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Edexcel Paper 2 2018 June Q1
1. $$\operatorname { g } ( x ) = \frac { 2 x + 5 } { x - 3 } \quad x \geqslant 5$$
  1. Find \(\mathrm { gg } ( 5 )\).
  2. State the range of g.
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\), stating its domain.
Edexcel Paper 2 2018 June Q2
  1. Relative to a fixed origin \(O\),
    the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
    the point \(B\) has position vector ( \(4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) ),
    and the point \(C\) has position vector ( \(a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\) ), where \(a\) is a constant and \(a < 0 D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).
    1. Find the position vector of \(D\).
    Given \(| \overrightarrow { A C } | = 4\)
  2. find the value of \(a\).
Edexcel Paper 2 2018 June Q3
  1. (a) "If \(m\) and \(n\) are irrational numbers, where \(m \neq n\), then \(m n\) is also irrational."
Disprove this statement by means of a counter example.
(b) (i) Sketch the graph of \(y = | x | + 3\)
(ii) Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).
Edexcel Paper 2 2018 June Q4
  1. (i) Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\)
    (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\)
Edexcel Paper 2 2018 June Q5
  1. The equation \(2 x ^ { 3 } + x ^ { 2 } - 1 = 0\) has exactly one real root.
    1. Show that, for this equation, the Newton-Raphson formula can be written
    $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 3 } + x _ { n } ^ { 2 } + 1 } { 6 x _ { n } ^ { 2 } + 2 x _ { n } }$$ Using the formula given in part (a) with \(x _ { 1 } = 1\)
  2. find the values of \(x _ { 2 }\) and \(x _ { 3 }\)
  3. Explain why, for this question, the Newton-Raphson method cannot be used with \(x _ { 1 } = 0\)
Edexcel Paper 2 2018 June Q6
6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$
    1. Calculate f(2)
    2. Write \(\mathrm { f } ( x )\) as a product of two algebraic factors. Using the answer to (a)(ii),
  2. prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
  3. deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$
Edexcel Paper 2 2018 June Q7
  1. (i) Solve, for \(0 \leqslant x < \frac { \pi } { 2 }\), the equation
$$4 \sin x = \sec x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$5 \sin \theta - 5 \cos \theta = 2$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 2018 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{580fc9b9-d78c-4a86-91fc-22638cb5186d-20_540_1465_294_301} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, \(H\) metres, has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.
  1. Find a quadratic equation linking \(H\) with \(x\) that models this situation. The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.
  2. Use your equation to find the greatest horizontal distance of the bar from \(O\).
  3. Give one limitation of the model.
Edexcel Paper 2 2018 June Q9
  1. Given that \(\theta\) is measured in radians, prove, from first principles, that
$$\frac { \mathrm { d } } { \mathrm {~d} \theta } ( \cos \theta ) = - \sin \theta$$ You may assume the formula for \(\cos ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)
(5)
Edexcel Paper 2 2018 June Q10
  1. A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm .
In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius. Using this model and all the information given,
  1. find an equation linking the radius of the mint and the time.
    (You should define the variables that you use.)
  2. Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.
  3. Suggest a limitation of the model.
Edexcel Paper 2 2018 June Q11
11. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$
  1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \quad x > 3$$
  2. Prove that \(\mathrm { f } ( x )\) is a decreasing function.
Edexcel Paper 2 2018 June Q12
  1. (a) Prove that
$$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.
Edexcel Paper 2 2018 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{580fc9b9-d78c-4a86-91fc-22638cb5186d-38_714_826_251_621} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , x > 0\)
The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
Show that the exact area of \(R\) is \(A e ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
(10)
Edexcel Paper 2 2018 June Q14
  1. A scientist is studying a population of mice on an island.
The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation $$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$
  1. Find the number of mice in the population at the start of the study.
  2. Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\) The rate of growth is a maximum after \(T\) months.
  3. Find, according to the model, the value of \(T\). According to the model, the maximum number of mice on the island is \(P\).
  4. State the value of \(P\).
Edexcel Paper 2 2019 June Q1
  1. Given
$$2 ^ { x } \times 4 ^ { y } = \frac { 1 } { 2 \sqrt { 2 } }$$ express \(y\) as a function of \(x\).
Edexcel Paper 2 2019 June Q2
  1. The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground.
The results are given in the table below with the time in seconds and the speed in \(\mathrm { ms } ^ { - 1 }\).
Time \(( \mathrm { s } )\)0510152025
Speed \(\left( \mathrm { m } \mathrm { s } ^ { - 1 } \right)\)2510182842
Using all of this information,
  1. estimate the length of runway used by the jet to take off. Given that the jet accelerated smoothly in these 25 seconds,
  2. explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.
Edexcel Paper 2 2019 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-06_490_458_248_806} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\), radius 5 cm and angle \(A O B = 40 ^ { \circ }\) The attempt of a student to find the area of the sector is shown below. $$\begin{aligned} \text { Area of sector } & = \frac { 1 } { 2 } r ^ { 2 } \theta
& = \frac { 1 } { 2 } \times 5 ^ { 2 } \times 40
& = 500 \mathrm {~cm} ^ { 2 } \end{aligned}$$
  1. Explain the error made by this student.
  2. Write out a correct solution.
Edexcel Paper 2 2019 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-08_620_679_251_740} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C _ { 1 }\) with parametric equations $$x = 10 \cos t , \quad y = 4 \sqrt { 2 } \sin t , \quad 0 \leqslant t < 2 \pi$$ meets the circle \(C _ { 2 }\) with equation $$x ^ { 2 } + y ^ { 2 } = 66$$ at four distinct points as shown in Figure 2.
Given that one of these points, \(S\), lies in the 4th quadrant, find the Cartesian coordinates of \(S\).
Edexcel Paper 2 2019 June Q5
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
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
  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
    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
    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
    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. \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 \(x _ { n }\)