OCR Further Pure Core AS (Further Pure Core AS) 2022 June

1

1. Determine whether the point \(( 19 , - 12,17 )\) lies on the line \(\mathbf { r } = \left( \begin{array} { r } 4
   - 2
   7 \end{array} \right) + \lambda \left( \begin{array} { r } 3
   - 2
   4 \end{array} \right)\). Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { r } 1
   - 2
   2 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 3
   6
   2 \end{array} \right)\).
2. 1. Find, in degrees, the angle between \(\mathbf { a }\) and \(\mathbf { b }\).
   2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).

2 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } a & 1
- 1 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } - 2 & 5
- 1 & 0 \end{array} \right)\) where \(a\) is a constant.

1. Find the following matrices.
   • \(\mathbf { A } + \mathbf { B }\)
   • AB
   • \(\mathbf { A } ^ { 2 }\)
   • 1. Given that the determinant of \(\mathbf { A }\) is 25 find the value of \(a\).
     2. You are given instead that the following system of equations does not have a unique solution.
   $$\begin{array} { r } a x + y = - 2
   - x + 3 y = - 6 \end{array}$$ Determine the value of \(a\).

3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).

4 Prove that \(3 ^ { n } > 10 n\) for all integers \(n \geqslant 4\).

5 In this question you must show detailed reasoning.

1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).

6 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \frac { 1 } { 13 } \left( \begin{array} { r r } 5 & 12
12 & - 5 \end{array} \right)\). You are given that \(\mathbf { A }\) represents the transformation T which is a reflection in a certain straight line. You are also given that this straight line, the mirror line, passes through the origin, \(O\).

1. Explain why there must be a line of invariant points for T . State the geometric significance of this line.
2. By considering the line of invariant points for T , determine the equation of the mirror line. Give your answer in the form \(y = m x + c\). The coordinates of the point \(P\) are \(( 1,5 )\).
3. By considering the image of \(P\) under the transformation T , or otherwise, determine the coordinates of the point on the mirror line which is closest to \(P\).
4. The line with equation \(y = a x + 2\) is an invariant line for T. Determine the value of \(a\).

7 In this question you must show detailed reasoning.
Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined as follows.
\(\mathrm { C } _ { 1 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } + 2 - \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\}\) and \(\mathrm { C } _ { 2 } = \left\{ \mathrm { z } : \arg ( \mathrm { z } - 2 - \sqrt { 3 } - 2 \mathrm { i } ) = \frac { 2 } { 3 } \pi \right\}\)
By considering the representations of \(C _ { 1 }\) and \(C _ { 2 }\) on an Argand diagram, determine the locus \(C _ { 1 } \cap C _ { 2 }\).

8 The line segment \(A B\) is a diameter of a sphere, \(S\). The point \(C\) is any point on the surface of \(S\).

1. Explain why \(\overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0\) for all possible positions of \(C\). You are now given that \(A\) is the point ( \(11,12 , - 14\) ) and \(B\) is the point ( \(9,13,6\) ).
2. Given that the coordinates of \(C\) have the form ( \(2 p , p , 1\) ), where \(p\) is a constant, determine the coordinates of the possible positions of \(C\). \section*{END OF QUESTION PAPER}