Edexcel CP AS (Core Pure AS) 2018 June

1. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & - 3
4 & - 2 & 1
3 & 5 & - 2 \end{array} \right)$$

1. Find \(\mathbf { M } ^ { - 1 }\) giving each element in exact form.
2. Solve the simultaneous equations $$\begin{array} { r } 2 x + y - 3 z = - 4
   4 x - 2 y + z = 9
   3 x + 5 y - 2 z = 5 \end{array}$$
3. Interpret the answer to part (b) geometrically.

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1. The cubic equation

$$z ^ { 3 } - 3 z ^ { 2 } + z + 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(2 \alpha + 1\) ), ( \(2 \beta + 1\) ) and ( \(2 \gamma + 1\) ), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.

1. (a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 1 - \mathrm { i } | \leqslant 3 \} \cap \quad z \in \mathbb { C } : \frac { \pi } { 4 } \leqslant \arg ( z - 2 ) \leqslant \frac { 3 \pi } { 4 }$$ The complex number \(w\) satisfies $$| w - 1 - \mathrm { i } | = 3 \text { and } \arg ( w - 2 ) = \frac { \pi } { 4 }$$ (b) Find, in simplest form, the exact value of \(| w | ^ { 2 }\)

1. Part of the mains water system for a housing estate consists of water pipes buried beneath the ground surface. The water pipes are modelled as straight line segments. One water pipe, \(W\), is buried beneath a particular road. With respect to a fixed origin \(O\), the road surface is modelled as a plane with equation \(3 x - 5 y - 18 z = 7\), and \(W\) passes through the points \(A ( - 1 , - 1 , - 3 )\) and \(B ( 1,2 , - 3 )\). The units are in metres.
   1. Use the model to calculate the acute angle between \(W\) and the road surface.
   A point \(C ( - 1 , - 2,0 )\) lies on the road. A section of water pipe needs to be connected to \(W\) from \(C\).
2. Using the model, find, to the nearest cm, the shortest length of pipe needed to connect \(C\) to \(W\).

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

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { A }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { B }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { B }\). Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { C }\), (c) find the matrix \(\mathbf { C }\).
3. Show that there is a real number \(k\) for which the point \(( 1 , k )\) is invariant under \(T\).

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

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { 1 } { 2 } n \left[ 6 n ^ { 2 } - 3 n - 1 \right]$$ for all positive integers \(n\).
(b) Hence find any values of \(n\) for which $$\sum _ { r = 5 } ^ { n } ( 3 r - 2 ) ^ { 2 } + 103 \sum _ { r = 1 } ^ { 28 } r \cos \left( \frac { r \pi } { 2 } \right) = 3 n ^ { 3 }$$

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\)
When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } 5 & - 8
2 & - 3 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 4 n + 1 & - 8 n
2 n & 1 - 4 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(A B C D E F G H A\) of the bottle with the measurements taken by the student. The horizontal cross-section between \(C F\) and \(D E\) is a circle of diameter 8 cm and the horizontal cross-section between \(B G\) and \(A H\) is a circle of diameter 2 cm . The student thinks that the curve \(G F\) could be modelled as a curve with equation $$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model.
2. Use the model to find the volume of water that the bottle can contain.
3. State a limitation of the model. The label on the bottle states that the bottle holds approximately \(750 \mathrm {~cm} ^ { 3 }\) of water.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.