Questions AS Paper 2 (315 questions)

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Edexcel AS Paper 2 Specimen Q4
7 marks Moderate -0.8
  1. Sara was studying the relationship between rainfall, \(r \mathrm {~mm}\), and humidity, \(h \%\), in the UK. She takes a random sample of 11 days from May 1987 for Leuchars from the large data set.
She obtained the following results.
\(h\)9386959786949797879786
\(r\)1.10.33.720.6002.41.10.10.90.1
Sara examined the rainfall figures and found $$Q _ { 1 } = 0.1 \quad Q _ { 2 } = 0.9 \quad Q _ { 3 } = 2.4$$ A value that is more than 1.5 times the interquartile range (IQR) above \(Q _ { 3 }\) is called an outlier.
  1. Show that \(r = 20.6\) is an outlier.
  2. Give a reason why Sara might:
    1. include
    2. exclude
      this day's reading. Sara decided to exclude this day's reading and drew the following scatter diagram for the remaining 10 days' values of \(r\) and \(h\). \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-08_988_1081_1555_420}
  3. Give an interpretation of the correlation between rainfall and humidity. The equation of the regression line of \(r\) on \(h\) for these 10 days is \(r = - 12.8 + 0.15 h\)
  4. Give an interpretation of the gradient of this regression line.
    1. Comment on the suitability of Sara's sampling method for this study.
    2. Suggest how Sara could make better use of the large data set for her study.
Edexcel AS Paper 2 Specimen Q5
9 marks Easy -1.2
5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
Edexcel AS Paper 2 Specimen Q6
4 marks Easy -1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f3dbcb4-3260-4493-a230-12577b4ed691-12_520_1072_616_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A car moves along a straight horizontal road. At time \(t = 0\), the velocity of the car is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then accelerates with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds. The car travels a distance \(D\) metres during these \(T\) seconds. Figure 1 shows the velocity-time graph for the motion of the car for \(0 \leqslant t \leqslant T\).
Using the graph, show that \(D = U T + 1 / 2 a T ^ { 2 }\).
(No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.)
Edexcel AS Paper 2 Specimen Q7
7 marks Standard +0.3
  1. A car is moving along a straight horizontal road with constant acceleration. There are three points \(A , B\) and \(C\), in that order, on the road, where \(A B = 22 \mathrm {~m}\) and \(B C = 104 \mathrm {~m}\). The car takes 2 s to travel from \(A\) to \(B\) and 4 s to travel from \(B\) to \(C\).
Find
  1. the acceleration of the car,
  2. the speed of the car at the instant it passes \(A\).
Edexcel AS Paper 2 Specimen Q8
9 marks Standard +0.3
  1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.
The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$
  1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
  2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.
Edexcel AS Paper 2 Specimen Q9
10 marks Moderate -0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f3dbcb4-3260-4493-a230-12577b4ed691-18_694_1262_223_406} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches the pulley. The balls are modelled as particles, the string is modelled as being light and inextensible, the pulley is modelled as being small and smooth and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Write down an equation of motion for \(A\).
    2. Write down an equation of motion for \(B\).
  2. Hence find the acceleration of \(B\).
  3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
  4. Suggest two improvements that could be made in the model.
Edexcel AS Paper 2 Specimen Q1
9 marks Moderate -0.8
  1. A company manager is investigating the time taken, \(t\) minutes, to complete an aptitude test. The human resources manager produced the table below of coded times, \(x\) minutes, for a random sample of 30 applicants.
Coded time ( \(x\) minutes)Frequency (f)Coded time midpoint (y minutes)
\(0 \leq x < 5\)32.5
\(5 \leq x < 10\)157.5
\(10 \leq x < 15\)212.5
\(15 \leq x < 25\)920
\(25 \leq x < 35\)130
(You may use \(\sum f y = 355\) and \(\sum f y ^ { 2 } = 5675\) )
  1. Use linear interpolation to estimate the median of the coded times.
  2. Estimate the standard deviation of the coded times. The company manager is told by the human resources manager that he subtracted 15 from each of the times and then divided by 2 , to calculate the coded times.
