OCR MEI Further Pure Core AS (Further Pure Core AS) Specimen

The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).

1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]

1. Describe fully the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). [2]
2. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). Explaining your reasoning, find the area of the image of the triangle following this transformation. [2]

1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. [2] \includegraphics{figure_3}
2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg(z - 2i) = \frac{\pi}{4}\). [2]

1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$x - 2y - z = 6$$ $$3x + y + 5z = -4$$ $$-4x + 2y - 3z = a$$ [4]
2. Determine whether the intersection of the three planes could be on the \(z\)-axis. [2]

The cubic equation \(x^3 - 4x^2 + px + q = 0\) has roots \(\alpha\), \(\frac{2}{\alpha}\) and \(\alpha + \frac{2}{\alpha}\). Find

• the values of the roots of the equation,
• the value of \(p\).

[7]

1. Show that, when \(n = 5\), \(\sum_{r=n+1}^{2n} r^2 = 330\). [1]
2. Find, in terms of \(n\), a fully factorised expression for \(\sum_{r=n+1}^{2n} r^2\). [4]

The plane \(\Pi\) has equation \(3x - 5y + z = 9\).

1. Show that \(\Pi\) contains
   and
   [4]
2. Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through \((4,1,2)\). [3]

In this question you must show detailed reasoning.

1. Explain why all cubic equations with real coefficients have at least one real root. [2]
2. Points representing the three roots of the equation \(z^3 + 9z^2 + 27z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices. [7]

You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).

1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]

It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).

1. Explain geometrically why this claim is true. [2]
2. Verify algebraically that this claim is true. [3]