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AQA Paper 2 2023 June Q8
10 marks Standard +0.3
8
  1. Given that \(\cos \theta \neq \pm 1\), prove the identity $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } \equiv 2 \operatorname { cosec } ^ { 2 } \theta$$ 8
  2. Hence, find the set of values of \(A\) for which the equation $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = A$$ has real solutions.
    Fully justify your answer.
    8
  3. Given that \(\theta\) is obtuse and $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = 16$$ find the exact value of \(\cot \theta\)
AQA Paper 2 2023 June Q9
6 marks Moderate -0.8
9
  1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + x ) ^ { - \frac { 1 } { 2 } }$$ 9
  2. A student substitutes \(x = 2\) into the expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) to find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Explain the mistake in the student's approach.
    [0pt] [1 mark] 9
  3. By substituting \(x = - \frac { 1 } { 4 }\) in your expansion for \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Give your answer to three significant figures.
AQA Paper 2 2023 June Q10
6 marks Standard +0.8
10

  1. 10

  2. 10
  3. Given that \(a\) and \(b\) are distinct positive numbers, use proof by contradiction to prove that $$\frac { a } { b } + \frac { b } { a } > 2$$ \section*{END OF SECTION A
    TURN OVER FOR SECTION B}
AQA Paper 2 2023 June Q11
1 marks Easy -2.0
11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N
AQA Paper 2 2023 June Q12
1 marks Easy -1.2
12 A particle moves in a straight line.
After the first 4 seconds of its motion, the displacement of the particle from its initial position is 0 metres. One of the graphs on the opposite page shows the velocity \(v \mathrm {~ms} ^ { - 1 }\) of the particle after time \(t\) seconds of its motion. Identify the correct graph.
Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-19_2249_896_260_484}
AQA Paper 2 2023 June Q13
5 marks Moderate -0.3
13 A ball falls freely towards the Earth.
The ball passes through two different fixed points \(M\) and \(N\) before reaching the Earth's surface. At \(M\) the ball has velocity \(u \mathrm {~ms} ^ { - 1 }\) At \(N\) the ball has velocity \(3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) It can be assumed that:
  • the motion is due to gravitational force only
  • the acceleration due to gravity remains constant throughout.
13
  1. Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\frac { 2 u } { g }\) seconds.
    [0pt] [2 marks] 13
  2. Point \(M\) is \(h\) metres above the Earth. Show that \(h > \frac { 4 u ^ { 2 } } { g }\) Fully justify your answer.
    The car is moving in a straight line.
    The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the car at time \(t\) seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where \(k\) is a constant.
    When \(t = 3\) the car has a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show that \(k = \frac { 1 } { 3 }\)
AQA Paper 2 2023 June Q14
4 marks Moderate -0.8
14 A car has an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) A particle, \(Q\), moves in a straight line across a rough horizontal surface.
A horizontal driving force of magnitude \(D\) newtons acts on \(Q\) \(Q\) moves with a constant acceleration of \(0.91 \mathrm {~ms} ^ { - 2 }\) \(Q\) has a weight of 0.65 N
The only resistance force acting on \(Q\) is due to friction.
The coefficient of friction between \(Q\) and the surface is 0.4 Find \(D\)
AQA Paper 2 2023 June Q15
4 marks Standard +0.3
15 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
AQA Paper 2 2023 June Q16
4 marks Moderate -0.8
16 A particle moves under the action of two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) It is given that $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 1.6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( k \mathbf { i } + 5 k \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(k\) is a constant.
The acceleration of the particle is \(( 3.2 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) Find \(k\) \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-25_2488_1716_219_153}
AQA Paper 2 2023 June Q17
6 marks Standard +0.3
17 A uniform plank \(P Q\), of length 7 metres, lies horizontally at rest, in equilibrium, on two fixed supports at points \(X\) and \(Y\) The distance \(P X\) is 1.4 metres and the distance \(Q Y\) is 2 metres as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_56_689_534_762} \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_225_830_607_694} 17
  1. The reaction force on the plank at \(X\) is \(4 g\) newtons.
    17
    1. (i) Show that the mass of the plank is 9.6 kilograms.
      17
    2. (ii) Find the reaction force, in terms of \(g\), on the plank at \(Y\) 17
    3. The support at \(Y\) is moved so that the distance \(Q Y = 1.4\) metres. The plank remains horizontally at rest in equilibrium.
      It is claimed that the reaction force at \(Y\) remains unchanged.
      Explain, with a reason, whether this claim is correct.
AQA Paper 2 2023 June Q18
6 marks Moderate -0.3
18 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors representing due east and due north respectively. A particle, \(T\), is moving on a plane at a constant speed.
The path followed by \(T\) makes the exact shape of a triangle \(A B C\). \(T\) moves around \(A B C\) in an anticlockwise direction as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925} On its journey from \(A\) to \(B\) the velocity vector of \(T\) is \(( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) 18
  1. Find the speed of \(T\) as it moves from \(A\) to \(B\) 18
  2. On its journey from \(B\) to \(C\) the velocity vector of \(T\) is \(( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Show that the acute angle \(A B C = 60 ^ { \circ }\) 18
  3. It is given that \(A B C\) is an equilateral triangle. \(T\) returns to its initial position after 9 seconds.
    Vertex \(B\) lies at position vector \(\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]\) metres with respect to a fixed origin \(O\) Find the position vector of \(C\)
AQA Paper 2 2023 June Q19
12 marks Moderate -0.3
19 A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod. The train engine has a mass of 1.5 kilograms.
The trailer has a mass 0.7 kilograms.
