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Edexcel CP AS 2020 June Q1
6 marks Standard +0.8
  1. A system of three equations is defined by
$$\begin{aligned} k x + 3 y - z & = 3 \\ 3 x - y + z & = - k \\ - 16 x - k y - k z & = k \end{aligned}$$ where \(k\) is a positive constant.
Given that there is no unique solution to all three equations,
  1. show that \(k = 2\) Using \(k = 2\)
  2. determine whether the three equations are consistent, justifying your answer.
  3. Interpret the answer to part (b) geometrically.
Edexcel CP AS 2020 June Q2
8 marks Moderate -0.8
  1. Given that
$$\begin{aligned} z _ { 1 } & = 2 + 3 \\ \left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 } \\ \arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers,
  1. write \(z _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) Give the exact value of \(r\) and give the value of \(\theta\) in radians to 4 significant figures.
  2. Find \(z _ { 2 }\) giving your answer in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are integers.
Edexcel CP AS 2020 June Q3
5 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09bd14c0-c368-4ae1-bee0-cc8bf82abecc-06_582_588_255_758} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle with radius \(r\) and centre at the origin.
The region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the part of the circle for which \(y > 0\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to create a sphere with volume \(V\) Use integration to show that \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
Edexcel CP AS 2020 June Q4
13 marks Standard +0.3
  1. All units in this question are in metres.
A lawn is modelled as a plane that contains the points \(L ( - 2 , - 3 , - 1 ) , M ( 6 , - 2,0 )\) and \(N ( 2,0,0 )\), relative to a fixed origin \(O\).
  1. Determine a vector equation of the plane that models the lawn, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
    1. Show that, according to the model, the lawn is perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 2 \\ - 10 \end{array} \right)\)
    2. Hence determine a Cartesian equation of the plane that models the lawn. There are two posts set in the lawn.
      There is a washing line between the two posts.
      The washing line is modelled as a straight line through points at the top of each post with coordinates \(P ( - 10,8,2 )\) and \(Q ( 6,4,3 )\).
  3. Determine a vector equation of the line that models the washing line.
  4. State a limitation of one of the models. The point \(R ( 2,5,2.75 )\) lies on the washing line.
  5. Determine, according to the model, the shortest distance from the point \(R\) to the lawn, giving your answer to the nearest cm. Given that the shortest distance from the point \(R\) to the lawn is actually 1.5 m ,
  6. use your answer to part (e) to evaluate the model, explaining your reasoning.
Edexcel CP AS 2020 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09bd14c0-c368-4ae1-bee0-cc8bf82abecc-12_351_655_246_705} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A block has length \(( r + 2 ) \mathrm { cm }\), width \(( r + 1 ) \mathrm { cm }\) and height \(r \mathrm {~cm}\), as shown in Figure 2.
In a set of \(n\) such blocks, the first block has a height of 1 cm , the second block has a height of 2 cm , the third block has a height of 3 cm and so on.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that the total volume, \(V\), of all \(n\) blocks in the set is given by $$V = \frac { n } { 4 } ( n + 1 ) ( n + 2 ) ( n + 3 ) \quad n \geqslant 1$$ Given that the total volume of all \(n\) blocks is $$\left( n ^ { 4 } + 6 n ^ { 3 } - 11710 \right) \mathrm { cm } ^ { 3 }$$
  2. determine how many blocks make up the set.
Edexcel CP AS 2020 June Q6
16 marks Standard +0.3
$$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.
Given that the matrix \(\mathbf { A }\) is self-inverse,
  1. determine the value of \(b\) and the possible values for \(a\). The matrix \(\mathbf { A }\) represents a linear transformation \(M\).
    Using the smaller value of \(a\) from part (a),
  2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line.
    (ii) $$\mathbf { P } = \left( \begin{array} { c c } p & 2 p \\ - 1 & 3 p \end{array} \right)$$ where \(p\) is a positive constant.
    The matrix \(\mathbf { P }\) represents a linear transformation \(U\).
    The triangle \(T\) has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of \(T\) under the linear transformation \(U\) is 15
  3. Determine the value of \(p\). The transformation \(V\) consists of a stretch scale factor 3 parallel to the \(x\)-axis with the \(y\)-axis invariant followed by a stretch scale factor - 2 parallel to the \(y\)-axis with the \(x\)-axis invariant. The transformation \(V\) is represented by the matrix \(\mathbf { Q }\).
  4. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is the transformation \(W\), which is represented by the matrix \(\mathbf { R }\), (c) find the matrix \(\mathbf { R }\).
Edexcel CP AS 2020 June Q7
6 marks Challenging +1.2
7. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 } , z _ { 3 }\) and \(z _ { 4 }\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).
Edexcel CP AS 2020 June Q8
6 marks Standard +0.3
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7
Edexcel CP AS 2020 June Q9
6 marks Standard +0.3
  1. The cubic equation
$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha , \beta\), and \(\gamma\).
