7 The plane $\Pi _ { 1 }$ has equation $r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )$.\\
(a) Obtain an equation of $\Pi _ { 1 }$ in the form $\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }$.\\

(b) The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4$.

Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.\\

The line $l$ passes through the point $A$ with position vector $a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }$ and is parallel to $( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }$, where $a$ and $b$ are positive constants.\\
(c) Given that the perpendicular distance from $A$ to $\Pi _ { 1 }$ is $\sqrt { 2 }$, find the value of $a$.\\

(d) Given that the obtuse angle between $l$ and $\Pi _ { 1 }$ is $\frac { 3 } { 4 } \pi$, find the exact value of $b$.\\

If you use the following page to complete the answer to any question, the question number must be clearly shown.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q7}}