Pre-U Pre-U 9795/1 Specimen — Question 8 10 marks

Exam BoardPre-U
ModulePre-U 9795/1 (Pre-U Further Mathematics Paper 1)
SessionSpecimen
Marks10
Topic3x3 Matrices
TypeFind inverse then solve system
DifficultyStandard +0.8 This is a multi-part question requiring matrix inversion (computationally demanding for 3×3), determinant calculation with parameter, and analysis of consistency conditions. While the techniques are standard Further Maths content, the computational load and the need to handle parameters across three connected parts elevates this above average difficulty.
Spec4.03o Inverse 3x3 matrix4.03r Solve simultaneous equations: using inverse matrix
8
  1. Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 5 a \\ 3 x + 8 y + a z & = b \end{aligned}$$ has a unique solution.
  2. By use of the inverse matrix of a suitable \(3 \times 3\) matrix, find the unique solution in the case \(a = 1\) and \(b = 2\).
  3. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.
(i) \(\begin{vmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & a \end{vmatrix} = 5a + 8 - 3(2a+3) + 4 = 0\) M1A1
\(\Rightarrow a = 3\), so \(a \neq 3 \Rightarrow\) unique solution A1 [3]
(ii) \(\mathbf{A} = \begin{pmatrix}1&3&4\\2&5&-1\\3&8&1\end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix}-5\\5\\2\end{pmatrix}\)
\(\mathbf{A}^{-1} = \frac{1}{2}\begin{pmatrix}13&29&-23\\-5&-11&9\\1&1&-1\end{pmatrix}\) M1A1
\(\begin{pmatrix}x\\y\\z\end{pmatrix} = \mathbf{A}^{-1}\mathbf{B} = \begin{pmatrix}17\\-6\\-1\end{pmatrix}\) A2 [4]
(Allow final A1 if one or two values correct)
(iii) EITHER: Eliminate one variable from first two equations, giving line of intersection of first two planes as \(\begin{pmatrix}x\\y\\z\end{pmatrix} = \frac{1}{9}\left\{\begin{pmatrix}55\\0\\-25\end{pmatrix} + t\begin{pmatrix}-23\\9\\-1\end{pmatrix}\right\}\) M1A1
Substitute in third equation and obtain \(b = 10\) A1 [3]
OR: \(\begin{bmatrix}1&3&4&-5\\2&5&-1&15\\3&8&3&b\end{bmatrix} \to \begin{bmatrix}1&3&4&-5\\3&8&3&10\\0&0&0&b-10\end{bmatrix}\) M1A1
\(\Rightarrow\) consistent if \(b = 10\) A1 (3)
**(i)** $\begin{vmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & a \end{vmatrix} = 5a + 8 - 3(2a+3) + 4 = 0$ M1A1

$\Rightarrow a = 3$, so $a \neq 3 \Rightarrow$ unique solution A1 **[3]**

**(ii)** $\mathbf{A} = \begin{pmatrix}1&3&4\\2&5&-1\\3&8&1\end{pmatrix}$, $\mathbf{B} = \begin{pmatrix}-5\\5\\2\end{pmatrix}$

$\mathbf{A}^{-1} = \frac{1}{2}\begin{pmatrix}13&29&-23\\-5&-11&9\\1&1&-1\end{pmatrix}$ M1A1

$\begin{pmatrix}x\\y\\z\end{pmatrix} = \mathbf{A}^{-1}\mathbf{B} = \begin{pmatrix}17\\-6\\-1\end{pmatrix}$ A2 **[4]**

(Allow final A1 if one or two values correct)

**(iii)** EITHER: Eliminate one variable from first two equations, giving line of intersection of first two planes as $\begin{pmatrix}x\\y\\z\end{pmatrix} = \frac{1}{9}\left\{\begin{pmatrix}55\\0\\-25\end{pmatrix} + t\begin{pmatrix}-23\\9\\-1\end{pmatrix}\right\}$ M1A1

Substitute in third equation and obtain $b = 10$ A1 **[3]**

OR: $\begin{bmatrix}1&3&4&-5\\2&5&-1&15\\3&8&3&b\end{bmatrix} \to \begin{bmatrix}1&3&4&-5\\3&8&3&10\\0&0&0&b-10\end{bmatrix}$ M1A1

$\Rightarrow$ consistent if $b = 10$ A1 **(3)**
8 (i) Show that if $a \neq 3$ then the system of equations

$$\begin{aligned}
x + 3 y + 4 z & = - 5 \\
2 x + 5 y - z & = 5 a \\
3 x + 8 y + a z & = b
\end{aligned}$$

has a unique solution.\\
(ii) By use of the inverse matrix of a suitable $3 \times 3$ matrix, find the unique solution in the case $a = 1$ and $b = 2$.\\
(iii) Given that $a = 3$, find the value of $b$ for which the equations are consistent.

\hfill \mbox{\textit{Pre-U Pre-U 9795/1  Q8 [10]}}
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