OCR MEI Further Pure Core AS (Further Pure Core AS) 2019 June

1 In this question you must show detailed reasoning.
Find \(\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\), giving your answer correct to 4 decimal places.

2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).

3 In this question you must show detailed reasoning.
\(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1
- 1 & 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1
   0 & 1 \end{array} \right)\), find \(\mathbf { B }\).

4

1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3
   - 1 & 1 & 2
   - 2 & 1 & 2 \end{array} \right)\).
2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19
   - x + y + 2 z & = 4
   - 2 x + y + 2 z & = k \end{aligned}$$
3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).

5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).

6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k
1 & \lambda - k \end{array} \right)\), and \(\lambda\)
and \(k\) are real constants. and \(k\) are real constants.

1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\).
   2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
   Find the possible values of \(\lambda\).

7

1. Sketch on a single Argand diagram
   1. the set of points for which \(| z - 1 - 3 i | = 3\),
   2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).

8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.

1. Show that \(a = - 2\) and \(b = 2\).
2. Find the other roots of this equation.