AQA AS Paper 2 (AS Paper 2) 2024 June

1 Line \(L\) has equation $$5 y = 4 x + 6$$ Find the gradient of a line parallel to line \(L\)
Circle your answer.
\(- \frac { 5 } { 4 }\)
\(- \frac { 4 } { 5 }\)
\(\frac { 4 } { 5 }\)
\(\frac { 5 } { 4 }\)

2 One of the equations below is true for all values of \(x\)
Identify the correct equation.
Tick \(( \checkmark )\) one box.
\(\cos ^ { 2 } x = - 1 - \sin ^ { 2 } x\) □
\(\cos ^ { 2 } x = - 1 + \sin ^ { 2 } x\)
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\(\cos ^ { 2 } x = 1 - \sin ^ { 2 } x\)
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-02_113_113_1813_790}
\(\cos ^ { 2 } x = 1 + \sin ^ { 2 } x\) □

3 It is given that $$3 \log _ { a } x = \log _ { a } 72 - 2 \log _ { a } 3$$ Solve the equation to find the value of \(x\)
Fully justify your answer.
[0pt] [4 marks]

4 Curve \(C\) has equation \(y = 8 \sin x\) 4

1. Curve \(C\) is transformed onto curve \(C _ { 1 }\) by a translation of vector \(\left[ \begin{array} { l } 0
   4 \end{array} \right]\)
   Find the equation of \(C _ { 1 }\) 4
2. \(\quad\) Curve \(C\) is transformed onto curve \(C _ { 2 }\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C _ { 2 }\) 4
3. Curve \(C\) is transformed onto curve \(C _ { 3 }\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C _ { 3 }\)

5 A student suggests that for any positive integer \(n\) the value of the expression $$4 n ^ { 2 } + 3$$ is always a prime number.
Prove that the student's statement is false by finding a counter example.
Fully justify your answer.

6 In the expansion of \(( 3 + a x ) ^ { n }\), where \(a\) and \(n\) are integers, the coefficient of \(x ^ { 2 }\) is 4860 6

1. Show that $$3 ^ { n } a ^ { 2 } n ( n - 1 ) = 87480$$ [3 marks]
   6
2. The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative. 6
3. 1. Verify that \(n = 6\)
      6
4. (ii) Find the value of \(a\)

7

1. Find the equation of the perpendicular bisector of \(A B\)
   7
2. \(\quad\) A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis.
   Find the equation of the circle.
   [0pt] [4 marks]

8 Prove that the graph of the curve with equation $$y = x ^ { 3 } + 15 x - \frac { 18 } { x }$$ has no stationary points.
[0pt] [5 marks]
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9 A curve has equation $$y = x - a \sqrt { x } + b$$ where \(a\) and \(b\) are constants. The curve intersects the line \(y = 2\) at points with coordinates \(( 1,2 )\) and \(( 9,2 )\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-12_696_807_641_605} 9

1. Show that \(a\) has the value 4 and find the value of \(b\)
   9
2. On the diagram, the region enclosed between the curve and the line \(y = 2\) is shaded. Show that the area of this shaded region is \(\frac { 16 } { 3 }\) Fully justify your answer.
   [0pt] [6 marks]

10

1. (ii) Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
   \hline \end{tabular} \end{center} 10
2. 1. Show that \(\frac { \mathrm { d } F } { \mathrm {~d} t } = k F\)
      10
3. (ii) Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released.
   10
4. The singer claims that 30 days after the song is released, the account will have more than a billion followers. Comment on the singer's claim.

11 The table below shows the daily salt intake, \(x\) grams, and the daily Vitamin C intake, \(y\) milligrams, for a group of 10 adults. A scatter diagram of the data is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-17_675_1150_1110_431} One of the adults is an outlier. Identify the letter of the adult that is the outlier.
Circle your answer below.
A
B
E
J Which one of the following is not a measure of spread?
Circle your answer.
median
range
standard deviation
variance

13 The headteacher of a school wishes to collect the opinions of the students on a new timetable structure. To do this, a random sample of size 50 , stratified by year group, will be selected.
The school has a total of 720 students. The number of students in each of the year groups at this school is shown below. 13

1. Find the number of students from each year group that should be selected in the stratified random sample.
   13
2. State one advantage of using a stratified random sample.

14 The discrete random variables \(X\) and \(Y\) can be modelled by the distributions $$\begin{gathered} X \sim \mathrm {~B} ( 40 , p )
Y \sim \mathrm {~B} ( 25,0.6 ) \end{gathered}$$ It is given that the mean of \(X\) is equal to the variance of \(Y\) 14

1. Find the value of \(p\)
   14
2. \(\quad\) Find \(\mathrm { P } ( Y = 17 )\)

15 The number of flowers which grow on a certain type of plant can be modelled by the discrete random variable \(X\) The probability distribution of \(X\) is given in the table below. 15

1. Find the value of \(p\)
   15
2. Two plants of this type are randomly selected from a large batch received from a local garden centre. Find the probability that the two plants will produce a total of three flowers.
   [0pt] [3 marks]
   15
3. 1. State one assumption necessary for the calculation in part (b) to be valid. 15
4. (ii) Comment on whether, in reality, this assumption is likely to be valid.

16

1. (ii) &

   16

   where \(n\) is the total number of cars which had a measured hydrocarbon emission in the Large Data Set.

   16

   Find the mean of \(X\)

   [1 mark]

   16

   
   \hline &
   \hline \end{tabular} \end{center} 16

(ii) State one type of emission where more than 80\% of the data is known for cars in the entire UK Department for Transport Stock Vehicle Database.
[0pt] [1 mark]

17 The proportion of vegans in a city is thought to be 8\% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than \(8 \%\) To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the \(5 \%\) level, whether this sample supports the owner's belief.
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