AQA AS Paper 2 (AS Paper 2) 2019 June

1 Find the gradient of the curve \(y = \mathrm { e } ^ { - 3 x }\) at the point where it crosses the \(y\)-axis. Circle your answer.
\(\begin{array} { l l l } - 3 & - 1 & 1 \end{array}\)

2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\)
Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □
\(( 2 , - 3 )\) □
\(( 2,3 )\) □

3 It is given that \(\sin \theta = - 0.1\) and \(180 ^ { \circ } < \theta < 270 ^ { \circ }\) Find the exact value of \(\cos \theta\)

4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$

5 A triangular prism has a cross section \(A B C\) as shown in the diagram below. Angle \(A B C = 25 ^ { \circ }\)
Angle \(A C B = 30 ^ { \circ }\)
\(B C = 40\) millimetres. The length of the prism is 300 millimetres.
Calculate the volume of the prism, giving your answer to three significant figures.

6 A curve has equation \(y = \frac { 2 } { x \sqrt { x } }\)
\includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-05_508_549_420_744} The region enclosed between the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = a\) has area 3 units. Given that \(a > 1\), find the value of \(a\).
Fully justify your answer.

7 The points \(A ( a , 3 )\) and \(B ( 10,6 )\) lie on a circle.
\(A B\) is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where \(c , d\) and \(e\) are rational numbers. Find the values of \(a , c , d\) and \(e\).

8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point \(R ( 2,3 )\)
The gradient of the curve at \(R\) is 8
8

1. Find the value of \(p\) and the value of \(q\).
   8
2. Calculate the area enclosed between the normal to the curve at \(R\) and the coordinate 8
3. axes.
   \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$

9

1. Find the exact coordinates of the turning points of \(C\).
   Determine the nature of each turning point.
   Fully justify your answer.
   9
2. State the coordinates of the turning points of the curve $$y = \mathrm { f } ( x + 1 ) - 4$$

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below. Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10

1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
3. Using Zena's results, calculate the values of \(A\) and \(k\).
   10
4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
   10
5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.

11 A survey is undertaken to find out the most popular political party in London.
The first 1100 available people from London are surveyed.
Identify the name of this type of sampling.
Circle your answer.
simple random
opportunity
stratified
quota

12 Manny is studying the price and number of pages of a random sample of books.
He calculates the value of the product moment correlation coefficient between the price and number of pages in each book as 1.05 Which of the following best describes the value 1.05 ?
Tick ( \(\checkmark\) ) one box.
definitely correct □
probably correct □
probably incorrect □
definitely incorrect □
\includegraphics[max width=\textwidth, alt={}, center]{b45dc98e-1699-47c9-9228-5abe0e5c9195-15_2488_1716_219_153}

13 Denzel wants to buy a car with a propulsion type other than petrol or diesel.
He takes a sample, from the Large Data Set, of the CO2 emissions, in \(\mathrm { g } / \mathrm { km }\), of cars with one particular propulsion type. The sample is as follows $$\begin{array} { l l l l l l l l } 82 & 13 & 96 & 49 & 96 & 92 & 70 & 81 \end{array}$$ 13

1. Using your knowledge of the Large Data Set, state which propulsion type this sample is for, giving a reason for your answer.
   13
2. Calculate the mean of the sample.
   13
3. Calculate the standard deviation of the sample.
   13
4. Denzel claims that the value 13 is an outlier. 13
5. 1. Any value more than 2 standard deviations from the mean can be regarded as an outlier. Verify that Denzel's claim is correct.
      13
6. (ii) State what effect, if any, removing the value 13 from the sample would have on the standard deviation.

14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where \(c\) is a constant.
14

1. Show that \(c = \frac { 1 } { 10 }\)
   14
2. Calculate \(\mathrm { P } ( X \geq 1 )\)

15 Two independent events, \(A\) and \(B\), are such that $$\begin{aligned} \mathrm { P } ( A ) & = 0.2
\mathrm { P } ( A \cup B ) & = 0.8 \end{aligned}$$ 15

1. 1. Find \(\mathrm { P } ( B )\)
      15
2. (ii) Find \(\mathrm { P } ( A \cap B )\)
   15
3. State, with a reason, whether or not the events \(A\) and \(B\) are mutually exclusive.
   \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

16
16

1. 
   \end{tabular} &

   Andrea is the manager of a company which makes mobile phone chargers.

   In the past, she had found that \(12 \%\) of all chargers are faulty.

   Andrea decides to move the manufacture of chargers to a different factory.

   Andrea tests 60 of the new chargers and finds that 4 chargers are faulty.

   Investigate, at the \(10 \%\) level of significance, whether the proportion of faulty chargers has reduced.

   [7 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   
   \hline \end{tabular} \end{center} 16
2. State, in context, two assumptions that are necessary for the distribution that you have used in part (a) to be valid.