Questions Paper 3 - A-Level Maths

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AQA Paper 3 2020 June Q18

Find the probability that:
18
1. a box contains exactly 5 shirts with a colour defect
  18
(ii) a box contains fewer than 15 shirts with a sewing defect
18
(iii) a box contains at least 20 shirts which do not have a fabric defect.
Question 18 continues on the next page 18
Tiana wants to investigate the proportion, $p$, of defective shirts with a fabric defect. She wishes to test the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.3
& \mathrm { H } _ { 1 } : p < 0.3 \end{aligned}$$ She takes a random sample of 60 shirts with a defect and finds that $x$ of them have a fabric defect. 18
1. Using a $5 \%$ level of significance, find the critical region for $x$.
  18
(ii) In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context.

AQA Paper 3 2021 June Q1

1 marks

1 The graph of $y = \arccos x$ is shown.
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-02_599_856_758_591} State the coordinates of the end point $P$.
Circle your answer.
[0pt] [1 mark] $$( - \pi , 1 ) \quad ( - 1 , \pi ) \quad \left( - \frac { \pi } { 2 } , 1 \right) \quad \left( - 1 , \frac { \pi } { 2 } \right)$$

AQA Paper 3 2021 June Q2

2 Simplify fully $$\frac { ( x + 3 ) ( 6 - 2 x ) } { ( x - 3 ) ( 3 + x ) } \quad \text { for } x \neq \pm 3$$ Circle your answer.
-2
2
$\frac { ( 6 - 2 x ) } { ( x - 3 ) }$
$\frac { ( 2 x - 6 ) } { ( x - 3 ) }$
$3 \mathrm { f } ( x ) = 3 x ^ { 2 }$
Obtain $\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$
Circle your answer. $$\frac { 3 h ^ { 2 } } { h } \quad x ^ { 3 } \quad \frac { 3 ( x + h ) ^ { 2 } - 3 x ^ { 2 } } { h } \quad 6 x$$

AQA Paper 3 2021 June Q4

Show that the first three terms, in descending powers of $x$, of the expansion of $$( 2 x - 3 ) ^ { 10 }$$ are given by $$1024 x ^ { 10 } + p x ^ { 9 } + q x ^ { 8 }$$ where $p$ and $q$ are integers to be found.
4
Find the constant term in the expansion of $$\left( 2 x - \frac { 3 } { x } \right) ^ { 10 }$$

AQA Paper 3 2021 June Q5

5 marks

5 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 $\pounds 1.80$ 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[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-05_544_579_676_730} 5

1. Find the area of this flowerbed.
  \section*{Question 5 continues on the next page} 5
(ii) Find the cost of the edging strip required for this flowerbed.
5
A flowerbed is to be made with an area of $20 \mathrm {~m} ^ { 2 }$
5
1. Show that the cost, $\pounds C$, 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.
  5
(ii) Hence, show that the minimum cost of the edging strip for this flowerbed occurs when $r \approx 4.5$ Fully justify your answer.
[0pt] [5 marks]

AQA Paper 3 2021 June Q6

4 marks

6 Given that $x > 0$ and $x \neq 25$, fully simplify $$\frac { 10 + 5 x - 2 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } } { 5 - \sqrt { x } }$$ Fully justify your answer.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-09_2488_1716_219_153}

AQA Paper 3 2021 June Q7

7 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. 7

$\quad$ Find $W _ { 2 }$
7
Explain why $$W _ { n } = A \times 0.98 ^ { n - 1 }$$ and state the value of $A$.
7
Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. 7
Assuming it does not start to rain again, find the maximum amount of water in the bucket.
7
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).

AQA Paper 3 2021 June Q8

6 marks

8 Given that $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } x \cos x d x = a \pi + b$$ find the exact value of $a$ and the exact value of $b$. Fully justify your answer.
[0pt] [6 marks]

AQA Paper 3 2021 June Q9

9 A function f is defined for all real values of $x$ as $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ The function has exactly two stationary points when $x = 0$ and $x = - \frac { 15 } { 4 }$
9

1. Find $\mathrm { f } ^ { \prime \prime } ( x )$
  9
(ii) Determine the nature of the stationary points.
Fully justify your answer.
9
State the range of values of $x$ for which $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ is an increasing function.
9
A second function g is defined for all real values of $x$ as $$\mathrm { g } ( x ) = x ^ { 4 } - 5 x ^ { 3 }$$ 9
1. State the single transformation which maps f onto g .
  9
(ii) State the range of values of $x$ for which g is an increasing function.

