Let λ ∈ ℤ, → a = λ ^ i + ^ j - ^ k and → b = 3 ^ i - ^ j + 2 ^ k ....

Given that

\overset{\to}{a} = \lambda\overset{^}{i} + \overset{^}{j} - \overset{^}{k}

\overset{\to}{b} = 3\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}

Also \left(\overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c}\right) \times \overset{\to}{c} = \overset{\to}{0}$</p><p>$⇒ k\left(\overset{\to}{a} + \overset{\to}{b}\right) = \overset{\to}{c}$</p><p>$\overset{\to}{a} \cdot \overset{\to}{c} = - 17$and$\overset{\to}{b} \cdot \overset{\to}{c} = - 20$</p><p>Now, Take$\overset{\to}{a} \cdot \overset{\to}{c} = - 17$</p><p>$⇒ k\left(\lambda\overset{^}{i} + \overset{^}{j} - \overset{^}{k}\right) \cdot \left(\lambda\overset{^}{i} + \overset{^}{j} - \overset{^}{k} + 3\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) = - 17$</p><p>$⇒ k\left({\lambda}^{2} + 3\lambda + 0 - 1\right) = - 17$</p><p>$⇒ k\left({\lambda}^{2} + 3\lambda - 1\right) = - 17 . . . . ( 1 )$</p><p>Similarly, on taking$\overset{\to}{b} \cdot \overset{\to}{c} = - 20$, we get, </p><p>$k\left(3\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) \cdot \left(\lambda\overset{^}{i} + \overset{^}{j} - \overset{^}{k} + 3\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) = - 20$</p><p>$⇒ k\left(3\lambda + 9 + 2\right) = - 20$</p><p>$⇒ k\left(3\lambda + 11\right) = - 20 . . . ( 2 )$</p><p>Now on solving equation \left(1\right) \text{\text{ and } } \left(2\right)$ we get,

$⇒ 20{\lambda}^{2} + 9\lambda - 207 = 0$

$⇒ \lambda = 3, \dfrac{- 69}{20}$

For $\lambda = 3, k = - 1$

$⇒ \overset{\to}{c} = - 1 \left(\overset{\to}{a} + \overset{\to}{b}\right)</p><p>$⇒ - \left(\left(\lambda + 3\right)\overset{^}{i} + \overset{^}{k}\right) = - 6\overset{^}{i} - \overset{^}{k}

⇒ \overset{\to}{c} \times \left(\lambda\overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right) = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ - 6 & 0 & - 1 \\ 3 & 1 & 1\end{cases}\right|= \overset{^}{i} - 3\overset{^}{j} + 6\overset{^}{k}

⇒ {\left|\overset{\to}{c} \times \left(\lambda\overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right)\right|}^{2} = 46

Hence this is the correct option.