AQA AS Paper 2 (AS Paper 2) Specimen

1 \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 3 x + a\), where \(a\) is a constant.
Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\)
Circle your answer.
[0pt] [1 mark]
\(- 9 - 339\)

2 The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h] \end{figure} State the equation of the graph shown in Figure 2. \begin{figure}[h] \end{figure} Circle your answer.
[0pt] [1 mark] $$y = \mathrm { f } ( 2 x ) \quad y = \mathrm { f } \left( \frac { x } { 2 } \right) \quad y = 2 \mathrm { f } ( x ) \quad y = \frac { 1 } { 2 } \mathrm { f } ( x )$$

3 Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
[0pt] [2 marks]

4 Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x ^ { 2 } - 5 x + a\), where \(a\) is a constant. Fully justify your answer.
[0pt] [3 marks]

5 The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
Find the possible values of the constant \(k\)
Fully justify your answer.
[0pt] [4 marks]

6 A curve has equation \(y = 6 x ^ { 2 } + \frac { 8 } { x ^ { 2 } }\) and is sketched below for \(x > 0\)
\includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-06_638_842_539_758} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2 a\), where \(a > 0\), giving your answer in terms of \(a\)
[0pt] [4 marks]

7 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 \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
Fully justify your answer.
[0pt] [5 marks]

8 Prove that the function \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 15 x - 1\) is an increasing function.
[0pt] [6 marks]

9 A curve has equation \(y = \mathrm { e } ^ { 2 x }\)
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.
[0pt] [5 marks]
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 $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.

10

1. Using David's model: 10
2. 1. state the population of rabbits on the island on 1 January 2016; 10
3. (ii) predict the population of rabbits on 1 January 2021. 10
4. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
   [0pt] [2 marks] 10
5. Give one reason why David's model may not be appropriate.
   [0pt] [1 mark] 10
6. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ 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.
   [0pt] [3 marks]

11 The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\). 11

1. 1. Write down the radius of the circle. 11
2. (ii) Write down the coordinates of \(C\).
   [0pt] [1 mark] 11
3. 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 \(y = m x + c\)
   [0pt] [4 marks] 11
4. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
   [0pt] [4 marks]

12

1. Given that \(n\) is an even number, prove that \(9 n ^ { 2 } + 6 n\) has a factor of 12
   [0pt] [3 marks]
   12
2. Determine if \(9 n ^ { 2 } + 6 n\) has a factor of 12 for any integer \(n\).
   END OF SECTION A

13 The number of pots of yoghurt, \(X\), consumed per week by adults in Milton is a discrete random variable with probability distribution given by Find \(\mathrm { P } ( 3 \leq X < 6 )\) Circle the correct answer. \(0.26 \quad 0.31 \quad 0.35 \quad 0.40\)

14 In the Large Data Set, the emissions of carbon dioxide are measured in what units? Circle your answer.
[0pt] [1 mark]
mg/litre
g/litre
g/km
mg/km 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. 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’.
[0pt] [2 marks]

16 The boxplot below represents the time spent in hours by students revising for a history exam.
\includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-18_373_753_427_778} 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.
   [0pt] [1 mark] 16
2. Use the information in the boxplot to explain why the distribution of revision times is negatively skewed.
   [0pt] [1 mark]

17 The table below is an extract from the Large Data Set. 17

1. 1. Calculate the standard deviation of the engine sizes in the table.
      [0pt] [1 mark] 17
2. (ii) The mean of the engine sizes is 2084
   Any value more than 2 standard deviations from the mean can be identified as an outlier. Using this definition of an outlier, show that the sample of engine sizes has exactly one outlier. Fully justify your answer.
   [0pt] [3 marks] 17
3. Rajan calculates the mean of the masses of the cars in this extract and states that it is 1094 kg. Use your knowledge of the Large Data Set to suggest what error Rajan is likely to have made in his calculation.
   [0pt] [1 mark] 17
4. Rajan claims there is an error in the data recorded in the table for one of the Toyotas from the South West, because there is no value for its carbon monoxide emissions. Use your knowledge of the Large Data Set to comment on Rajan's claim.
   [0pt] [1 mark]

18 Neesha wants to open an Indian restaurant in her town.
Her cousin, Ranji, has an Indian restaurant in a neighbouring town. To help Neesha plan her menu, she wants to investigate the dishes chosen by a sample of Ranji's customers. Ranji has a list of the 750 customers who dined at his restaurant during the past month and the dish that each customer chose. Describe how Neesha could obtain a simple random sample of size 50 from Ranji's customers.
[0pt] [4 marks]

19 Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution \(\mathrm { B } ( n , 0.20 )\). 19

1. There are 10 students in Ellie's statistics class.
   Using the distributional model suggested by Ellie, find the probability that, of the students in her class: 19
2. 1. two or fewer eat at least five portions of fruit and vegetables every day;
      [0pt] [1 mark] 19
3. (ii) at least one but fewer than four eat at least five portions of fruit and vegetables every day;
   [0pt] [2 marks] 19
4. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day. 19
5. 1. Name the sampling method used by Declan. 19
6. (ii) Describe one weakness of this sampling method.
   19
7. (iii) Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the \(5 \%\) significance level to investigate whether Declan's belief is supported by this evidence.
   [0pt] [6 marks]