SPS SPS FM Pure 2026 January — Question 12 3 marks

Exam BoardSPS
ModuleSPS FM Pure (SPS FM Pure)
Year2026
SessionJanuary
Marks3
TopicVectors 3D & Lines
TypePosition vectors and magnitudes
DifficultyStandard +0.3 This is a straightforward vectors question requiring finding points at a given distance from the origin in a direction parallel to a given line. Students need to find the unit vector in the direction of lā‚‚, scale it by āˆš2, and consider both positive and negative directions. This is a standard textbook exercise with clear steps and no novel insight required, making it slightly easier than average.
Spec1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.03q Inverse transformations4.03r Solve simultaneous equations: using inverse matrix
12 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 3
4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).\\[0pt] Q7.\\ The simultaneous equations $$\begin{aligned} & 2 x - y = 1
& 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).
  1. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists.
    Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
  2. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
    (A) when \(k = 5\) and \(b = 21\),
    (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
    (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\).
    [0pt] Q8.
    You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(\mathbf { M }\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
    [0pt]
  3. By investigating the invariant points of the reflection, find the equation of the mirror line. [3]
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { r r } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
    [0pt] Q9.
    1. Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
      (4 marks)
    2. A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 i | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
      (3 marks)
    3. The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).
      (5 marks)
      [0pt]
      [0pt]
      [0pt]
      [0pt]
12
\end{array} \right) + \mu \left( \begin{array} { r } 
5 \\
- 3 \\
4
\end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$

The point $A$ does not lie on $l _ { 2 }$. Given that the vector $\overrightarrow { O A }$ is parallel to the line $l _ { 2 }$ and $| \overrightarrow { O A } | = \sqrt { 2 }$ units, calculate the possible position vectors of the point $A$.\\[0pt]

Q7.\\
The simultaneous equations

$$\begin{aligned}
& 2 x - y = 1 \\
& 3 x + k y = b
\end{aligned}$$

are represented by the matrix equation $\mathbf { M } \binom { x } { y } = \binom { 1 } { b }$.\\
\begin{enumerate}[label=(\roman*)]
\item 

State the value of $k$ for which $\mathbf { M } ^ { - 1 }$ does not exist and find $\mathbf { M } ^ { - 1 }$ in terms of $k$ when $\mathbf { M } ^ { - 1 }$ exists.\\
Use $\mathbf { M } ^ { - 1 }$ to solve the simultaneous equations when $k = 5$ and $b = 21$.
\item 

The two equations can be interpreted as representing two lines in the $x - y$ plane. Describe the relationship between these two lines\\
(A) when $k = 5$ and $b = 21$,\\
(B) when $k = - \frac { 3 } { 2 }$ and $b = 1$,\\
(C) when $k = - \frac { 3 } { 2 }$ and $b = \frac { 3 } { 2 }$.\\[0pt]

Q8.\\
You are given the matrix $\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)$.
\end{enumerate}\\
\begin{enumerate}[label=(\roman*)]
\item Calculate $\mathbf { M } ^ { 2 }$.

You are now given that the matrix $\mathbf { M }$ represents a reflection in a line through the origin.
\item Explain how your answer to part (i) relates to this information.\\[0pt]
\item By investigating the invariant points of the reflection, find the equation of the mirror line. [3]
\item Describe fully the transformation represented by the matrix $\mathbf { P } = \left( \begin{array} { r r } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)$.
\item A composite transformation is formed by the transformation represented by $\mathbf { P }$ followed by the transformation represented by $\mathbf { M }$. Find the single matrix that represents this composite transformation.
\item The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?\\[0pt]

Q9.\\
(a) Two points, $A$ and $B$, on an Argand diagram are represented by the complex numbers $2 + 3 \mathrm { i }$ and $- 4 - 5 \mathrm { i }$ respectively. Given that the points $A$ and $B$ are at the ends of a diameter of a circle $C _ { 1 }$, express the equation of $C _ { 1 }$ in the form $\left| z - z _ { 0 } \right| = k$.\\
(4 marks)\\
(b) A second circle, $C _ { 2 }$, is represented on the Argand diagram by the equation $| z - 5 + 4 i | = 4$. Sketch on one Argand diagram both $C _ { 1 }$ and $C _ { 2 }$.\\
(3 marks)\\
(c) The points representing the complex numbers $z _ { 1 }$ and $z _ { 2 }$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively and are such that $\left| z _ { 1 } - z _ { 2 } \right|$ has its maximum value. Find this maximum value, giving your answer in the form $a + b \sqrt { 5 }$.\\
(5 marks)\\[0pt]
\\[0pt]
\\[0pt]
\\[0pt]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q12 [3]}}
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