Questions — Edexcel (9671 questions)

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Edexcel CP1 2024 June Q1
9 marks Standard +0.3
1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where \(a\) and \(b\) are real constants.
Given that \(2 + 5 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. determine the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  2. Show all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
Edexcel CP1 2024 June Q2
8 marks Moderate -0.5
  1. The roots of the equation
$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are \(\alpha , \beta\) and \(\gamma\)
Without solving the equation,
  1. write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
  2. Use the answers to part (a) to determine the value of
    1. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }\)
    2. \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
    3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
Edexcel CP1 2024 June Q3
8 marks Challenging +1.2
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc3e8e46-c60b-4263-9652-d7c2a322cfae-10_563_561_395_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,
  1. determine the value of \(a\).
  2. Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).
Edexcel CP1 2024 June Q4
10 marks Standard +0.8
  1. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.
    1. Show that
    $$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ where \(n\) is a positive integer.
  2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
  3. Hence, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)
Edexcel CP1 2024 June Q5
9 marks Standard +0.3
  1. A raindrop falls from rest from a cloud. The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, of the raindrop, \(t\) seconds after the raindrop starts to fall, is modelled by the differential equation
$$( t + 2 ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 v = k ( t + 2 ) - 3 \quad t \geqslant 0$$ where \(k\) is a positive constant.
  1. Solve the differential equation to show that $$v = \frac { k } { 4 } ( t + 2 ) - 1 + \frac { 4 ( 2 - k ) } { ( t + 2 ) ^ { 3 } }$$ Given that \(v = 4\) when \(t = 2\)
  2. determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.
  3. Comment on the validity of the model for very large values of \(t\)
Edexcel CP1 2024 June Q6
6 marks Standard +0.3
  1. Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
Edexcel CP1 2024 June Q7
10 marks Standard +0.3
  1. The line \(l _ { 1 }\) has equation
$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)
  1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
  2. Hence determine a Cartesian equation of \(\Pi\)
  3. Hence determine the value of \(p\) Given that
    • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
    • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
    • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)
Edexcel CP1 2024 June Q8
15 marks Challenging +1.2
  1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.
At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
  3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
  4. determine the value of \(T\), giving your answer to 2 decimal places.
  5. Suggest a limitation of the model.
Edexcel CP1 Specimen Q1
5 marks Standard +0.8
  1. Prove that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$ where \(a\) and \(b\) are constants to be found.
Edexcel CP1 Specimen Q2
6 marks Standard +0.3
  1. Prove by induction that for all positive integers \(n\),
$$f ( n ) = 2 ^ { 3 n + 1 } + 3 \left( 5 ^ { 2 n + 1 } \right)$$ is divisible by 17
Edexcel CP1 Specimen Q3
9 marks Standard +0.3
3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.
Edexcel CP1 Specimen Q4
9 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-08_492_1063_214_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 4 + \cos 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 9 } { 2 }\)
The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.
Edexcel CP1 Specimen Q5
10 marks Standard +0.8
  1. A pond initially contains 1000 litres of unpolluted water.
The pond is leaking at a constant rate of 20 litres per day.
It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water. It is assumed that the pollutant instantly dissolves throughout the pond upon entry.
Given that there are \(x\) grams of the pollutant in the pond after \(t\) days,
  1. show that the situation can be modelled by the differential equation, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 50 - \frac { 4 x } { 200 + t }$$
  2. Hence find the number of grams of pollutant in the pond after 8 days.
  3. Explain how the model could be refined.
Edexcel CP1 Specimen Q6
9 marks Standard +0.3
6. $$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$
  1. Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
  2. Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is $$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
  3. Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of $$\mathrm { f } ( x ) + \ln k$$ where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).
Edexcel CP1 Specimen Q7
8 marks
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-14_259_327_214_868} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the image of a gold pendant which has height 2 cm . The pendant is modelled by a solid of revolution of a curve \(C\) about the \(y\)-axis. The curve \(C\) has parametric equations $$x = \cos \theta + \frac { 1 } { 2 } \sin 2 \theta , \quad y = - ( 1 + \sin \theta ) \quad 0 \leqslant \theta \leqslant 2 \pi$$
  1. Show that a Cartesian equation of the curve \(C\) is $$x ^ { 2 } = - \left( y ^ { 4 } + 2 y ^ { 3 } \right)$$
  2. Hence, using the model, find, in \(\mathrm { cm } ^ { 3 }\), the volume of the pendant.
Edexcel CP1 Specimen Q8
7 marks Challenging +1.8
  1. The line \(l _ { 1 }\) has equation \(\frac { x - 2 } { 4 } = \frac { y - 4 } { - 2 } = \frac { z + 6 } { 1 }\)
The plane \(\Pi\) has equation \(x - 2 y + z = 6\)
The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi\).
