AQA AS Paper 2 (AS Paper 2) 2024 June

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

One of the equations below is true for all values of \(x\) Identify the correct equation. Tick (\(\checkmark\)) one box. [1 mark] \(\cos^2 x = -1 - \sin^2 x\) \(\square\) \(\cos^2 x = -1 + \sin^2 x\) \(\square\) \(\cos^2 x = 1 - \sin^2 x\) \(\square\) \(\cos^2 x = 1 + \sin^2 x\) \(\square\)

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. [4 marks]

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

1. Curve \(C\) is transformed onto curve \(C_1\) by a translation of vector \(\begin{pmatrix} 0 \\ 4 \end{pmatrix}\) Find the equation of \(C_1\) [1 mark]
2. Curve \(C\) is transformed onto curve \(C_2\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C_2\) [1 mark]
3. Curve \(C\) is transformed onto curve \(C_3\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C_3\) [1 mark]

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

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

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

Point \(A\) has coordinates \((4, 1)\) and point \(B\) has coordinates \((-8, 5)\)

1. Find the equation of the perpendicular bisector of \(AB\) [5 marks]
2. 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. [4 marks]

Prove that the graph of the curve with equation $$y = x^3 + 15x - \frac{18}{x}$$ has no stationary points. [5 marks]

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{figure_1}

1. Show that \(a\) has the value 4 and find the value of \(b\) [3 marks]
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. [6 marks]

A singer has a social media account with a number of followers. The singer releases a new song and the number of followers grows exponentially. The number of followers, \(F\), may be modelled by the formula $$F = ae^{kt}$$ where \(t\) is the number of days since the song was released and \(a\) and \(k\) are constants. • Two days after the song is released the account has 2050 followers. • Five days after the song is released the account has 9200 followers. On the graph below ln \(F\) has been plotted against \(t\) for these two pieces of data. A line has been drawn passing through these two data points. \includegraphics{figure_2}

1. 1. Show that \(\ln F = \ln a + kt\) [2 marks]
   2. Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
2. 1. Show that \(\frac{dF}{dt} = kF\) [2 marks]
   2. Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released. [2 marks]
3. 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. [1 mark]

The table below shows the daily salt intake, \(x\) grams, and the daily Vitamin C intake, \(y\) milligrams, for a group of 10 adults.

Adult	A	B	C	D	E	F	G	H	I	J
\(x\)	5.3	6.2	3.6	10.4	2.4	9.4	6	5	7.1	11.2
\(y\)	90	145	88	48	114	44	80	95	55	41

A scatter diagram of the data is shown below. \includegraphics{figure_3} One of the adults is an outlier. Identify the letter of the adult that is the outlier. Circle your answer below. [1 mark] A \(\qquad\) B \(\qquad\) E \(\qquad\) J

Which one of the following is not a measure of spread? Circle your answer. [1 mark] median \(\qquad\) range \(\qquad\) standard deviation \(\qquad\) variance

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.

Year group	10	11	12	13
Number of students	200	240	150	130

1. Find the number of students from each year group that should be selected in the stratified random sample. [3 marks]
2. State one advantage of using a stratified random sample. [1 mark]

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

1. Find the value of \(p\) [3 marks]
2. Find P(\(Y = 17\)) [1 mark]

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.

\(x\)	0	1	2	3	4	5 or more
P(\(X = x\))	0.03	0.15	0.22	0.31	0.09	\(p\)

1. Find the value of \(p\) [2 marks]
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. [3 marks]
3. 1. State one assumption necessary for the calculation in part (b) to be valid. [1 mark]
   2. Comment on whether, in reality, this assumption is likely to be valid. [1 mark]

An investigation into the hydrocarbon emissions, \(X\) g/km, from cars in the Large Data Set was carried out. The results are summarised below. $$\sum x = 128.657 \qquad \sum x^2 = 8.701 \, 707 \qquad n = 2405$$ where \(n\) is the total number of cars which had a measured hydrocarbon emission in the Large Data Set.

1. 1. Find the mean of \(X\) [1 mark]
   2. Find the standard deviation of \(X\) [2 marks]
2. 1. The Large Data Set is a sample taken from the entire UK Department for Transport Stock Vehicle Database. It is claimed that the values in part (a)(i) and part (a)(ii) obtained from the Large Data Set should be reliable estimates for the mean and standard deviation of the hydrocarbon emissions for the entire UK Department for Transport Stock Vehicle Database. State, with a reason, whether this claim is likely to be correct. [1 mark]
   2. 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. [1 mark]

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. [5 marks]