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Edexcel Paper 2 2019 June Q5
3 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-10_890_958_260_550} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = \sqrt { x }\) The point \(P ( x , y )\) lies on the curve.
The rectangle, shown shaded on Figure 3, has height \(y\) and width \(\delta x\).
Calculate $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 9 } \sqrt { x } \delta x$$
Edexcel Paper 2 2019 June Q6
10 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-12_728_1086_246_493} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = \begin{cases} ( x - 2 ) ^ { 2 } + 1 & x \leqslant 2 \\ 4 x - 7 & x > 2 \end{cases}$$
  1. Find the value of \(\operatorname { gg } ( 0 )\).
  2. Find all values of \(x\) for which $$\mathrm { g } ( x ) > 28$$ The function h is defined by $$\mathrm { h } ( x ) = ( x - 2 ) ^ { 2 } + 1 \quad x \leqslant 2$$
  3. Explain why h has an inverse but g does not.
  4. Solve the equation $$\mathrm { h } ^ { - 1 } ( x ) = - \frac { 1 } { 2 }$$
Edexcel Paper 2 2019 June Q7
7 marks Moderate -0.8
  1. A small factory makes bars of soap.
On any day, the total cost to the factory, \(\pounds y\), of making \(x\) bars of soap is modelled to be the sum of two separate elements:
  • a fixed cost
  • a cost that is proportional to the number of bars of soap that are made that day
    1. Write down a general equation linking \(y\) with \(x\), for this model.
The bars of soap are sold for \(\pounds 2\) each.
On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of \(\pounds 80\) Using the above information,
  • show that \(y = 0.84 x + 428\)
  • With reference to the model, interpret the significance of the value 0.84 in the equation. Assuming that each bar of soap is sold on the day it is made,
  • find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.
    •
    Edexcel Paper 2 2019 June Q8
    6 marks Standard +0.8
    1. (i) Find the value of
    $$\sum _ { r = 4 } ^ { \infty } 20 \times \left( \frac { 1 } { 2 } \right) ^ { r }$$ (3)
    (ii) Show that $$\sum _ { n = 1 } ^ { 48 } \log _ { 5 } \left( \frac { n + 2 } { n + 1 } \right) = 2$$
    Edexcel Paper 2 2019 June Q9
    9 marks Moderate -0.3
    1. A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions.
    The engineer measures and records the braking distance, \(d\) metres, when the brakes are applied from a speed of \(V \mathrm { kmh } ^ { - 1 }\). Graphs of \(d\) against \(V\) and \(\log _ { 10 } d\) against \(\log _ { 10 } V\) were plotted.
    The results are shown below together with a data point from each graph. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-24_631_659_699_285} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-24_684_684_644_1101} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure}
    1. Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula $$d = k V ^ { n } \quad \text { where } k \text { and } n \text { are constants }$$ with \(k \approx 0.017\) Using the information given in Figure 5, with \(k = 0.017\)
    2. find a complete equation for the model giving the value of \(n\) to 3 significant figures. Sean is driving this car at \(60 \mathrm { kmh } ^ { - 1 }\) in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.
    3. Use your formula to find out if Sean will be able to stop before reaching the puddle.
    Edexcel Paper 2 2019 June Q10
    6 marks Standard +0.3
    10. Figure 7 Figure 7 shows a sketch of triangle \(O A B\).
    The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O A }\).
    The point \(M\) is the midpoint of \(A B\).
    The straight line through \(C\) and \(M\) cuts \(O B\) at the point \(N\).
    Given \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O B } = \mathbf { b }\)
    1. Find \(\overrightarrow { C M }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\)
    2. Show that \(\overrightarrow { O N } = \left( 2 - \frac { 3 } { 2 } \lambda \right) \mathbf { a } + \frac { 1 } { 2 } \lambda \mathbf { b }\), where \(\lambda\) is a scalar constant.
    3. Hence prove that \(O N : N B = 2 : 1\)
    Edexcel Paper 2 2019 June Q11
    11 marks Standard +0.8
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-32_589_771_248_648} \captionsetup{labelformat=empty} \caption{Figure 8}
    \end{figure} Figure 8 shows a sketch of the curve \(C\) with equation \(y = x ^ { x } , x > 0\)
    1. Find, by firstly taking logarithms, the \(x\) coordinate of the turning point of \(C\).
      (Solutions based entirely on graphical or numerical methods are not acceptable.) The point \(P ( \alpha , 2 )\) lies on \(C\).
    2. Show that \(1.5 < \alpha < 1.6\) A possible iteration formula that could be used in an attempt to find \(\alpha\) is $$x _ { n + 1 } = 2 x _ { n } ^ { 1 - x _ { n } }$$ Using this formula with \(x _ { 1 } = 1.5\)
    3. find \(x _ { 4 }\) to 3 decimal places,
    4. describe the long-term behaviour of \(x _ { n }\)
    Edexcel Paper 2 2019 June Q12
    7 marks Challenging +1.3
    1. (a) Prove
    $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } \equiv 2 \cot 2 \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } = 4$$ giving any solutions to one decimal place.
