Let → a = ^ i + 4 ^ j + 2 ^ k , → b = 3 ^ i - 2 ^ j + 7 ^ k and → c...

We have been given that

$\overset{\to}{d} \times \overset{\to}{b} = \overset{\to}{c} \times \overset{\to}{b}$

$⇒ ( \overset{\to}{d} - \overset{\to}{c} ) \times \overset{\to}{b} = 0$

$⇒ \overset{\to}{d} - \overset{\to}{c} = \lambda\overset{\to}{b} ( ∵ \overset{\to}{d} \ne \overset{\to}{c} )$

$⇒ \overset{\to}{d} = \overset{\to}{c} + \lambda\overset{\to}{b}$

It is given that $\overset{\to}{a} \cdot \overset{\to}{d} = 24$

$⇒ \overset{\to}{a} \cdot \overset{\to}{c} + \lambda\overset{\to}{b} \cdot \overset{\to}{a} = 24$

$⇒ 6 + \lambda ( 3 - 8 + 14 ) = 24$

$⇒ 9\lambda = 18$

$⇒ \lambda = 2$

$∴ \overset{\to}{d} = \overset{\to}{c} + 2\overset{\to}{b}$

⇒ \overset{\to}{d} = 8\overset{^}{i} - 5\overset{^}{j} + 18\overset{^}{k}

$| \overset{\to}{d}{|}^{2} = 64 + 25 + 324$$= 413$

Hence, this is the correct option.