Questions AS Paper 2 (315 questions)

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OCR MEI AS Paper 2 Specimen Q7
7 marks Easy -1.2
7 A farmer has 200 apple trees. She is investigating the masses of the crops of apples from individual trees. She decides to select a sample of these trees and find the mass of the crop for each tree.
  1. Explain how she can select a random sample of 10 different trees from the 200 trees. The masses of the crops from the 10 trees, measured in kg, are recorded as follows. \(\begin{array} { l l l l l l l l l l } 23.5 & 27.4 & 26.2 & 29.0 & 25.1 & 27.4 & 26.2 & 28.3 & 38.1 & 24.9 \end{array}\)
  2. For these data find
OCR MEI AS Paper 2 Specimen Q8
7 marks Moderate -0.8
8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-07_1144_1541_497_276} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.
  1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
  2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
  3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
    1. \(\quad t = 0\),
    2. \(t = 20\).
  4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.
OCR MEI AS Paper 2 Specimen Q9
7 marks Easy -1.3
9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan) \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-08_202_1595_1153_299} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
  2. The birth rates for all the countries in Australasia are shown below.
    CountryBirth rate per 1000
    Australia12.19
    New Zealand13.4
    Papua New Guinea24.89
    1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
    2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
  3. Describe the correlation in the scatter diagram.
  4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.
OCR MEI AS Paper 2 Specimen Q10
9 marks Moderate -0.3
10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.
  1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
    1. exactly 28 trains arrive on time,
    2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
  2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,
OCR MEI AS Paper 2 Specimen Q12
3 marks Standard +0.3
12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}
OCR MEI AS Paper 2 2024 June Q11
6 marks Moderate -0.8
  1. Verify that the curve cuts the \(x\)-axis at \(x = 4\) and at \(x = 9\). The curve does not cut or touch the \(x\)-axis at any other points.
  2. Determine the exact area bounded by the curve and the \(x\)-axis.
OCR MEI AS Paper 2 2021 November Q10
6 marks Standard +0.3
  1. Show that PQ is perpendicular to QR . A circle passes through \(\mathrm { P } , \mathrm { Q }\) and R .
  2. Determine the coordinates of the centre of the circle.
OCR MEI AS Paper 2 2019 June Q10
10 marks Standard +0.8
10 In this question you must show detailed reasoning. The equation of a curve is \(y = \frac { x ^ { 2 } } { 4 } + \frac { 2 } { x } + 1\). A tangent and a normal to the curve are drawn at the point where \(x = 2\). Calculate the area bounded by the tangent, the normal and the \(x\)-axis. \section*{END OF QUESTION PAPER}
OCR MEI AS Paper 2 2023 June Q14
7 marks Moderate -0.8
14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).
OCR MEI AS Paper 2 2024 June Q8
4 marks Moderate -0.3
8 In this question you must show detailed reasoning. Determine the coordinates of the point of intersection of the line with equation \(y = 2 x + 3\) and the curve with equation \(y ^ { 2 } - 4 x ^ { 2 } = 33\).
OCR MEI AS Paper 2 2020 November Q7
8 marks Moderate -0.3
7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.
OCR MEI AS Paper 2 2021 November Q3
3 marks Moderate -0.8
3 In this question you must show detailed reasoning. You are given that \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Explain why \(\tan 690 ^ { \circ } = - \frac { 1 } { \sqrt { 3 } }\).
OCR MEI AS Paper 2 Specimen Q11
6 marks Standard +0.8
11 In this question you must show detailed reasoning. Fig. 11 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic function. Fig. 11 also shows the coordinates of the turning points and the points of intersection with the axes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-11_805_620_543_317} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Show that the tangent to \(y = \mathrm { f } ( x )\) at \(x = t\) is parallel to the tangent to \(y = \mathrm { f } ( x )\) at \(x = - t\) for all values of \(t\).
AQA AS Paper 2 2019 June Q1
1 marks Easy -1.8
1 Find the gradient of the curve \(y = \mathrm { e } ^ { - 3 x }\) at the point where it crosses the \(y\)-axis. Circle your answer. \(\begin{array} { l l l } - 3 & - 1 & 1 \end{array}\)
AQA AS Paper 2 2019 June Q2
1 marks Easy -1.3
2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\) Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □ \(( 2 , - 3 )\) □ \(( 2,3 )\) □
AQA AS Paper 2 2019 June Q3
2 marks Moderate -0.8
3 It is given that \(\sin \theta = - 0.1\) and \(180 ^ { \circ } < \theta < 270 ^ { \circ }\) Find the exact value of \(\cos \theta\)
AQA AS Paper 2 2019 June Q4
4 marks Moderate -0.8
4 Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv 2 \left( - 1 + \log _ { 10 } 3 x \right)$$
AQA AS Paper 2 2019 June Q5
4 marks Moderate -0.3
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.
AQA AS Paper 2 2019 June Q6
5 marks Moderate -0.3
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.
AQA AS Paper 2 2019 June Q7
6 marks Moderate -0.3
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 \(e\).
AQA AS Paper 2 2019 June Q8
10 marks Standard +0.3
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 (b) axes. \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$
AQA AS Paper 2 2019 June Q9
10 marks Moderate -0.3
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 $$y = \mathrm { f } ( x + 1 ) - 4$$
AQA AS Paper 2 2019 June Q10
10 marks Moderate -0.3
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.
Time (minutes)1020
Temperature (degrees Celsius)3012
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.
AQA AS Paper 2 2019 June Q11
1 marks Easy -2.0
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
AQA AS Paper 2 2019 June Q12
1 marks Easy -1.8
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}