3. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors.]

Three forces, $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$, are given by

$$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$

where $a$ and $b$ are constants.\\
The forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a particle $P$ of mass 4 kg .\\
Given that $P$ rests in equilibrium on a smooth horizontal surface under the action of these three forces,
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\item find the size of the angle between the direction of $\mathbf { F } _ { 3 }$ and the direction of $- \mathbf { j }$.

The force $\mathbf { F } _ { 3 }$ is now removed and replaced by the force $\mathbf { F } _ { 4 }$ given by $\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )$ N, where $\lambda$ is a positive constant.

When the three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 4 }$ act on $P$, the acceleration of $P$ has magnitude $3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
\item Find the value of $\lambda$.\\

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\hfill \mbox{\textit{Edexcel M1 2021 Q3 [9]}}