AQA Paper 3 (Paper 3) 2024 June

1 Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent.
[0pt] [1 mark] Tick \(( \checkmark )\) one box.
\(0.1 + 0.2 + 0.4 + 0.8 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_113_113_858_927}
\(1 - 1 + 1 - 1 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_117_117_1014_927}
\(128 - 64 + 32 - 16 + \ldots\)
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-02_118_117_1169_927}
\(1 + 2 + 4 + 8 + \ldots\) □

2 The quadratic equation $$4 x ^ { 2 } + b x + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer.
\(b = 0\)
\(b = \pm 12\)
\(b = \pm 13\)
\(b = \pm 36\)

4 A curve has equation \(y = x ^ { 4 } + 2 ^ { x }\) Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)

5 The diagram below shows a sector of a circle \(O A B\). The chord \(A B\) divides the sector into a triangle and a shaded segment. Angle \(A O B\) is \(\frac { \pi } { 6 }\) radians.
The radius of the sector is 18 cm .
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-06_467_428_614_790} Show that the area of the shaded segment is $$k ( \pi - 3 ) \mathrm { cm } ^ { 2 }$$ where \(k\) is an integer to be found.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-07_2491_1753_173_123}

6

1. Find \(\int \left( 6 x ^ { 2 } - \frac { 5 } { \sqrt { x } } \right) \mathrm { d } x\) 6
2. 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.

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

1. 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
2. Find the exact coordinates of \(A\)

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

1. Find the room temperature.
   8
2. Find the value of \(T\)
   [0pt] [2 marks]
   Question 8 continues on the next page 8
3. The oven reaches a temperature of \(86 ^ { \circ } \mathrm { C }\) one minute after it is switched on. 8
4. 1. Find the value of \(k\).
      8
5. (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 $$x ^ { 2 } + y ^ { 2 } = 12 y - 8 x - 27$$

9

1. 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
2. State the coordinates of \(P\) 9
3. Find the \(y\)-coordinate of \(Q\)
   \section*{Question 9 continues on the next page} 9
4. 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 $$f ^ { \prime } ( x ) = 15 x ^ { 2 } + 1$$

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}

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. Find the interquartile range of the number of websites visited by these 84 students.
Circle your answer.
[0pt] [1 mark]
341942 Identify this Venn diagram. Tick ( ✓ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_506_501_584_374}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_111_580_897}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_504_580_1203}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_117_120_580_1710}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_508_501_1135_374}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_111_1133_897}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_505_506_1133_1201}
\includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-23_112_109_1133_1717} Turn over for the next question

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

1. Calculate the mean of \(X\). 14
2. Calculate the standard deviation of \(X\).
   14
3. 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.

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

1. Find the mean of \(X\)
   [0pt] [1 mark] 15
2. Show that the variance of \(X\) is 6.93
   [0pt] [1 mark] 15
3. Find \(\mathrm { P } ( X < 10 )\)
   [0pt] [1 mark] 15
4. \(\quad\) Find \(\mathrm { P } ( X \geq 6 )\)
   [0pt] [2 marks]
   15
5. \(\quad\) Find \(\mathrm { P } ( 9 \leq X \leq 15 )\)
   [0pt] [2 marks] L
   15
6. 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.

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

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

1. 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
2. State the probability that the length of a new-born baby is less than 50 cm .
   17
3. Find the probability that the length of a new-born baby is more than 56 cm .
   17
4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm .
   17
5. Determine the length exceeded by 95\% of all new-born babies at the clinic.
   17
6. 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 .

18

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

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. 1. State suitable null and alternative hypotheses for Talat's test. 19
2. (ii) Using a 10\% level of significance, find the critical region for Talat's test.
   19
3. (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
4. 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]