OCR Further Pure Core 1 (Further Pure Core 1) 2021 June

1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\). Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).

3 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\). You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
3. For each of the values of \(a\) found in part (b) determine whether the equations have
   A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t\) where \(x\) is the displacement of the particle below the equilibrium position at time \(t\).
   When \(t = 0\) the particle is stationary and its displacement is 2 .
   1. Find the particular solution of the differential equation.
   2. Write down an approximate equation for the displacement when \(t\) is large.