu$ and standard deviation $\sigma$
1. Given that $P(Y < 346) = 0.975$, show that $$346 - \mu = 1.96\sigma$$ Fully justify your answer. [3 marks]
2. Given further that $$P(Y < 336) = 0.14$$ find $\mu$ and $\sigma$ [4 marks]