Edexcel CP AS (Core Pure AS) 2023 June

1. $$\left( \begin{array} { l l } x & 9
y & z \end{array} \right) - 3 \left( \begin{array} { l l } z & y
z & y \end{array} \right) = k \mathbf { I }$$ where \(x , y , z\) and \(k\) are constants.
Determine the value of \(x\), the value of \(y\) and the value of \(z\).

1. \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad\) where \(a\) and \(b\) are real constants

Given that \(- 3 + 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(a\) and the value of \(b\).
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Write down the roots of the equation \(\mathrm { f } ( \mathrm { z } + 2 ) = 0\)

3. $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Describe fully the single geometric transformation \(A\) represented by the matrix \(\mathbf { A }\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & 0
   \sqrt { 3 } & 0 & 5 \sqrt { 3 }
   1 & 2 & 0 \end{array} \right)$$ The transformation \(B\) is represented by the matrix \(\mathbf { B }\).
   The transformation \(A\) followed by the transformation \(B\) is the transformation \(C\), which is represented by the matrix \(\mathbf { C }\). To determine matrix \(\mathbf { C }\), a student attempts the following matrix multiplication. $$\left( \begin{array} { c c c } 1 & 0 & 0
   0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
   0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right) \left( \begin{array} { c c c } 1 & 3 & 0
   \sqrt { 3 } & 0 & 5 \sqrt { 3 }
   1 & 2 & 0 \end{array} \right)$$
2. State the error made by the student.
3. Determine the correct matrix \(\mathbf { C }\).

1. (i) (a) Show that

$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$ where \(k\) is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.) Given that

• \(n\) is a positive integer
• \(\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }\) is a real number
  (b) use the answer to part (a) to write down the smallest possible value of \(n\).
  (ii) The complex number \(z = a + b \mathrm { i }\) where \(a\) and \(b\) are real constants.

Given that

• \(\left| z ^ { 10 } \right| = 59049\)
• \(\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }\)
  determine the value of \(a\) and the value of \(b\).

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A large pile of concrete waste is created on a building site.
Figure 1 shows a central vertical cross-section of the concrete waste.
The curve \(C\), shown in Figure 2, has equation $$y + x ^ { 2 } = 2 \quad 0 \leqslant x \leqslant \sqrt { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis and the curve \(C\). The volume of concrete waste is modelled by the volume of revolution formed when \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. The units are metres. The density of the concrete waste is \(900 \mathrm { kgm } ^ { - 3 }\)

1. Use the model to estimate the mass of the concrete waste. Give your answer to 2 significant figures.
2. Give a limitation of the model. The mass of the concrete waste is approximately 5500 kg .
3. Use this information and your answer to part (a) to evaluate the model, giving a reason for your answer.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 2
   2
   0 \end{array} \right) + \lambda \left( \begin{array} { l } 3
   0
   1 \end{array} \right)\) where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) is parallel to \(\left( \begin{array} { r } 1
2
- 3 \end{array} \right)\)

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular. The plane \(\Pi\) contains the line \(l _ { 1 }\) and is perpendicular to \(\left( \begin{array} { r } 1
   2
   - 3 \end{array} \right)\)
2. Determine a Cartesian equation of \(\Pi\)
3. Verify that the point \(A ( 3,1,1 )\) lies on \(\Pi\) Given that
   • the point of intersection of \(\Pi\) and \(l _ { 2 }\) has coordinates \(( 2,3,2 )\)
   • the point \(B ( p , q , r )\) lies on \(l _ { 2 }\)
   • the distance \(A B\) is \(2 \sqrt { 5 }\)
   • \(p , q\) and \(r\) are positive integers
   • determine the coordinates of \(B\).

1. (i) Shade, on an Argand diagram, the set of points for which

$$| z - 3 | \leqslant | z + 6 i |$$ (ii) Determine the exact complex number \(w\) which satisfies both $$\arg ( w - 2 ) = \frac { \pi } { 3 } \quad \text { and } \quad \arg ( w + 1 ) = \frac { \pi } { 6 }$$

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { n } { 3 } \left( a n ^ { 2 } - 1 \right)$$ where \(a\) is a constant to be determined.
(b) Hence determine the sum of the squares of all positive odd three-digit integers.

1. (i)

$$\mathbf { P } = \left( \begin{array} { r r r } k & - 2 & 7
- 3 & - 5 & 2
k & k & 4 \end{array} \right)$$ where \(k\) is a constant Show that \(\mathbf { P }\) is non-singular for all real values of \(k\).
(ii) $$\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1
- 3 & 0 \end{array} \right)$$ The matrix \(\mathbf { Q }\) represents a linear transformation \(T\)
Under \(T\), the point \(A ( a , 2 )\) and the point \(B ( 4 , - a )\), where \(a\) is a constant, are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\) respectively. Given that the distance \(A ^ { \prime } B ^ { \prime }\) is \(\sqrt { 58 }\), determine the possible values of \(a\).

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. The quartic equation $$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$ has roots \(p , q , r\) and \(s\).
   Without solving the equation, determine the quartic equation whose roots are $$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$ Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found.
2. The roots of the cubic equation $$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$ are \(\alpha , 2 \alpha\) and \(\alpha - \beta\)
   Determine
   (a) the values of the roots of the equation,
   (b) the value of \(n\).