OCR Further Pure Core AS (Further Pure Core AS) 2023 June

1 The roots of the equation \(4 x ^ { 4 } - 2 x ^ { 3 } - 3 x + 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). By using a suitable substitution, find a quartic equation whose roots are \(\alpha + 2 , \beta + 2 , \gamma + 2\) and \(\delta + 2\) giving your answer in the form \(a t ^ { 4 } + b t ^ { 3 } + c t ^ { 2 } + d t + e = 0\), where \(a , b , c , d\), and \(e\) are integers.

2 The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the following equations.
\(L _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 5
6
15 \end{array} \right) + \lambda \left( \begin{array} { c } 5
- 2
- 2 \end{array} \right)\)
\(L _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 24
1
- 5 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 4 \end{array} \right)\)

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, giving the position vector of the point of intersection.
2. Find the equation of the line which intersects \(L _ { 1 }\) and \(L _ { 2 }\) and is perpendicular to both. Give your answer in cartesian form.

3 In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval \([ 0,2 \pi )\).
Complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are defined by \(z _ { 1 } = 3 + 4 \mathrm { i }\) and \(z _ { 2 } = - 5 + 12 \mathrm { i }\).

1. Determine \(z _ { 1 } z _ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
2. Express \(z _ { 2 }\) in modulus-argument form.
3. Verify, by direct calculation, that \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).

4 The vector \(\mathbf { p }\), all of whose components are positive, is given by \(\mathbf { p } = \left( \begin{array} { c } a ^ { 2 }
a - 5
26 \end{array} \right)\) where \(a\) is a constant.
You are given that \(\mathbf { p }\) is perpendicular to the vector \(\left( \begin{array} { c } 2
6
- 3 \end{array} \right)\).
Determine the value of \(a\).

5 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 2 } - 3 x + 12 = 0\) are \(\alpha\) and \(\beta\). By considering the symmetric functions of the roots, \(\alpha + \beta\) and \(\alpha \beta\), determine the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).

6 Prove by induction that \(4 \times 8 ^ { n } + 66\) is divisible by 14 for all integers \(n \geqslant 0\).

7 In this question you must show detailed reasoning. Matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3
a + 9 & a & 4
0 & - 13 & a - 1 \end{array} \right)\) where \(a\) is a constant.
Find all possible values of \(a\) for which \(\operatorname { det } \mathbf { A }\) has the same value as it has when \(a = 2\).

8

1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
3. The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
   1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
   2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.

9 Matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b
0 & 1 & 0
b & 0 & a \end{array} \right)\) where \(a\) and \(b\) are constants.

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). The constants \(a\) and \(b\) are given by \(a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )\) and \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
2. By determining exact expressions for \(a b\) and \(a ^ { 2 } - b ^ { 2 }\) and using the result from part (a), show that \(\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1
   0 & 2 & 0
   1 & 0 & \sqrt { 3 } \end{array} \right)\) where \(k\) is a real number whose value is to be determined.
3. Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
4. Describe fully the transformation represented by \(\mathbf { R }\). \section*{END OF QUESTION PAPER}