Questions — Edexcel (10514 questions)

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Edexcel Paper 1 Specimen Q3
4 marks Standard +0.8
  1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 3 \\ a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$
  1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
  2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)
Edexcel Paper 1 Specimen Q4
5 marks Moderate -0.8
Relative to a fixed origin \(O\),
the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
Given that \(A B C D\) is a parallelogram,
  1. find the position vector of point \(D\). The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
    Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
  2. find the position vector of \(X\).
Edexcel Paper 1 Specimen Q5
10 marks Standard +0.3
  1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant
Given that \(\mathrm { f } ( - 6 ) = 0\)
    1. show that \(a = 4\)
    2. express \(\mathrm { f } ( x )\) as a product of two algebraic factors. Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
  2. show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
  3. hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.
Edexcel Paper 1 Specimen Q6
6 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_780_246_223} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_706_246_1133} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve \(B C A\) used to model this entrance.
The points \(A , O , B\) and \(C\) are assumed to lie in the same vertical plane and the ground \(A O B\) is assumed to be horizontal.
  1. Find an equation for curve \(B C A\). A coach has height 4.1 m and width 2.4 m .
  2. Determine whether or not it is possible for the coach to enter the tunnel.
  3. Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.
Edexcel Paper 1 Specimen Q7
5 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-14_604_1063_251_502} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 1 Specimen Q8
5 marks Standard +0.8
  1. There were 2100 tonnes of wheat harvested on a farm during 2017.
The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.
  1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
    • £5.15 per tonne to harvest the first 2000 tonnes of wheat
    • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
    • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)
Edexcel Paper 1 Specimen Q9
5 marks Moderate -0.5
  1. The curve \(C\) has equation
$$y = 2 x ^ { 3 } + 5$$ The curve \(C\) passes through the point \(P ( 1,7 )\).
Use differentiation from first principles to find the value of the gradient of the tangent to \(C\) at \(P\).
Edexcel Paper 1 Specimen Q10
13 marks Moderate -0.3
  1. The function f is defined by
$$f : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - 1$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Show that $$\mathrm { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \pm 1$$ where \(a\) is an integer to be found. The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
  3. Find the value of \(\mathrm { fg } ( 2 )\).
  4. Find the range of g.
  5. Explain why the function \(g\) does not have an inverse.
Edexcel Paper 1 Specimen Q11
10 marks Standard +0.8
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-22_760_1182_248_443} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f ^ { \prime } ( x ) = k - 4 x - 3 x ^ { 2 }$$ where \(k\) is constant,
  1. show that \(C\) has a point of inflection at \(x = - \frac { 2 } { 3 }\) Given also that the distance \(A B = 4 \sqrt { 2 }\)
  2. find, showing your working, the integer value of \(k\).
Edexcel Paper 1 Specimen Q12
7 marks Standard +0.8
  1. Show that
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$
Edexcel Paper 1 Specimen Q13
11 marks Standard +0.3
13.
  1. Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom's model to
  2. find the depth of water at 00:00 hours on 18th October 2017,
  3. find the maximum depth of water,
  4. find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
    By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
    1. explain why Jolene's model is not correct,
    2. hence find a suitable model for \(H\) in terms of \(x\).
Edexcel Paper 1 Specimen Q14
5 marks Standard +0.8
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-30_659_1232_248_420} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , y = 2 \sin t , \quad 0 < t \leqslant 2 \pi$$ Show that a Cartesian equation of \(C\) can be written in the form $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found.
Edexcel Paper 2 2018 June Q1
6 marks Standard +0.3
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
5 marks Moderate -0.3
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
5 marks Moderate -0.8
  1. "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.
    1. Sketch the graph of \(y = | x | + 3\)
    2. Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).
Edexcel Paper 2 2018 June Q4
7 marks Moderate -0.3
  1. Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\)
  2. 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
6 marks Moderate -0.5
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 marks Standard +0.8
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
9 marks Standard +0.8
  1. Solve, for \(0 \leqslant x < \frac { \pi } { 2 }\), the equation $$4 \sin x = \sec x$$
  2. 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
7 marks Moderate -0.8
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
5 marks Standard +0.8
  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
8 marks Standard +0.3
  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
7 marks Standard +0.3
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
9 marks Standard +0.3
  1. Prove that $$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$
  2. 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
10 marks Challenging +1.2
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)