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Edexcel Paper 2 Specimen Q3
4 marks Moderate -0.3
  1. A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation
$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$ where \(A\) is a constant. The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.
  1. Find a complete equation for the model.
  2. Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place. Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\). Using this information,
  3. evaluate the model, explaining your reasoning.
Edexcel Paper 2 Specimen Q4
6 marks Moderate -0.3
  1. Sketch the graph with equation $$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
  2. Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$
  3. Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.
Edexcel Paper 2 Specimen Q5
5 marks Standard +0.3
  1. The line \(l\) has equation
$$3 x - 2 y = k$$ where \(k\) is a real constant.
Given that the line \(l\) intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for \(k\).
Edexcel Paper 2 Specimen Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-12_624_1057_258_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 2.
  1. Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
  2. Show that the \(x\) coordinate of \(Q\) satisfies $$x = \frac { 8 } { 1 + \ln x }$$
  3. Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with \(x _ { 1 } = 3.5\) to
    1. find the value of \(x _ { 5 }\) to 4 decimal places,
    2. find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.
Edexcel Paper 2 Specimen Q7
12 marks Moderate -0.8
  1. A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.
A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.
  1. Show that the scientist's model for \(p\) leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where \(a\) and \(k\) are constants. The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
  2. Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
  3. Hence show that the model for \(p\) can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant \(b\). With reference to this model,
    1. interpret the value of the constant \(a\),
    2. interpret the value of the constant \(b\).
  5. State a long term limitation of the model for \(p\).
Edexcel Paper 2 Specimen Q8
7 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-18_367_709_280_676} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl. When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)
  1. Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\) Given that the flow of water into the bowl is increased to a constant rate of \(300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(12 < h \leqslant 24\)
  2. find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 20\)
Edexcel Paper 2 Specimen Q9
9 marks Standard +0.3
A circle with centre \(A ( 3 , - 1 )\) passes through the point \(P ( - 9,8 )\) and the point \(Q ( 15 , - 10 )\)
  1. Show that \(P Q\) is a diameter of the circle.
  2. Find an equation for the circle. A point \(R\) also lies on the circle. Given that the length of the chord \(P R\) is 20 units,
  3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  4. Find the size of angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.
Edexcel Paper 2 Specimen Q10
9 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-22_554_862_260_603} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$
  1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
  2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).
Edexcel Paper 2 Specimen Q11
5 marks Standard +0.8
  1. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.
Show that the sum of the first \(n\) terms of the sequence is a square number.
Edexcel Paper 2 Specimen Q12
7 marks Standard +0.8
  1. A curve \(C\) is given by the equation
$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q13
10 marks Standard +0.8
13.
  1. Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q14
8 marks Standard +0.3
  1. Kayden claims that $$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
  2. Prove that \(\sqrt { 3 }\) is an irrational number.
Edexcel Paper 3 2019 June Q1
6 marks Moderate -0.8
  1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]
At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).
  1. Find the acceleration of \(P\) when \(t = 4\)
  2. Find the position vector of \(P\) when \(t = 4\)
Edexcel Paper 3 2019 June Q2
8 marks Standard +0.3
  1. A particle, \(P\), moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0\), the particle is at the point \(A\) and is moving with velocity ( \(- \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(3 \mathbf { i } - 4 \mathbf { j }\) )
  1. Find the value of \(T\). At time \(t = 4\) seconds, \(P\) is at the point \(B\).
  2. Find the distance \(A B\).
Edexcel Paper 3 2019 June Q3
12 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-06_339_812_242_628} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley, \(P\), fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(B\) hangs freely below \(P\), as shown in Figure 1. The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\) The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.
  1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
  2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
  3. Suggest two refinements to the model that would make it more realistic.
Edexcel Paper 3 2019 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-10_417_844_244_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ramp, \(A B\), of length 8 m and mass 20 kg , rests in equilibrium with the end \(A\) on rough horizontal ground. The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as \(A\). The point of contact between the ramp and the drum is \(C\), where \(A C = 5 \mathrm {~m}\), as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) The ramp is modelled as a uniform rod.
  1. Explain why the reaction from the drum on the ramp at point \(C\) acts in a direction which is perpendicular to the ramp.
  2. Find the magnitude of the resultant force acting on the ramp at \(A\). The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of mass of the ramp is assumed to be closer to \(A\) than to \(B\),
  3. state how this would affect the magnitude of the normal reaction between the ramp and the drum at \(C\).
Edexcel Paper 3 2019 June Q5
13 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-14_223_855_239_605} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The points \(A\) and \(B\) lie 50 m apart on horizontal ground.
At time \(t = 0\) two small balls, \(P\) and \(Q\), are projected in the vertical plane containing \(A B\).
Ball \(P\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(30 ^ { \circ }\) to \(A B\).
