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9. [Maximum mark: 8]

A line $L_1$ has vector equation $\mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ where $t \in \mathbb{R}$.
The plane $\Pi_1$ contains the line $L_1$ and passes through the point (2, 1, 5).

(a) Show that the Cartesian equation of the plane $\Pi_1$ is $x + y - z = -2$. [4]

Consider the three planes:
$\Pi_1 : x + y - z = -2$
$\Pi_2 : 2x + by - z = 3$
$\Pi_3 : x - y + 2z = d$
where $b, d \in \mathbb{Q}^+$.
The three planes intersect in a line.

(b) Find the value of $b$ and the value of $d$. [4]