Questions AS Paper 2 (315 questions)

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AQA AS Paper 2 2024 June Q9
9 marks Standard +0.3
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]
AQA AS Paper 2 2024 June Q10
11 marks Moderate -0.3
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. 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]
    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]
AQA AS Paper 2 2024 June Q11
1 marks Easy -1.8
The table below shows the daily salt intake, \(x\) grams, and the daily Vitamin C intake, \(y\) milligrams, for a group of 10 adults.
AdultABCDEFGHIJ
\(x\)5.36.23.610.42.49.4657.111.2
\(y\)9014588481144480955541
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
AQA AS Paper 2 2024 June Q12
1 marks Easy -2.5
Which one of the following is not a measure of spread? Circle your answer. [1 mark] median \(\qquad\) range \(\qquad\) standard deviation \(\qquad\) variance
AQA AS Paper 2 2024 June Q13
4 marks Easy -1.8
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 group10111213
Number of students200240150130
  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]
AQA AS Paper 2 2024 June Q14
4 marks Moderate -0.8
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]
AQA AS Paper 2 2024 June Q15
7 marks Moderate -0.3
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.
\(x\)012345 or more
P(\(X = x\))0.030.150.220.310.09\(p\)
  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]
    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]
AQA AS Paper 2 2024 June Q16
5 marks Easy -1.8
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. Find the mean of \(X\) [1 mark]
    2. Find the standard deviation of \(X\) [2 marks]
    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]
AQA AS Paper 2 2024 June Q17
5 marks Moderate -0.3
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]
AQA AS Paper 2 Specimen Q1
1 marks Easy -1.8
\(p(x) = x^3 - 5x^2 + 3x + a\), where \(a\) is a constant. Given that \(x - 3\) is a factor of \(p(x)\), find the value of \(a\) Circle your answer. [1 mark] \(-9\) \quad\quad \(-3\) \quad\quad \(3\) \quad\quad \(9\)
AQA AS Paper 2 Specimen Q2
1 marks Easy -1.8
The graph of \(y = f(x)\) is shown in Figure 1. \includegraphics{figure_1} State the equation of the graph shown in Figure 2. \includegraphics{figure_2} Circle your answer. [1 mark] \(y = f(2x)\) \quad\quad \(y = f\left(\frac{x}{2}\right)\) \quad\quad \(y = 2f(x)\) \quad\quad \(y = \frac{1}{2}f(x)\)
AQA AS Paper 2 Specimen Q3
2 marks Easy -1.2
Find the value of \(\log_a(a^5) + \log_a\left(\frac{1}{a}\right)\) [2 marks]
AQA AS Paper 2 Specimen Q4
3 marks Moderate -0.8
Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x^2 - 5x + a\), where \(a\) is a constant. Fully justify your answer. [3 marks]
AQA AS Paper 2 Specimen Q5
4 marks Moderate -0.3
The quadratic equation \(3x^2 + 4x + (2k - 1) = 0\) has real and distinct roots. Find the possible values of the constant \(k\) Fully justify your answer. [4 marks]
AQA AS Paper 2 Specimen Q6
4 marks Moderate -0.3
A curve has equation \(y = 6x^2 + \frac{8}{x^2}\) and is sketched below for \(x > 0\) \includegraphics{figure_6} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2a\), where \(a > 0\), giving your answer in terms of \(a\) [4 marks]
AQA AS Paper 2 Specimen Q7
5 marks Standard +0.3
Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]
AQA AS Paper 2 Specimen Q8
6 marks Moderate -0.5
Prove that the function \(f(x) = x^3 - 3x^2 + 15x - 1\) is an increasing function. [6 marks]
AQA AS Paper 2 Specimen Q9
5 marks Moderate -0.3
A curve has equation \(y = e^{2x}\) Find the coordinates of the point on the curve where the gradient of the curve is \(\frac{1}{2}\) Give your answer in an exact form. [5 marks]
AQA AS Paper 2 Specimen Q10
8 marks Moderate -0.3
David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50e^{0.5t}$$ where \(t\) is the time in years after 1 January 2016.
  1. Using David's model:
    1. state the population of rabbits on the island on 1 January 2016; [1 mark]
    2. predict the population of rabbits on 1 January 2021. [1 mark]
  2. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures. [2 marks]
  3. Give one reason why David's model may not be appropriate. [1 mark]
  4. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000e^{-0.1t}$$ where \(t\) is the time in years after 1 January 2016. Using the two models, find the year during which the population of rabbits first exceeds the population of crickets. [3 marks]
AQA AS Paper 2 Specimen Q11
10 marks Moderate -0.3
The circle with equation \((x - 7)^2 + (y + 2)^2 = 5\) has centre C.
    1. Write down the radius of the circle. [1 mark]
    2. Write down the coordinates of C. [1 mark]
  2. The point \(P(5, -1)\) lies on the circle. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = mx + c\) [4 marks]
  3. The point Q(3, 3) lies outside the circle and the point T lies on the circle such that QT is a tangent to the circle. Find the length of QT. [4 marks]
AQA AS Paper 2 Specimen Q12
4 marks Moderate -0.3
  1. Given that \(n\) is an even number, prove that \(9n^2 + 6n\) has a factor of 12 [3 marks]
  2. Determine if \(9n^2 + 6n\) has a factor of 12 for any integer \(n\). [1 mark]
AQA AS Paper 2 Specimen Q13
1 marks Easy -1.8
The number of pots of yoghurt, \(X\), consumed per week by adults in Milton is a discrete random variable with probability distribution given by
\(\boldsymbol{x}\)01234567 or more
\(\mathbf{P(X = x)}\)0.300.100.050.070.030.160.090.20
Find \(P(3 \leq X < 6)\) Circle the correct answer. [1 mark] 0.26 \quad\quad 0.31 \quad\quad 0.35 \quad\quad 0.40
AQA AS Paper 2 Specimen Q14
1 marks Easy -2.5
In the Large Data Set, the emissions of carbon dioxide are measured in what units? Circle your answer. [1 mark] mg/litre \quad\quad g/litre \quad\quad g/km \quad\quad mg/km
AQA AS Paper 2 Specimen Q15
2 marks Moderate -0.3
A school took 225 children on a trip to a theme park. After the trip the children had to write about their favourite ride at the park from a choice of three. The table shows the number of children who wrote about each ride.
Ride written about
The DropThe BeanstalkThe GiantTotal
Year 724452392
Year 836172275
Year 920132558
Total807570225
Three children were randomly selected from those who went on the trip. Calculate the probability that one wrote about 'The Drop', one wrote about 'The Beanstalk' and one wrote about The Giant'. [2 marks]
AQA AS Paper 2 Specimen Q16
2 marks Easy -1.8
The boxplot below represents the time spent in hours by students revising for a history exam. \includegraphics{figure_16}
  1. Use the information in the boxplot to state the value of a measure of central tendency of the revision times, stating clearly which measure you are using. [1 mark]
  2. Use the information in the boxplot to explain why the distribution of revision times is negatively skewed. [1 mark]