Propulsion Type	Region	Engine Size	Mass	$\mathrm { CO } _ { 2 }$	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 $\mathrm { CO } _ { 2 }$ emissions in the table.
  [0pt] [2 marks]
  13
(ii) 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 $\mathrm { CO } _ { 2 }$ emissions. Fully justify your answer.
13
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.
$14 \quad A$ and $B$ are two events such that $$\begin{aligned} & \mathrm { P } ( A \cap B ) = 0.1
& \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.2
& \mathrm { P } ( B ) = 2 \mathrm { P } ( A ) \end{aligned}$$

AQA Paper 3 2021 June Q14

$\quad$ Find $\mathrm { P } ( A )$
14
$\quad$ Find $\operatorname { P } ( B \mid A )$
14
Determine if $A$ and $B$ are independent events.

AQA Paper 3 2021 June Q15

15 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.

AQA Paper 3 2021 June Q16

16 The discrete random variable $X$ has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where $c$ and $k$ are constants.
16

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

AQA Paper 3 2021 June Q17

17 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.
17

State two assumptions, in context, for $Y$ to be modelled as $B ( 10,0.6 )$
17
$\quad$ Find $\mathrm { P } ( Y = 4 )$
17
$\quad$ Find $\mathrm { P } ( Y \geq 4 )$
17
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 a $5 \%$ significance level whether James's claim is correct.
\begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{\begin{tabular}{l}

AQA Paper 3 2021 June Q18

7 marks

18 (b)
The weight, $Y$ grams, of marmalade in a jar can be modelled as a normal variable with mean $\mu$ and standard deviation $\sigma$
18 (b) (i)
18 (b) (i) $\_\_\_\_$ $\_\_\_\_$ $346 - \mu = 1.96 \sigma$
Fully justify your answer. $\_\_\_\_$
[0pt] [3 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$
\end{tabular}}
\hline \end{tabular} \end{center} 18 (b) (ii) Given further that $$\mathrm { P } ( Y < 336 ) = 0.14$$ find $\mu$ and $\sigma$
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-28_2492_1721_217_150}

AQA Paper 3 2022 June Q1

1 marks

1 State the range of values of $x$ for which the binomial expansion of $$\sqrt { 1 - \frac { x } { 4 } }$$ is valid. Circle your answer.
[0pt] [1 mark] $$| x | < \frac { 1 } { 4 } \quad | x | < 1 \quad | x | < 2 \quad | x | < 4$$

AQA Paper 3 2022 June Q2

2 The shaded region, shown in the diagram below, is defined by $$x ^ { 2 } - 7 x + 7 \leq y \leq 7 - 2 x$$

\includegraphics[max width=\textwidth, alt={}]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-03_801_830_456_607}

Identify which of the following gives the area of the shaded region.
Tick $( \checkmark )$ one box. $$\begin{array} { l c } \int ( 7 - 2 x ) \mathrm { d } x - \int \left( x ^ { 2 } - 7 x + 7 \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 5 x \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( 5 x - x ^ { 2 } \right) \mathrm { d } x & \square
\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 9 x + 14 \right) \mathrm { d } x & \square \end{array}$$

AQA Paper 3 2022 June Q3

3 The function f is defined by $$f ( x ) = 2 x + 1$$ Solve the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ Circle your answer. $$x = - 1 \quad x = 0 \quad x = 1 \quad x = 2$$

AQA Paper 3 2022 June Q4

4 Find $$\int \left( x ^ { 2 } + x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x$$

AQA Paper 3 2022 June Q5

Sketch the graph of $$y = \sin 2 x$$ for $0 ^ { \circ } \leq x \leq 360 ^ { \circ }$
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-05_1095_1246_534_402} 5
The equation $$\sin 2 x = A$$ has exactly two solutions for $0 ^ { \circ } \leq x \leq 360 ^ { \circ }$
State the possible values of $A$.

AQA Paper 3 2022 June Q6

6 A design for a surfboard is shown in Figure 1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_415_1403_447_322}

\end{figure} The curve of the top half of the surfboard can be modelled by the parametric equations $$\begin{aligned} & x = - 2 t ^ { 2 }
& y = 9 t - 0.7 t ^ { 2 } \end{aligned}$$ for $0 \leq t \leq 9.5$ as shown in Figure 2, where $x$ and $y$ are measured in centimetres. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-06_383_1342_1379_351}

\end{figure} 6

Find the length of the surfboard.
6
1. Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
  6
(ii) Hence, show that the width of the surfboard is approximately one third of its length.
$7 \quad$ A planet takes $T$ days to complete one orbit of the Sun.
$T$ is known to be related to the planet's average distance $d$, in millions of kilometres, from the Sun. A graph of $\log _ { 10 } T$ against $\log _ { 10 } d$ is shown with data for Mercury and Uranus labelled.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-08_752_1447_580_296}

AQA Paper 3 2022 June Q7

1. Find the equation of the straight line in the form $$\log _ { 10 } T = a + b \log _ { 10 } d$$ where $a$ and $b$ are constants to be found.
  7
(ii) Show that $$T = \mathrm { K } d ^ { \mathrm { n } }$$ where K and n are constants to be found.
7
Neptune takes approximately 60000 days to complete one orbit of the Sun.
Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun.

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