  3. Calculate an estimate for the median and the standard deviation of \(t\).
    (3) The following year, the company has 25 positions available. The company manager decides not to offer a position to any applicant who takes 35 minutes or more to complete the aptitude test. The company has 60 applicants.
  4. Comment on whether or not the company manager's decision will result in the company being able to fill the 25 positions available from these 60 applicants. Give a reason for your answer.
Edexcel AS Paper 2 Specimen Q2
8 marks Standard +0.3
2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)
  1. Find \(\mathrm { P } ( 5 \leq X < 12 )\). Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
  2. Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
  3. State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test. \section*{(Total for Question 2 is 8 marks)}
Edexcel AS Paper 2 Specimen Q3
6 marks Standard +0.3
  1. Pete is investigating the relationship between daily rainfall, \(w \mathrm {~mm}\), and daily mean pressure, \(p\) hPa , in Perth during 2015. He used the large data set to take a sample of size 12.
He obtained the following results.
\(p\)100710121013100910191010101010101013101110141022
\(w\)102.063.063.038.438.035.034.232.030.428.028.015
Pete drew the following scatter diagram for the values of \(w\) and \(p\) and calculated the quartiles.
Q 1Q 2Q 3
\(p\)10101011.51013.5
\(w\)29.234.650.7
\includegraphics[max width=\textwidth, alt={}]{b29b0411-8401-420b-9227-befe25c245d8-04_818_1081_989_477}
An outlier is a value which is more than 1.5 times the interquartile range above Q3 or more than 1.5 times the interquartile range below Q1.
  1. Show that the 3 points circled on the scatter diagram above are outliers.
    (2)
  2. Describe the effect of removing the 3 outliers on the correlation between daily rainfall and daily mean pressure in this sample.
    (1) John has also been studying the large data set and believes that the sample Pete has taken is not random.
  3. From your knowledge of the large data set, explain why Pete's sample is unlikely to be a random sample. John finds that the equation of the regression line of \(w\) on \(p\), using all the data in the large data set, is $$w = 1023 - 0.223 p$$
  4. Give an interpretation of the figure - 0.223 in this regression line. John decided to use the regression line to estimate the daily rainfall for a day in December when the daily mean pressure is 1011 hPa .
  5. Using your knowledge of the large data set, comment on the reliability of John's estimate.
    (Total for Question 3 is 6 marks)
Edexcel AS Paper 2 Specimen Q4
7 marks Moderate -0.3
4. Alyona, Dawn and Sergei are sometimes late for school. The events \(A , D\) and \(S\) are as follows:
A Alyona is late for school
D Dawn is late for school
S Sergei is late for school The Venn diagram below shows the three events \(A , D\) and \(S\) and the probabilities associated with each region of \(D\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(D\). \includegraphics[max width=\textwidth, alt={}, center]{b29b0411-8401-420b-9227-befe25c245d8-06_624_1068_845_479}
  1. Write down 2 of the events \(A , D\) and \(S\) that are mutually exclusive. Give a reason for your answer. The probability that Sergei is late for school is 0.2 . The events \(A\) and \(D\) are independent.
  2. Find the value of \(r\).
    (4) Dawn and Sergei's teacher believes that when Sergei is late for school, Dawn tends to be late for school.
  3. State whether or not \(D\) and \(S\) are independent, giving a reason for your answer.
    (1)