A string inclined at an angle of \(40 ^ { \circ }\) above the horizontal is attached to the front of the train engine. The tension in the string is 2 newtons.
As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration \(0.06 \mathrm {~ms} ^ { - 2 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-30_373_789_904_756} As it moves the train engine experiences a total resistance force of 0.8 N
19
  1. Show that the total resistance force experienced by the trailer is approximately 0.6 N
    19
  2. At the instant that the toy reaches a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks. As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of \(h\) metres. It can be assumed that the resistance forces remain unchanged.
    19 (b) (i) Find the tension in the rod after the string has broken.
    19 (b) (ii) Find \(h\)Do not write outside the box
    \includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-33_2488_1716_219_153}
    Nell and her pet dog Maia are visiting the beach.
    The beach surface can be assumed to be level and horizontal. Nell and Maia are initially standing next to each other.
    Nell throws a ball forward, from a height of 1.8 metres above the surface of the beach, at an angle of \(60 ^ { \circ }\) above the horizontal with a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Exactly 0.2 seconds after the ball is thrown, Maia sets off from Nell and runs across the surface of the beach, in a straight line with a constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Maia catches the ball when it is 0.3 metres above ground level as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-34_778_1287_1027_463}
AQA Paper 2 2023 June Q20
7 marks Moderate -0.8
20 In this question use \(g = 9.8 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} Find \(a\)
\includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-36_2488_1719_219_150}
Edexcel AS Paper 2 2018 June Q1
3 marks Moderate -0.8
  1. A company is introducing a job evaluation scheme. Points ( \(x\) ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( \(\pounds y\) ) will then be allocated to each job according to the number of points awarded.
Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was \(y = 4.5 x - 47\)
  1. Describe the correlation between points and pay.
  2. Give an interpretation of the gradient of this regression line.
  3. Explain why this model might not be appropriate for all jobs in the company.
Edexcel AS Paper 2 2018 June Q2
4 marks Moderate -0.3
  1. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty.
Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.
  1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
  2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
Edexcel AS Paper 2 2018 June Q3
7 marks Moderate -0.3
Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\) Naasir and his friends play the game 15 times.
  1. Find the probability that Naasir wins
    1. exactly 2 games,
    2. more than 5 games. Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
  2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.
Edexcel AS Paper 2 2018 June Q4
8 marks Moderate -0.8
  1. Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.
\begin{table}[h]
Windspeed\(\mathrm { n } / \mathrm { a }\)67891112131416
Frequency13232231212
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Calculate the mean for these data.
  2. Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]
    Month\(A\)\(B\)\(C\)\(D\)\(E\)
    Mean7.588.268.578.5711.57
    Standard Deviation2.933.893.463.874.64
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  3. Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
    1. State the meaning of the * symbol on some of the box plots.
    2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}
Edexcel AS Paper 2 2018 June Q5
8 marks Moderate -0.3
5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)
  1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
  2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
  3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)
Edexcel AS Paper 2 2018 June Q6
4 marks Moderate -0.8
  1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).
Edexcel AS Paper 2 2018 June Q7
7 marks Moderate -0.3
  1. A train travels along a straight horizontal track between two stations, \(A\) and \(B\).
In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. For this model of the motion of the train between \(A\) and \(B\),
    1. state the value of the constant velocity of the train,
    2. state the time for which the train is decelerating,
    3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
  2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
  3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.
Edexcel AS Paper 2 2018 June Q8
10 marks Standard +0.3
A particle, \(P\), moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement, \(x\) metres, of \(P\) from the origin \(O\), is given by \(x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)\)
  1. Find the times when \(P\) is instantaneously at rest.
  2. Find the total distance travelled by \(P\) in the time interval \(0 \leqslant t \leqslant 2\)
  3. Show that \(P\) will never move along the negative \(x\)-axis.
Edexcel AS Paper 2 2018 June Q9
9 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,
  1. find, in terms of \(m\) and \(g\), the tension in the string,
  2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
  3. find the value of \(k\).
  4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).
Edexcel AS Paper 2 Specimen Q1
4 marks Easy -1.8
  1. Sara is investigating the variation in daily maximum gust, \(t \mathrm { kn }\), for Camborne in June and July 1987.
She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.
  1. State the sampling technique Sara used.
  2. From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
  3. Calculate the standard deviation.
Edexcel AS Paper 2 Specimen Q2
5 marks Moderate -0.8
  1. The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}
Delay (minutes)Number of motorists
4-66
7-8
917
10-1245
13-159
16-20
Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.
Edexcel AS Paper 2 Specimen Q3
5 marks Easy -1.2
  1. The Venn diagram shows the probabilities for students at a college taking part in various sports. \(A\) represents the event that a student takes part in Athletics. \(T\) represents the event that a student takes part in Tennis. \(C\) represents the event that a student takes part in Cricket. \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}
The probability that a student selected at random takes part in Athletics or Tennis is 0.75
  1. Find the value of \(p\).
  2. State, giving a reason, whether or not the events \(A\) and \(T\) are statistically independent. Show your working clearly.
  3. Find the probability that a student selected at random does not take part in Athletics or Cricket.