Answer
work done $\int_C \textbf F \cdot \textbf{dr} = 30$ units
Work Step by Step
The given path C can be written as
$\textbf r(t) = 5t \textbf i +3t\textbf j +2t \textbf k, 0\le t 1\\
= x(t) \textbf i +y(t)\textbf j +z(t) \textbf k$
It follows that $x(t) = 5t, y(t) = 3t, z(t) = 2t$
$\textbf F(x,y,z) = yz \textbf i + xz \textbf j + xy \textbf k\\
\Rightarrow \textbf F = 6t^2 \textbf i + 10t^2 \textbf j + 15t^2 \textbf k\\
\textbf r' = 5\textbf i + 3\textbf j+ 2\textbf k$
Work done = $\int_C \textbf F \cdot d\textbf r = \int_{t=0}^1\textbf F \cdot \textbf r' dt\\
= \int_{t=0}^1 (6t^2 \textbf i + 10t^2 \textbf j + 15t^2 \textbf k)\cdot (5\textbf i + 3\textbf j+ 2\textbf k)dt\\
= \int_0^1 (30t^2+30t^2+30t^2)dt\\
= [30t^3]_0^1 = 30$
work done = $30$ units
