SPS SPS FM Pure (SPS FM Pure) 2025 February

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = (1 \quad 3).$$ Calculate all possible products formed from two of these three matrices. [4]

The complex number \(z\) satisfies the equation \(z^2 - 4iz + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

Prove by mathematical induction that \(\sum_{r=1}^{n} (r \times r!) = (n+1)! - 1\) for all positive integers \(n\). [6]

The cubic equation $$2x^3 + 6x^2 - 3x + 12 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the cubic equation whose roots are \((\alpha + 3)\), \((\beta + 3)\) and \((\gamma + 3)\), giving your answer in the form \(pw^3 + qw^2 + rw + s = 0\), where \(p\), \(q\), \(r\) and \(s\) are integers to be found. [5]

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \quad b \in \mathbb{R}$$ [3]

The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\).

1. Find the complex numbers \(z_1\) and \(z_2\). [6]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{21}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]

Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$

1. Write the equation of line \(l_1\) in the form $$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$ where \(\lambda\) is a parameter and \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be found. [2 marks]

1. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
   1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
   2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
   3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]

$$f(z) = z^3 - 8z^2 + pz - 24$$ where \(p\) is a real constant. Given that the equation \(f(z) = 0\) has distinct roots $$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$

1. solve completely the equation \(f(z) = 0\) [6]
2. Hence find the value of \(p\). [2]