Edexcel CP AS (Core Pure AS) 2021 June

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. 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 \(\mathbf { R }\).
3. 1. Evaluate the determinant of \(\mathbf { R }\).
   2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).

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.

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 $$\sum _ { r = 1 } ^ { k } ( 5 r - 2 ) ^ { 2 } = 94 k ^ { 2 }$$

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.

5. \begin{figure}[h] \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.

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.

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. 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 \(\mathrm { f } ( - 2 \mathrm { z } ) = 0\)

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.

9. \begin{figure}[h] \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.