Without solving the cubic equation,
  1. determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
  2. find a cubic equation that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\), giving your answer in the form \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
Edexcel CP AS 2020 June Q10
7 marks Challenging +1.2
  1. Given that there are two distinct complex numbers \(z\) that satisfy
$$\{ z : | z - 3 - 5 i | = 2 r \} \cap \quad z : \arg ( z - 2 ) = \frac { 3 \pi } { 4 }$$ determine the exact range of values for the real constant \(r\).
Edexcel CP AS 2021 June Q1
7 marks Easy -1.2
1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$
    1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
    2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
  2. Determine \(\mathbf { R }\).
    1. Evaluate the determinant of \(\mathbf { R }\).
    2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).
Edexcel CP AS 2021 June Q2
5 marks Standard +0.3
  1. The cubic equation
$$9 x ^ { 3 } - 5 x ^ { 2 } + 4 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(3 \alpha - 2\) ), ( \(3 \beta - 2\) ) and ( \(3 \gamma - 2\) ), giving your answer in the form \(a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be determined.
Edexcel CP AS 2021 June Q3
9 marks Moderate -0.3
  1. (a) Use the standard results for summations to show that for all positive integers \(n\)
$$\sum _ { r = 1 } ^ { n } ( 5 r - 2 ) ^ { 2 } = \frac { 1 } { 6 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence determine the value of \(k\) for which $$\sum _ { r = 1 } ^ { k } ( 5 r - 2 ) ^ { 2 } = 94 k ^ { 2 }$$
Edexcel CP AS 2021 June Q4
7 marks Standard +0.3
4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4 \\ k & 2 & - 2 \\ 4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10 \\ 2 & - 20 & 24 \\ - 3 & 2 & - 1 \end{array} \right)$$ where \(k\) is a constant.
  1. Determine, in simplest form in terms of \(k\), the matrix \(\mathbf { M N }\).
  2. Given that \(k = 5\)
    1. write down \(\mathbf { M N }\)
    2. hence write down \(\mathbf { M } ^ { - 1 }\)
  3. Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2 \\ & 5 x + 2 y - 2 z = 3 \\ & 4 x + y - 2 z = - 1 \end{aligned}$$
  4. Interpret the answer to part (c) geometrically.
Edexcel CP AS 2021 June Q5
10 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-12_584_830_246_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows an Argand diagram.
The set \(P\), of points that lie within the shaded region including its boundaries, is defined by $$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$ where \(a\), \(b\), \(c\) and \(d\) are integers.
  1. Write down the values of \(a , b , c\) and \(d\). The set \(Q\) is defined by $$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
  2. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form.
Edexcel CP AS 2021 June Q6
11 marks Standard +0.3
  1. A mining company has identified a mineral layer below ground.
The mining company wishes to drill down to reach the mineral layer and models the situation as follows. With respect to a fixed origin \(O\),
  • the ground is modelled as a horizontal plane with equation \(z = 0\)
  • the mineral layer is modelled as part of the plane containing the points \(A ( 10,5 , - 50 ) , B ( 15,30 , - 45 )\) and \(C ( - 5,20 , - 60 )\), where the units are in metres
    1. Determine an equation for the plane containing \(A , B\) and \(C\), giving your answer in the form r.n \(= d\)
    2. Determine, according to the model, the acute angle between the ground and the plane containing the mineral layer. Give your answer to the nearest degree.
The mining company plans to drill vertically downwards from the point \(( 5,12,0 )\) on the ground to reach the mineral layer.
  • Using the model, determine, in metres to 1 decimal place, the distance the mining company will need to drill in order to reach the mineral layer.
  • State a limitation of the assumption that the mineral layer can be modelled as a plane.
    •
    Edexcel CP AS 2021 June Q7
    9 marks Standard +0.3
    7. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + p z ^ { 2 } + q z + r$$ where \(p , q\) and \(r\) are real constants.
    The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\) where \(\alpha = 3\) and \(\beta = 2 + \mathrm { i }\) Given that \(\gamma\) is a complex root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
      1. write down the root \(\gamma\),
      2. explain why \(\delta\) must be real.
    2. Determine the value of \(\delta\).
    3. Hence determine the values of \(p , q\) and \(r\).
    4. Write down the roots of the equation \(\mathrm { f } ( - 2 \mathrm { z } ) = 0\)
    Edexcel CP AS 2021 June Q8
    9 marks Standard +0.8
    1. (a) Prove by induction that, for all positive integers \(n\),
    $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n + 2 )$$ (b) Hence, show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.
    Edexcel CP AS 2021 June Q9
    13 marks Standard +0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d7dcb9f-510c-42c7-bcac-6d6ab3ed6468-28_639_517_255_774} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the vertical cross-section, \(A O B C D E\), through the centre of a wax candle.
    In a model, the candle is formed by rotating the region bounded by the \(y\)-axis, the line \(O B\), the curve \(B C\), and the curve \(C D\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(B\) has coordinates \(( 3,0 )\) and the point \(C\) has coordinates \(( 5,15 )\).
    The units are in centimetres.
    The curve \(B C\) is represented by the equation $$y = \frac { \sqrt { 225 x ^ { 2 } - 2025 } } { a } \quad 3 \leqslant x < 5$$ where \(a\) is a constant.