    Edexcel Paper 2 2019 June Q13
    10 marks Standard +0.3
    13. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-40_501_401_242_831} \captionsetup{labelformat=empty} \caption{Figure 9}
    \end{figure} [A sphere of radius \(r\) has volume \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(4 \pi r ^ { 2 }\) ]
    A manufacturer produces a storage tank.
    The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness.
    The cylinder has radius \(r\) metres and height \(h\) metres and the hemisphere has radius \(r\) metres.
    The volume of the tank is \(6 \mathrm {~m} ^ { 3 }\).
    1. Show that, according to the model, the surface area of the tank, in \(\mathrm { m } ^ { 2 }\), is given by $$\frac { 12 } { r } + \frac { 5 } { 3 } \pi r ^ { 2 }$$ The manufacturer needs to minimise the surface area of the tank.
    2. Use calculus to find the radius of the tank for which the surface area is a minimum.
      (4)
    3. Calculate the minimum surface area of the tank, giving your answer to the nearest integer.
    Edexcel Paper 2 2019 June Q14
    15 marks Challenging +1.2
    1. (a) Use the substitution \(u = 4 - \sqrt { h }\) to show that
    $$\int \frac { \mathrm { d } h } { 4 - \sqrt { h } } = - 8 \ln | 4 - \sqrt { h } | - 2 \sqrt { h } + k$$ where \(k\) is a constant A team of scientists is studying a species of slow growing tree.
    The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { t ^ { 0.25 } ( 4 - \sqrt { h } ) } { 20 }$$ where \(h\) is the height in metres and \(t\) is the time, measured in years, after the tree is planted.
    (b) Find, according to the model, the range in heights of trees in this species. One of these trees is one metre high when it is first planted.
    According to the model,
    (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.
    Edexcel Paper 2 2022 June Q1
    4 marks Moderate -0.8
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-02_671_759_383_653} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 3 - 2 x |\) Solve $$| 3 - 2 x | = 7 + x$$
    Edexcel Paper 2 2022 June Q2
    4 marks Easy -1.2
    1. (a) Sketch the curve with equation
    $$y = 4 ^ { x }$$ stating any points of intersection with the coordinate axes.
    (b) Solve $$4 ^ { x } = 100$$ giving your answer to 2 decimal places.
    Edexcel Paper 2 2022 June Q3
    4 marks Moderate -0.8
    1. A sequence of terms \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
    $$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = 8 - a _ { n } \end{aligned}$$
      1. Show that this sequence is periodic.
      2. State the order of this periodic sequence.
    2. Find the value of $$\sum _ { n = 1 } ^ { 85 } a _ { n }$$
    Edexcel Paper 2 2022 June Q4
    3 marks Moderate -0.8
    1. Given that
    $$y = 2 x ^ { 2 }$$ use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x$$
    Edexcel Paper 2 2022 June Q5
    6 marks Standard +0.3
    1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } 2 x\) The values of \(y\) are given to 2 decimal places as appropriate.
    \(x\)34.567.59
    \(y\)1.6322.262.462.63
    1. Using the trapezium rule with all the values of \(y\) in the table, find an estimate for $$\int _ { 3 } ^ { 9 } \log _ { 3 } 2 x \mathrm {~d} x$$ Using your answer to part (a) and making your method clear, estimate
      1. \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( 2 x ) ^ { 10 } \mathrm {~d} x\)
      2. \(\int _ { 3 } ^ { 9 } \log _ { 3 } 18 x \mathrm {~d} x\)
    Edexcel Paper 2 2022 June Q6
    7 marks Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-12_634_741_251_662} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 8 \sin \left( \frac { 1 } { 2 } x \right) - 3 x + 9 \quad x > 0$$ and \(x\) is measured in radians.
    The point \(P\), shown in Figure 2, is a local maximum point on the curve.
    Using calculus and the sketch in Figure 2,
    1. find the \(x\) coordinate of \(P\), giving your answer to 3 significant figures. The curve crosses the \(x\)-axis at \(x = \alpha\), as shown in Figure 2 .
      Given that, to 3 decimal places, \(f ( 4 ) = 4.274\) and \(f ( 5 ) = - 1.212\)
    2. explain why \(\alpha\) must lie in the interval \([ 4,5 ]\)
    3. Taking \(x _ { 0 } = 5\) as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show your method and give your answer to 3 significant figures.
    Edexcel Paper 2 2022 June Q7
    5 marks Standard +0.3
    1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
    $$\sqrt { 4 - 9 x }$$ writing each term in simplest form. A student uses this expansion with \(x = \frac { 1 } { 9 }\) to find an approximation for \(\sqrt { 3 }\) Using the answer to part (a) and without doing any calculations,
    (b) state whether this approximation will be an overestimate or an underestimate of \(\sqrt { 3 }\) giving a brief reason for your answer.
    Edexcel Paper 2 2022 June Q8
    6 marks Standard +0.3
    1. In this question you must show all stages of your working.
    Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-18_633_730_386_669} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of part of a curve with equation $$y = \frac { ( x - 2 ) ( x - 4 ) } { 4 \sqrt { x } } \quad x > 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
    Find the exact area of \(R\), writing your answer in the form \(a \sqrt { 2 } + b\), where \(a\) and \(b\) are constants to be found.
