Edexcel CP AS (Core Pure AS) 2020 June

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.

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.

3. \begin{figure}[h] \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 \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)

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 \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
2. 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.

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

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a
a - 4 & b \end{array} \right)$$ 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 \(\mathbf { R }\).

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 \(d\).

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

$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7

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.

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 \(r\).