    1. Determine the value of \(a\) according to this model. The curve \(C D\) is represented by the equation $$y = 16 - 0.04 x ^ { 2 } \quad 0 \leqslant x < 5$$
    2. Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle.
    3. State a limitation of the model. When the candle was manufactured, \(700 \mathrm {~cm} ^ { 3 }\) of wax were required.
    4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.
    Edexcel CP AS 2022 June Q1
    7 marks Moderate -0.8
    1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1 \\ 7 & 2 \\ - 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2 \\ - 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1 \\ 4 & 3 & 8 \\ - 6 & 11 & 2 \end{array} \right)$$ Given that \(\mathbf { I }\) is the \(3 \times 3\) identity matrix,
      1. show that there is an integer \(k\) for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of \(k\)
      2. explain why there can be no constant \(m\) such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
      1. Show how the matrix \(\mathbf { C }\) can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14 \\ 4 x + 3 y + 8 z & = 3 \\ - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
      2. Hence use your calculator to solve these equations.
    Edexcel CP AS 2022 June Q2
    10 marks Standard +0.3
    1. (a) Express the complex number \(w = 4 \sqrt { 3 } - 4 \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) (b) Show, on a single Argand diagram,
      1. the point representing \(w\)
      2. the locus of points defined by \(\arg ( z + 10 i ) = \frac { \pi } { 3 }\) (c) Hence determine the minimum distance of \(w\) from the locus \(\arg ( z + 10 i ) = \frac { \pi } { 3 }\)
    Edexcel CP AS 2022 June Q3
    8 marks Moderate -0.3
    1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about the } \\ y \text {-axis is represented by the matrix } \end{array} \right]\) \(\left( \begin{array} { c c c } \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ - \sin \theta & 0 & \cos \theta \end{array} \right)\)
    The point \(P\) has coordinates (8, 3, 2)
    The point \(Q\) is the image of \(P\) under the transformation reflection in the plane \(y = 0\)
    1. Write down the coordinates of \(Q\) The point \(R\) is the image of \(P\) under the transformation rotation through \(120 ^ { \circ }\) anticlockwise about the \(y\)-axis, with respect to the right-hand rule.
    2. Determine the exact coordinates of \(R\)
    3. Hence find \(| \overrightarrow { P R } |\) giving your answer as a simplified surd.
    4. Show that \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) are perpendicular.
    5. Hence determine the exact area of triangle \(P Q R\), giving your answer as a surd in simplest form.
    Edexcel CP AS 2022 June Q4
    9 marks Standard +0.8
    1. The roots of the quartic equation
    $$3 x ^ { 4 } + 5 x ^ { 3 } - 7 x + 6 = 0$$ are \(\alpha , \beta , \gamma\) and \(\delta\) Making your method clear and without solving the equation, determine the exact value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\)
    2. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma } + \frac { 2 } { \delta }\)
    3. \(( 3 - \alpha ) ( 3 - \beta ) ( 3 - \gamma ) ( 3 - \delta )\)
    Edexcel CP AS 2022 June Q5
    12 marks Standard +0.3
    1. (a) Use the standard summation formulae to show that, for \(n \in \mathbb { N }\),
    $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = n \left( n ^ { 2 } - A n - B \right)$$ where \(A\) and \(B\) are integers to be determined.
    (b) Explain why, for \(k \in \mathbb { N }\), $$\sum _ { r = 1 } ^ { 3 k } r \tan ( 60 r ) ^ { \circ } = - k \sqrt { 3 }$$ Using the results from part (a) and part (b) and showing all your working,
    (c) determine any value of \(n\) that satisfies $$\sum _ { r = 5 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = 15 \left[ \sum _ { r = 6 } ^ { 3 n } r \tan ( 60 r ) ^ { \circ } \right] ^ { 2 }$$
    Edexcel CP AS 2022 June Q6
    13 marks Standard +0.3
    1. The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and
    • i and j are unit vectors directed across the width and length of the court respectively
    • \(\quad \mathbf { k }\) is a unit vector directed vertically upwards
    • units are metres
    After being hit, a tennis ball, modelled as a particle, moves along the path with equation $$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$ where \(\lambda\) is a scalar parameter with \(\lambda \geqslant 0\) Assuming that the tennis ball continues on this path until it hits the ground,
    1. find the value of \(\lambda\) at the point where the ball hits the ground. The direction in which the tennis ball is moving at a general point on its path is given by $$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
    2. Write down the direction in which the tennis ball is moving as it hits the ground.
    3. Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place. The net of the tennis court lies in the plane \(\mathbf { r } . \mathbf { j } = 0\)
    4. Find the position of the tennis ball at the point where it is in the same plane as the net. The maximum height above the court of the top of the net is 0.9 m .
      Modelling the top of the net as a horizontal straight line,
    5. state whether the tennis ball will pass over the net according to the model, giving a reason for your answer. With reference to the model,
    6. decide whether the tennis ball will actually pass over the net, giving a reason for your answer.