Find a vector equation of the line \(l _ { 2 }\)
Edexcel CP1 Specimen Q9
12 marks Standard +0.3
  1. A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line.
The vertical displacement, \(x\) metres, of the top of the capsule below its initial position at time \(t\) seconds is modelled by the differential equation, $$m \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 200 \cos t , \quad t \geqslant 0$$ where \(m\) is the mass of the capsule including its passengers, in thousands of kilograms.
The maximum permissible weight for the capsule, including its passengers, is 30000 N .
Taking the value of \(g\) to be \(10 \mathrm {~ms} ^ { - 2 }\) and assuming the capsule is at its maximum permissible weight,
    1. explain why the value of \(m\) is 3
    2. show that a particular solution to the differential equation is $$x = 40 \sin t - 20 \cos t$$
    3. hence find the general solution of the differential equation.
  2. Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released.
Edexcel CP2 2019 June Q1
10 marks Challenging +1.2
  1. (a) Prove that
$$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$ stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation $$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$
Edexcel CP2 2019 June Q2
8 marks Standard +0.3
  1. The roots of the equation
$$x ^ { 3 } - 2 x ^ { 2 } + 4 x - 5 = 0$$ are \(p , q\) and \(r\).
Without solving the equation, find the value of
  1. \(\frac { 2 } { p } + \frac { 2 } { q } + \frac { 2 } { r }\)
  2. \(( p - 4 ) ( q - 4 ) ( r - 4 )\)
  3. \(p ^ { 3 } + q ^ { 3 } + r ^ { 3 }\)
Edexcel CP2 2019 June Q3
6 marks
3. $$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$
  1. Using a substitution, that should be stated clearly, show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
  2. Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).
Edexcel CP2 2019 June Q4
8 marks Challenging +1.2
  1. The infinite series C and S are defined by
$$\begin{aligned} & \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots \\ & \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots \end{aligned}$$ Given that the series C and S are both convergent,
  1. show that $$\mathrm { C } + \mathrm { iS } = \frac { 2 \mathrm { e } ^ { \mathrm { i } \theta } } { 2 - \mathrm { e } ^ { 4 \mathrm { i } \theta } }$$
  2. Hence show that $$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$
Edexcel CP2 2019 June Q5
12 marks Standard +0.3
  1. An engineer is investigating the motion of a sprung diving board at a swimming pool.
Let \(E\) be the position of the end of the diving board when it is at rest in its equilibrium position and when there is no diver standing on the diving board.
A diver jumps from the diving board.
The vertical displacement, \(h \mathrm {~cm}\), of the end of the diving board above \(E\) is modelled by the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} h } { \mathrm {~d} t } + 37 h = 0$$ where \(t\) seconds is the time after the diver jumps.
  1. Find a general solution of the differential equation. When \(t = 0\), the end of the diving board is 20 cm below \(E\) and is moving upwards with a speed of \(55 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
  2. Find, according to the model, the maximum vertical displacement of the end of the diving board above \(E\).
  3. Comment on the suitability of the model for large values of \(t\).
Edexcel CP2 2019 June Q6
9 marks Standard +0.8
  1. In an Argand diagram, the points \(A , B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2 \mathrm { i }\).
    1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The points \(D , E\) and \(F\) are the midpoints of the sides of triangle \(A B C\).
  2. Find the exact area of triangle \(D E F\).
Edexcel CP2 2019 June Q7
11 marks Standard +0.8
7. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 3 & k & 4 \\ 3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p \\ & 3 x - 6 y + 4 z = 1 \\ & 3 x + 2 y - z = 0 \end{aligned}$$
    1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1 \\ & 3 x - 5 y + 4 z = q \\ & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.
Edexcel CP2 2019 June Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e63eca0e-e2e9-4660-855f-0696c680a8f9-24_359_552_274_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e63eca0e-e2e9-4660-855f-0696c680a8f9-24_323_387_283_1238} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the central vertical cross section \(A B C D\) of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain. Using these measurements, the curve \(B D\) is modelled by the equation $$y = \ln ( 3.6 x - k ) \quad 1 \leqslant x \leqslant 1.18$$ as shown in Figure 2.
  1. Find the value of \(k\).
  2. Find the depth of the paddling pool according to this model. The pool is being filled with water from a tap.
  3. Find, in terms of \(h\), the volume of water in the pool when the pool is filled to a depth of \(h \mathrm {~m}\). Given that the pool is being filled at a constant rate of 15 litres every minute,
  4. find, in \(\mathrm { cmh } ^ { - 1 }\), the rate at which the water level is rising in the pool when the depth of the water is 0.2 m .