Mathematics Question from JEE Main 2025 · 28 January, Shift 1

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{d} = \vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $\left|\vec{c} - 2\vec{a}\right|^2 = 8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $\left|10 - 3\,\vec{b} \cdot \vec{c}\right| + \left|\vec{d} \times \vec{c}\right|^2$ is equal to: