OCR MEI Further Pure Core AS (Further Pure Core AS) 2020 November

1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.

2 Fig. 2 shows two complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) represented on an Argand diagram. \begin{figure}[h] \end{figure}

1. On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.
   • \(\mathrm { Z } _ { 1 } { } ^ { * }\)
   • \(z _ { 2 } - z _ { 1 }\)
   • In this question you must show detailed reasoning.
   In the case where \(z _ { 1 } = 1 + 2 \mathrm { i }\) and \(z _ { 2 } = 3 + \mathrm { i }\), find \(\frac { z _ { 2 } - z _ { 1 } } { z _ { 1 } ^ { * } }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.

3 In this question you must show detailed reasoning.
The roots of the equation \(x ^ { 2 } - 2 x + 4 = 0\) are \(\alpha\) and \(\beta\).

1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.

4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0
1 & 0 & 0
0 & 0 & 1 \end{array} \right)\).

1. 1. Calculate \(\operatorname { det } \mathbf { M }\).
   2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
2. Describe fully the transformation associated with \(\mathbf { M }\).

5 You are given that \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 n + 4\).
Prove by induction that \(u _ { n } = n ^ { 2 } + 3 n + 1\) for all positive integers \(n\).

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2
2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1
1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.

1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
   1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
   2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
   3. Hence verify your answer to part (i).

7 In the quartic equation \(2 x ^ { 4 } - 20 x ^ { 3 } + a x ^ { 2 } + b x + 250 = 0\), the coefficients \(a\) and \(b\) are real. One root of the equation is \(2 + \mathrm { i }\). Find the other roots.

8

1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1
   - 1 & 0 \end{array} \right)\).
   1. Find \(\mathbf { M } ^ { 2 }\).
   2. Write down the transformation represented by \(\mathbf { M }\).
   3. Hence state the geometrical significance of the result of part (i).
2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0
   k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.

9 Three planes have equations
\(k x + y - 2 z = 0\)
\(2 x + 3 y - 6 z = - 5\)
\(3 x - 2 y + 5 z = 1\)
where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.

1. \(k = - 1\)
2. \(k = \frac { 2 } { 3 }\)

10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}