  4. Comment on the teacher's belief in the light of your answer to part (c).
    (1)
    (Total for Question 4 is 7 marks) \section*{Pearson Edexcel Level 3} \section*{GCE Mathematics} \section*{Paper 2: Mechanics}
    Specimen paper
    Time: \(\mathbf { 3 5 }\) minutes
    		Paper Reference(s)
    \(\mathbf { 8 M A 0 } / \mathbf { 0 2 }\)
    You must have:
    Mathematical Formulae and Statistical Tables, calculator
    Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
    \section*{Information}
    \section*{Advice}
AQA AS Paper 2 2018 June Q1
1 marks Easy -1.8
Given that \(\frac{dy}{dx} = \frac{1}{6x^2}\), find \(y\). Circle your answer. \(\frac{-1}{3x^3} + c\) \quad \(\frac{1}{2x^3} + c\) \quad \(\frac{-1}{6x} + c\) \quad \(\frac{-1}{3x} + c\) [1 mark]
AQA AS Paper 2 2018 June Q2
1 marks Easy -1.8
Figure 1 shows \(y = f(x)\). \includegraphics{figure_1} Which figure below shows \(y = f(2x)\)? Tick one box. \includegraphics{figure_2} \quad \includegraphics{figure_3} \quad \includegraphics{figure_4} \quad \includegraphics{figure_5} [1 mark]
AQA AS Paper 2 2018 June Q3
2 marks Easy -1.2
Express as a single logarithm \(2\log_a 6 - \log_a 3\) [2 marks]
AQA AS Paper 2 2018 June Q4
4 marks Moderate -0.3
Solve the equation \(\tan^2 2\theta - 3 = 0\) giving all the solutions for \(0° \leq \theta \leq 360°\) [4 marks]
AQA AS Paper 2 2018 June Q5
4 marks Standard +0.3
\(f'(x) = \left(2x - \frac{3}{x}\right)^2\) and \(f(3) = 2\) Find \(f(x)\). [4 marks]
AQA AS Paper 2 2018 June Q6
6 marks Standard +0.3
Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).
  1. Prove that \(ABCD\) is a rectangle. [4 marks]
  2. Find the area of \(ABCD\). [2 marks]
AQA AS Paper 2 2018 June Q7
6 marks Moderate -0.8
  1. Express \(2x^2 - 5x + k\) in the form \(a(x - b)^2 + c\) [3 marks]
  2. Find the values of \(k\) for which the curve \(y = 2x^2 - 5x + k\) does not intersect the line \(y = 3\) [3 marks]
AQA AS Paper 2 2018 June Q8
4 marks Moderate -0.3
A circle of radius 6 passes through the points \((0, 0)\) and \((0, 10)\).
  1. Sketch the two possible positions of the circle. [1 mark]
  2. Find the equations of the two circles. [3 marks]
AQA AS Paper 2 2018 June Q9
3 marks Standard +0.3
It is given that \(\cos 15° = \frac{1}{2}\sqrt{2 + \sqrt{3}}\) and \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\) Show that \(\tan^2 15°\) can be written in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. Fully justify your answer. [3 marks]
AQA AS Paper 2 2018 June Q10
5 marks Standard +0.3
In the binomial expansion of \((1 + x)^n\), where \(n \geq 4\), the coefficient of \(x^4\) is \(\frac{1}{2}\) times the sum of the coefficients of \(x^2\) and \(x^3\) Find the value of \(n\). [5 marks]
AQA AS Paper 2 2018 June Q11
9 marks Standard +0.8
Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. She bends a rectangle of steel to make an open cylinder and welds the joint. She then welds this cylinder to the circumference of a circular base. \includegraphics{figure_11} The planter must have a capacity of \(8000\text{cm}^3\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum. Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy. [9 marks]
AQA AS Paper 2 2018 June Q12
8 marks Standard +0.3
Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.
  1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
  2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
  3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
  4. Comment on the long-term accuracy of the model. [1 mark]
AQA AS Paper 2 2018 June Q13
1 marks Easy -1.8
The table below shows the probability distribution for a discrete random variable \(X\).
\(x\)01234 or more
P(X = x)0.350.25\(k\)0.140.1
Find the value of \(k\). Circle your answer. 0.14 \quad 0.16 \quad 0.18 \quad 1 [1 mark]
AQA AS Paper 2 2018 June Q14
1 marks Easy -1.8
Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]
AQA AS Paper 2 2018 June Q15
6 marks Moderate -0.8
Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.
  1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
  2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
  3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]