Ball \(Q\) is projected from \(B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 3.
At time \(t = 2\) seconds, \(P\) and \(Q\) collide.
Until they collide, the balls are modelled as particles moving freely under gravity.
  1. Find the velocity of \(P\) at the instant before it collides with \(Q\).
  2. Find
    1. the size of angle \(\theta\),
    2. the value of \(u\).
  3. State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.
Edexcel Paper 3 2019 June Q1
8 marks Moderate -0.8
  1. Three bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.
Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only
Sasha selects at random one marble from bag \(A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(C\).
  1. Draw a tree diagram to represent this information.
  2. Find the probability that Sasha selects 3 green marbles.
  3. Find the probability that Sasha selects at least 1 marble of each colour.
  4. Given that Sasha selects a red marble, find the probability that he selects it from bag \(B\).
Edexcel Paper 3 2019 June Q2
11 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-06_321_1822_294_127} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than \(1.5 \times\) IQR below \(Q _ { 1 }\) or
more than \(1.5 \times\) IQR above \(Q _ { 3 }\) The three lowest air temperatures in the data set are \(7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }\) and \(9.1 ^ { \circ } \mathrm { C }\) The highest air temperature in the data set is \(32.5 ^ { \circ } \mathrm { C }\)
  1. Complete the box plot in Figure 1 showing clearly any outliers.
  2. Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, \(x ^ { \circ } \mathrm { C }\), for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
  3. Show that, to 3 significant figures, the standard deviation is \(5.19 ^ { \circ } \mathrm { C }\) Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
  4. Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
  5. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer. \includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
    (Total for Question 2 is 11 marks)
Edexcel Paper 3 2019 June Q3
9 marks Standard +0.3
3. Barbara is investigating the relationship between average income (GDP per capita), \(x\) US dollars, and average annual carbon dioxide ( \(\mathrm { CO } _ { 2 }\) ) emissions, \(y\) tonnes, for different countries. She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual \(\mathrm { CO } _ { 2 }\) emissions and average income to be 0.446
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero. Barbara believes that a non-linear model would be a better fit to the data.
    She codes the data using the coding \(m = \log _ { 10 } x\) and \(c = \log _ { 10 } y\) and obtains the model \(c = - 1.82 + 0.89 m\) The product moment correlation coefficient between \(c\) and \(m\) is found to be 0.882
  2. Explain how this value supports Barbara's belief.
  3. Show that the relationship between \(y\) and \(x\) can be written in the form \(y = a x ^ { n }\) where \(a\) and \(n\) are constants to be found.
Edexcel Paper 3 2019 June Q4
9 marks Moderate -0.3
  1. Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.
Daily mean total cloud cover (oktas)012345678
Frequency (number of days)01471030525228
One of the 184 days is selected at random.
  1. Find the probability that it has a daily mean total cloud cover of 6 or greater. Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution. She uses the random variable \(X\) to denote the daily mean total cloud cover and believes that \(X \sim \mathrm {~B} ( 8,0.76 )\) Using Magali's model,
    1. find \(\mathrm { P } ( X \geqslant 6 )\)
    2. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7
  3. Explain whether or not your answers to part (b) support the use of Magali's model. There were 28 days that had a daily mean total cloud cover of 8 For these 28 days the daily mean total cloud cover for the following day is shown in the table below.
    Daily mean total cloud cover (oktas)012345678
    Frequency (number of days)001121599
  4. Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
  5. Comment on Magali's model in light of your answer to part (d).
Edexcel Paper 3 2019 June Q5
13 marks Standard +0.3
  1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml
Given that 15\% of bottles contain less than 24.63 ml
  1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
  2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
  3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.
Edexcel Paper 3 2022 June Q1
8 marks Moderate -0.3
  1. \hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) where
$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$
  1. Find the speed of \(P\) at time \(t = 2\) seconds.
  2. Find an expression, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\), for the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\) At time \(t = 4\) seconds, the position vector of \(P\) is ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
  3. Find the position vector of \(P\) at time \(t = 1\) second.
Edexcel Paper 3 2022 June Q2
10 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-04_282_627_246_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) A small block \(B\) of mass 5 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 1. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane. The block \(B\) is modelled as a particle.
The magnitude of the normal reaction of the plane on \(B\) is 68.6 N .
Using the model,
    1. find the magnitude of the frictional force acting on \(B\),
    2. state the direction of the frictional force acting on \(B\). The horizontal force of magnitude \(X\) newtons is now removed and \(B\) moves down the plane. Given that the coefficient of friction between \(B\) and the plane is 0.5
  2. find the acceleration of \(B\) down the plane.
Edexcel Paper 3 2022 June Q3
9 marks Standard +0.3
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )
  1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
    Given that \(\lambda = 2\)
  2. find the length of \(A B\).