Explanation

(d)
Let the plane $\overrightarrow{r} \bullet \left( \overrightarrow{i} - 2\overrightarrow{j} + 3\overrightarrow{k} \right) = 17\$divide the line joining the points$\ - 2\overrightarrow{i} + 4\overrightarrow{j} + 7\overrightarrow{k}\$and$\ 3\overrightarrow{i} - 5\overrightarrow{j} + 8\overrightarrow{k}\$ in the ratio $t:1$ at point $P$
Therefore, point $P$ is
$\frac{3t - 2}{t + 1}\overrightarrow{i} + \frac{- 5t + 4}{t + 1}\overrightarrow{j} + \frac{8t + 7}{t + 1}\overrightarrow{k}$
This lies on the given plane
$\therefore\ \frac{3t - 2}{t + 1}.(1) + \frac{- 5t + 4}{t + 1}( - 2) + \frac{8t + 7}{t + 1}(3) = 17$
Solving, we get
$t = \frac{3}{10}$