\begin{enumerate}
  \item Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are acting on an object of mass 3 kg such that
\end{enumerate}

$$\begin{aligned}
& \mathbf { F } _ { 1 } = ( \mathbf { i } + 8 c \mathbf { j } + 11 c \mathbf { k } ) \mathrm { N } , \\
& \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } - c \mathbf { j } - 12 \mathbf { k } ) \mathrm { N } , \\
& \mathbf { F } _ { 3 } = ( ( 15 c + 1 ) \mathbf { i } + 2 c \mathbf { j } - 5 c \mathbf { k } ) \mathrm { N } ,
\end{aligned}$$

where $c$ is a constant. The acceleration of the object is parallel to the vector $( \mathbf { i } + \mathbf { j } )$.\\
(a) Find the value of the constant $c$ and hence show that the acceleration of the object is $( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$.\\

(b) When $t = 0$ seconds, the object has position vector $\mathbf { r } _ { 0 } \mathrm {~m}$ and is moving with velocity $( - 17 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. When $t = 4$ seconds, the object has position vector $( - 13 \mathbf { i } + 84 \mathbf { j } ) \mathrm { m }$. Find the vector $\mathbf { r } _ { 0 }$.\\

\hfill \mbox{\textit{WJEC Unit 4 2024 Q8 [7]}}