Explanation

(a)
Given, 2 $\overrightarrow{\mathbf{a}} + 3\overrightarrow{\mathbf{b}} - 5\overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
$\Longrightarrow \ \frac{2\overrightarrow{\mathbf{a}} + 3\overrightarrow{\mathbf{b}}}{5} = \overrightarrow{\mathbf{c}}$
$\Longrightarrow \frac{2\overrightarrow{\mathbf{a}} + 3\overrightarrow{\mathbf{b}}}{2 + 3} = \overrightarrow{\mathbf{c}}$
$\Longrightarrow \ \frac{\overrightarrow{\mathbf{a}} + \frac{3}{2}\overrightarrow{\mathbf{b}}}{1 + \frac{3}{2}} = \overrightarrow{\mathbf{c}}\ \ \ \ \ \ldots\ldots\ldots(i)$
Let $\overrightarrow{\mathbf{c}}$divides $\overrightarrow{\mathbf{AB}}\$in the ratio λ:1
$Then,\ \ \overrightarrow{\mathbf{c}}\frac{\overrightarrow{\mathbf{a}} + \lambda\ \overrightarrow{\mathbf{b}}}{1 + \lambda}\ \ \ \ldots.(ii)$
On comparing Eqs.(i)and (ii), we get
$\ \lambda = \frac{3}{2}$
$\therefore\$Required ratio is 3:2 internally.
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