Questions — Edexcel (9685 questions)

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Edexcel Paper 2 Specimen Q14
9 marks Standard +0.3
  1. A company decides to manufacture a soft drinks can with a capacity of 500 ml .
The company models the can in the shape of a right circular cylinder with radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). In the model they assume that the can is made from a metal of negligible thickness.
  1. Prove that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the can is given by $$S = 2 \pi r ^ { 2 } + \frac { 1000 } { r }$$ Given that \(r\) can vary,
  2. find the dimensions of a can that has minimum surface area.
  3. With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.
Edexcel Paper 2 Specimen Q15
10 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-22_796_974_244_548} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation $$y = 5 x ^ { \frac { 3 } { 2 } } - 9 x + 11 , x \geqslant 0$$ The point \(P\) with coordinates \(( 4,15 )\) lies on \(C\).
The line \(l\) is the tangent to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis. Show that the area of \(R\) is 24 , making your method clear.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q16
12 marks Standard +0.8
  1. (a) Express \(\frac { 1 } { P ( 11 - 2 P ) }\) in partial fractions.
A population of meerkats is being studied.
The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began,
(b) determine the time taken, in years, for this population of meerkats to double,
(c) show that $$P = \frac { A } { B + C \mathrm { e } ^ { - \frac { 1 } { 2 } t } }$$ where \(A , B\) and \(C\) are integers to be found.
Edexcel Paper 2 Specimen Q1
4 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-02_364_369_374_849} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the major sector, shown shaded in Figure 1, is \(135 \mathrm {~cm} ^ { 2 }\). The reflex angle \(A O B\) is 4.8 radians. Find the exact length, in cm, of the minor arc \(A B\), giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are integers to be found.
(4)
Edexcel Paper 2 Specimen Q2
5 marks Moderate -0.8
  1. (a) Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that
$$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a).
Adele's working is shown below. Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\) (b) (i) Identify the mistake made by Adele in her working.
(ii) Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) (2)
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. (a) Sketch the graph with equation
$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) 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
  1. 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,
  2. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  3. 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. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) 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. (i) Kayden claims that
$$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
(ii) 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.