Explanation

(d)
Given, $\overrightarrow{\mathbf{a}} = (1,p,1),\overrightarrow{\mathbf{b}} = (q,2,2)$
$\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = r$ and $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = (0, - 3, - 3)$
Now, $\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = \left( \widehat{\dot{\mathbf{i}}} + p\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \bullet \left( q\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right)$
$\Longrightarrow q + 2p + 2 = r$ [given]….(i)
$Now,\ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}\left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
1 & p & 1 \\
q & 2 & 2
\end{matrix} \right|$
$\Longrightarrow (2p - 2)\widehat{\dot{\mathbf{i}}} + (q - 2)\widehat{\dot{\mathbf{j}}} + (2 - pq)\widehat{\mathbf{k}}$
$= \{ 0\widehat{\dot{\mathbf{i}}} + ( - 3)\widehat{\dot{\mathbf{j}}} + (3)\widehat{\mathbf{k}}$ [given]
$\Longrightarrow 2p - 2 = 0;q - 2 = - 3;2 - pq = 3$
$\Longrightarrow \ p = 1,\ q = - 1$
From Eqs. (i),
$- 1 + 2 + 2 = r$
$= r = 3$