Questions — WJEC Further Unit 1 (55 questions)

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WJEC Further Unit 1 Specimen Q4
7 marks Standard +0.8
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]
WJEC Further Unit 1 Specimen Q5
9 marks Standard +0.8
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]
WJEC Further Unit 1 Specimen Q6
9 marks Standard +0.8
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]
WJEC Further Unit 1 Specimen Q7
9 marks Standard +0.3
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]
WJEC Further Unit 1 Specimen Q8
12 marks Standard +0.3
The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).
    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]