WJEC Further Unit 1 (Further Unit 1) Specimen

Use mathematical induction to prove that \(4^n + 2\) is divisible by 6 for all positive integers \(n\). [7]

Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).

1. Give your answer in Cartesian form [7]
2. Give your answer in modulus-argument form. [4]

Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series $$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$ Express your answer as a product of linear factors. [6]

The roots of the equation $$x^3 - 4x^2 + 14x - 20 = 0$$ are denoted by \(\alpha\), \(\beta\), \(\gamma\).

1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = -12.$$ Explain why this result shows that exactly one of the roots of the above cubic equation is real. [3]
2. Given that one of the roots is \(1 + 3i\), find the other two roots. Explain your method for each root. [4]

The complex number \(z\) is represented by the point \(P(x, y)\) in an Argand diagram and $$|z - 3| = 2|z + i|.$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre. [9]

The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \((x, y)\) is transformed to the point \((x + 1, y - 2)\), followed by an anticlockwise rotation through \(90°\) about the origin.

1. Find the \(3 \times 3\) matrix representing \(T\). [6]
2. Show that \(T\) has no fixed points. [3]

The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P(x, y)\) and \(Q(u,v)\) in Argand diagrams and $$w = z(1 + z)$$

1. Show that $$v = y(1 + 2x)$$ and find an expression for \(u\) in terms of \(x\) and \(y\). [4]
2. The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = au^2 + bu\), where \(a\) and \(b\) are constants whose values are to be determined. [5]

The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).

1. 1. Find the vector \(\overrightarrow{AB}\).
   2. Write down the vector equation of the line \(L\). [3]
2. The plane \(\Pi\) has equation \(x + 3y - 2z = 5\).
   1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
   2. Find the acute angle between \(L\) and \(\Pi\). [9]