OCR FP1 (Further Pure Mathematics 1) 2016 June

1 Find \(\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )\), giving your answer in a fully factorised form.

2 The complex number \(z\) has modulus \(2 \sqrt { 3 }\) and argument \(- \frac { 1 } { 3 } \pi\). Giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers, and showing clearly how you obtain them, find

1. \(z\),
2. \(\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }\).

3 The quadratic equation \(k x ^ { 2 } + x + k = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)\) in terms of \(k\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r } 6
4
- 1 \end{array} \right)\). Find

1. \(5 \mathbf { A } - 3 \mathbf { B }\),
2. BC,
3. CA .

5 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 5 \text { and } u _ { n + 1 } = 3 u _ { n } + 2 \text { for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 2 \times 3 ^ { n } - 1\).

6 In an Argand diagram the points \(A\) and \(B\) represent the complex numbers \(5 + 4 \mathrm { i }\) and \(1 + 2 \mathrm { i }\) respectively.

1. Given that \(A\) and \(B\) are the ends of a diameter of a circle \(C\), find the equation of \(C\) in complex number form. The perpendicular bisector of \(A B\) is denoted by \(l\).
2. Sketch \(C\) and \(l\) on a single Argand diagram.
3. Find the complex numbers represented by the points of intersection of \(C\) and \(l\).

7 The matrix \(\left( \begin{array} { l l } 1 & 3
0 & 1 \end{array} \right)\) represents a transformation P .

1. Describe fully the transformation P . The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1
   - 1 & 0 \end{array} \right)\).
2. Given that \(\mathbf { M }\) represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .

1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.
3. Find \(\sum _ { r = n } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2
   0 & a & 5
   1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
2. Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1
   3 y + 5 z = 5
   x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
3. Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
4. Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
5. By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.