OCR MEI Further Pure Core AS (Further Pure Core AS) 2018 June

1 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1

3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices. 2 Find, to the nearest degree, the angle between the vectors \(\left( \begin{array} { r } 1
0
- 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 2
3
- 3 \end{array} \right)\). 3 Find real numbers \(a\) and \(b\) such that \(( a - 3 i ) ( 5 - i ) = b - 17 i\).

4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.

5 A transformation of the \(x - y\) plane is represented by the matrix \(\left( \begin{array} { r r } \cos \theta & 2 \sin \theta
2 \sin \theta & - \cos \theta \end{array} \right)\), where \(\theta\) is a positive acute angle.

1. Write down the image of the point \(( 2,3 )\) under this transformation.
2. You are given that this image is the point ( \(a , 0\) ). Find the value of \(a\).

6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0
4 & - 1 \end{array} \right)\).

7

1. Express \(\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }\) as a single fraction.
2. Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.

8 Prove by induction that \(\left( \begin{array} { l l } 1 & 1
0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1
0 & 2 ^ { n } \end{array} \right)\) for all positive integers \(n\).

9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h] \end{figure}

1. Find, in modulus-argument form, the complex number represented by the point P .
2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   • \(3 + 5 \mathrm { i }\)
   • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\)
     lie within this region.

10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0
2 x - y - z & = 0
x + y & = a \end{aligned}$$ where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   • when \(a = 0\);
   • when \(a \neq 0\).
   • Chris claims that the position vectors \(- \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , 2 \mathbf { i } - \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + \mathbf { j }\) lie in a plane. Determine whether or not Chris is correct.