Let a → = i ^ + 2 j ^ + λ k ^ , b → = 3 i ^ - 5 j ^ - λ k ^ , a → · c…

Given:

$\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{c}$

$\Rightarrow \overrightarrow{a}\times \overrightarrow{c}-\overrightarrow{b}\times \overrightarrow{c}=0$

$\Rightarrow \left(\overrightarrow{a}-\overrightarrow{b}\right)\times \overrightarrow{c}=0$

So, $\left(\overrightarrow{a}-\overrightarrow{b}\right)\parallel \overrightarrow{c}$, hence

$\overrightarrow{a}-\overrightarrow{b}=\mu \overrightarrow{c}$

where, $\mu$ is any scalar.

$\Rightarrow -2\hat{i}+7\hat{j}+2\lambda \hat{k}=\mu \overrightarrow{c} ....\left(1\right)$

Taking dot product with $\overrightarrow{a}$, we get

$\Rightarrow -2+14+2{\lambda }^2=7\mu$

$\Rightarrow 7\mu =2{\lambda }^2+12 ...\left(2\right)$

Taking dot product with $\overrightarrow{b}$ in $\left(1\right)$, we get

$$$-6-35-2{\lambda }^2=-\frac{43}{2}\mu$

$\Rightarrow 82+4{\lambda }^2=43\mu ....\left(3\right)$

Solving $\left(2\right) \& \left(3\right)$, we get

${\lambda }^2=1, \mu =2$

So,

$\overrightarrow{a}·\overrightarrow{b}=3-10-{\lambda }^2=-8$

$\Rightarrow \left|\overrightarrow{a}·\overrightarrow{b}\right|=8$