OCR FP1 AS (Further Pure 1 AS) 2018 March

1

1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.

2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the values of the following expressions.
   (a) \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
   (b) \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }

3
3
- 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1
3
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1
a
1 \end{array} \right) + \mu \left( \begin{array} { c } 2
2
- 3 \end{array} \right) \end{aligned}$$

1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the value of \(a\).

4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).

5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0
0 & a ^ { 2 } & 0
0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0
- b & 0.6 & 0
0 & 0 & 1 \end{array} \right)\).

1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
2. Hence find the possible values of \(a\).
3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
5. Hence find the possible values of \(b\).

6 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { l l } 1 & 2
1 & a \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c } 2 & 1
- 1 & b \end{array} \right)\).

1. Find the matrix \(\mathbf { A B }\).
2. State the conditions on \(a\) and \(b\) for \(\mathbf { A B }\) to be a singular matrix.
   \(P Q R S\) is a quadrilateral and the vertices \(P , Q , R\) and \(S\) are in clockwise order. A transformation, T , is represented by the matrix \(\mathbf { A B }\).
3. State the effect on both the area and also the orientation of the image of \(P Q R S\) under T in each of the following cases.
   (a) \(\quad a = 1\) and \(b = 1\)
   (b) \(\quad a = 2\) and \(b = 3\)

7 In this question you must show detailed reasoning.

1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).

8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by

• \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1
  1
  1 \end{array} \right)\)
• \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3
  1
  2 \end{array} \right)\).
  1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1
     1
     1 \end{array} \right)\). for all integers \(n \geqslant 1\).
  2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).

\section*{END OF QUESTION PAPER}