$\overrightarrow{ b }=\hat{ i }-\lambda \hat{ j }+2 \hat{ k }$ $\Rightarrow(\overrightarrow{ b }-\overrightarrow{ a }) \times((\overrightarrow{ a }+\overrightarrow{ b }) \times(\overrightarrow{ a } \times \overrightarrow{ b }))=8 \hat{ i }-40 \hat{ j }-24 \hat{ k }$
$\Rightarrow((\overrightarrow{ a }-\overrightarrow{ b }) \cdot(\overrightarrow{ a }+\overrightarrow{ b }))(\overrightarrow{ a } \times \overrightarrow{ b })=8 \hat{ i }-40 j -24 \hat{ k }$
$\Rightarrow 8(\overrightarrow{ a } \times \overrightarrow{ b })=8 \hat{ i }-40 \hat{ j }-24 \hat{ k }$
$\text { Now, } \overrightarrow{ a } \times \overrightarrow{ b }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ \lambda & 2 & -3 \\ 1 & -\lambda & 2\end{array}\right|$
$=(4-3 \lambda) \hat{ i }-(2 \lambda+3) \hat{ j }+\left(-\lambda^2-2\right) \hat{ k }$
$\Rightarrow \lambda=1$
$\therefore \overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$
$\overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }$
$\Rightarrow \lambda=1$
$\therefore \overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$
$\overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }$
$\Rightarrow \overrightarrow{ a }+\overrightarrow{ b }=2 \hat{ i }+\hat{ j }-\hat{ k }, \overrightarrow{ a }-\overrightarrow{ b }=3 \hat{ j }-5 \hat{ k }$ $\Rightarrow(\overrightarrow{ a }+\overrightarrow{ b }) \times(\overrightarrow{ a }-\overrightarrow{ b })=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & -1 \\ 0 & 3 & -5\end{array}\right|=2$
$\hat{ i }+10 \hat{ j }+6 \hat{ k }$
$\therefore \text { required answer }=4+100+36=140$
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| $x=x$ | $0$ | $1$ | $2$ | $3$ |
| $p(x)$ | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{2}$ | $\frac{1}{8}$ |