OCR FP1 AS (Further Pure 1 AS) 2017 Specimen

**In this question you must show detailed reasoning.** The equation \(x^2 + 2x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x^2 + px + q = 0\) has roots \(\alpha^2\) and \(\beta^2\). Find the values of \(p\) and \(q\). [3]

**In this question you must show detailed reasoning.** Given that \(z_1 = 3 + 2i\) and \(z_2 = -1 - i\), find the following, giving each in the form \(a + bi\).

1. \(z_1^* z_2\) [2]
2. \(\frac{z_1 + 2z_2}{z_2}\) [2]

1. You are given two matrices, **A** and **B**, where $$\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} -1 & 2 \\ 2 & -1 \end{pmatrix}.$$ Show that \(\mathbf{AB} = m\mathbf{I}\), where \(m\) is a constant to be determined. [2]
2. You are given two matrices, **C** and **D**, where $$\mathbf{C} = \begin{pmatrix} 2 & 1 & 5 \\ 1 & 1 & 3 \\ -1 & 2 & 2 \end{pmatrix} \text{ and } \mathbf{D} = \begin{pmatrix} -4 & 8 & -2 \\ -5 & 9 & -1 \\ 3 & -5 & 1 \end{pmatrix}.$$ Show that \(\mathbf{C}^{-1} = k\mathbf{D}\) where \(k\) is a constant to be determined. [2]
3. The matrices **E** and **F** are given by \(\mathbf{E} = \begin{pmatrix} k & k^2 \\ 3 & 0 \end{pmatrix}\) and \(\mathbf{F} = \begin{pmatrix} 2 \\ k \end{pmatrix}\) where \(k\) is a constant. Determine any matrix **F** for which \(\mathbf{EF} = \begin{pmatrix} -2k \\ 6 \end{pmatrix}\). [5]

Draw the region of the Argand diagram for which \(|z - 3 - 4i| \leq 5\) and \(|z| \leq |z - 2|\). [4]

The matrix **M** is given by \(\mathbf{M} = \begin{pmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{pmatrix}\).

1. The diagram in the Printed Answer Booklet shows the unit square \(OABC\). The image of the unit square under the transformation represented by **M** is \(OA'B'C'\). Draw and clearly label \(OA'B'C'\). [3]
2. Find the equation of the line of invariant points of this transformation. [3]
3. 1. Find the determinant of **M**. [1]
   2. Describe briefly how this value relates to the transformation represented by **M**. [2]

At the beginning of the year John had a total of £2000 in three different accounts. He has twice as much money in the current account as in the savings account. • The current account has an interest rate of 2.5% per annum. • The savings account has an interest rate of 3.7% per annum. • The supersaver account has an interest rate of 4.9% per annum. John has predicted that he will earn a total interest of £92 by the end of the year.

1. Model this situation as a matrix equation. [2]
2. Find the amount that John had in each account at the beginning of the year. [2]
3. In fact, the interest John will receive is £92 **to the nearest pound**. Explain how this affects the calculations. [2]

**In this question you must show detailed reasoning.** It is given that \(f(z) = z^3 - 13z^2 + 65z - 125\). The points representing the three roots of the equation \(f(z) = 0\) are plotted on an Argand diagram. Show that these points lie on the circle \(|z| = k\), where \(k\) is a real number to be determined. [9]

Prove that \(n! > 2^n\) for \(n \geq 4\). [5]

1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]
2. Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\). Find the point of intersection of \(l_1\) and \(l_2\). [4]
3. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]