SPS SPS FM Pure (SPS FM Pure) 2025 February

1. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows:

$$\mathbf { A } = \left( \begin{array} { l } 1

3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices.
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2. The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\). Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
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3. Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
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4. The cubic equation $$2 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 12 = 0$$ 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 \(p w ^ { 3 } + q w ^ { 2 } + r w + s = 0\), where \(p , q , r\) and \(s\) are integers to be found.
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5. In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) 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 $$| z - a | = b \quad a \in \mathbb { C } , b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 \mathrm { i } | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
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6. $$\mathrm { f } ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram,
2. find the value of \(p\) and the value of \(q\).
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7. Line \(l _ { 1 }\) has Cartesian equation $$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$

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.
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2. Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
3. 1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
4. (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.
5. (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that $$\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }$$ where \(p\) and \(q\) are constants to be found.
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8. Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0
4 & - 1 \end{array} \right)\).
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9. $$\mathrm { f } ( z ) = z ^ { 3 } - 8 z ^ { 2 } + p z - 24$$ where \(p\) is a real constant.
Given that the equation \(\mathrm { f } ( z ) = 0\) has distinct roots $$\alpha , \beta \text { and } \left( \alpha + \frac { 12 } { \alpha } - \beta \right)$$

1. solve completely the equation \(\mathrm { f } ( z ) = 0\)
2. Hence find the value of \(p\).
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