    Edexcel Paper 2 2022 June Q9
    5 marks Moderate -0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-22_419_569_301_226} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-22_522_927_239_917} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 4 shows a sketch of a Ferris wheel.
    The height above the ground, \(H \mathrm {~m}\), of a passenger on the Ferris wheel, \(t\) seconds after the wheel starts turning, is modelled by the equation $$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right|$$ where \(A\), \(b\) and \(\alpha\) are constants.
    Figure 5 shows a sketch of the graph of \(H\) against \(t\), for one revolution of the wheel.
    Given that
    • the maximum height of the passenger above the ground is 50 m
    • the passenger is 1 m above the ground when the wheel starts turning
    • the wheel takes 720 seconds to complete one revolution
      1. find a complete equation for the model, giving the exact value of \(A\), the exact value of \(b\) and the value of \(\alpha\) to 3 significant figures.
      2. Explain why an equation of the form
    $$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right| + d$$ where \(d\) is a positive constant, would be a more appropriate model.
    Edexcel Paper 2 2022 June Q10
    8 marks Standard +0.3
    1. The function f is defined by
    $$f ( x ) = \frac { 8 x + 5 } { 2 x + 3 } \quad x > - \frac { 3 } { 2 }$$
    1. Find \(\mathrm { f } ^ { - 1 } \left( \frac { 3 } { 2 } \right)\)
    2. Show that $$\mathrm { f } ( x ) = A + \frac { B } { 2 x + 3 }$$ where \(A\) and \(B\) are constants to be found. The function \(g\) is defined by $$g ( x ) = 16 - x ^ { 2 } \quad 0 \leqslant x \leqslant 4$$
    3. State the range of \(\mathrm { g } ^ { - 1 }\)
    4. Find the range of \(\mathrm { fg } ^ { - 1 }\)
    Edexcel Paper 2 2022 June Q11
    4 marks Easy -1.2
    1. Prove, using algebra, that
    $$n \left( n ^ { 2 } + 5 \right)$$ is even for all \(n \in \mathbb { N }\).
    Edexcel Paper 2 2022 June Q12
    6 marks Standard +0.8
    1. The function f is defined by
    $$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.
    1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point, (b) find the range of possible values of \(k\).
    Edexcel Paper 2 2022 June Q13
    6 marks Standard +0.3
    1. Relative to a fixed origin \(O\)
    • the point \(A\) has position vector \(4 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
    • the point \(B\) has position vector \(4 \mathbf { j } + 6 \mathbf { k }\)
    • the point \(C\) has position vector \(- 16 \mathbf { i } + p \mathbf { j } + 10 \mathbf { k }\) where \(p\) is a constant.
      Given that \(A , B\) and \(C\) lie on a straight line,
      1. find the value of \(p\).
    The line segment \(O B\) is extended to a point \(D\) so that \(\overrightarrow { C D }\) is parallel to \(\overrightarrow { O A }\) (b) Find \(| \overrightarrow { O D } |\), writing your answer as a fully simplified surd.
    Edexcel Paper 2 2022 June Q14
    10 marks Standard +0.3
    1. (a) Express \(\frac { 3 } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.
    When chemical \(A\) and chemical \(B\) are mixed, oxygen is produced.
    A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. The total volume of oxygen produced, \(V \mathrm {~m} ^ { 3 } , t\) hours after the chemicals were mixed, is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 3 V } { ( 2 t - 1 ) ( t + 1 ) } \quad V \geqslant 0 \quad t \geqslant k$$ where \(k\) is a constant.
    Given that exactly 2 hours after the chemicals were mixed, a total volume of \(3 \mathrm {~m} ^ { 3 }\) of oxygen had been produced,
    (b) solve the differential equation to show that $$V = \frac { 3 ( 2 t - 1 ) } { ( t + 1 ) }$$ The scientist noticed that
    • there was a time delay between the chemicals being mixed and oxygen being produced
    • there was a limit to the total volume of oxygen produced
    Deduce from the model
    (c) (i) the time delay giving your answer in minutes,
    (ii) the limit giving your answer in \(\mathrm { m } ^ { 3 }\)
    Edexcel Paper 2 2022 June Q15
    10 marks Challenging +1.8
    1. In this question you must show all stages of your working.
    \section*{Solutions relying on calculator technology are not acceptable.} Given that the first three terms of a geometric series are $$12 \cos \theta \quad 5 + 2 \sin \theta \quad \text { and } \quad 6 \tan \theta$$
    1. show that $$4 \sin ^ { 2 } \theta - 52 \sin \theta + 25 = 0$$ Given that \(\theta\) is an obtuse angle measured in radians,
    2. solve the equation in part (a) to find the exact value of \(\theta\)
    3. show that the sum to infinity of the series can be expressed in the form $$k ( 1 - \sqrt { 3 } )$$ where \(k\) is a constant to be found.