UPSC NDA I 2021 Mathematics Solved Paper

Concept:The scalar triple product measures the volume of a parallelepiped and equals zero when vectors are coplanar.Explanation:Given $\vec{a}, \vec{b}, \vec{c}$ are coplanar, so $[\vec{a}\,\vec{b}\,\vec{c}] = 0$.The expression is $(2\vec{a}\times 3\vec{b})\cdot 4\vec{c} + (5\vec{b}\times 3\vec{c})\cdot 6\vec{align}$. Each term equals a scalar triple product: $(2\vec{a}\times 3\vec{b})\cdot 4\vec{c} = [2\vec{a}\;3\vec{b}\;4\vec{c}]$ and $(5\vec{b}\times 3\vec{c})\cdot 6\vec{a} = [5\vec{b}\;3\vec{c}\;6\vec{a}]$. We can cyclically permute the second triple product: $[5\vec{b}\;3\vec{c}\;6\vec{a}] = [6\vec{a}\;5\vec{b}\;3\vec{c}]$. Extract constant factors: $[2\vec{a}\;3\vec{b}\;4\vec{c}] = (2)(3)(4)[\vec{a}\,\vec{b}\,\vec{c}] = 24[\vec{a}\,\vec{b}\,\vec{c}]$ and $[6\vec{a}\;5\vec{b}\;3\vec{c}] = (6)(5)(3)[\vec{a}\,\vec{b}\,\vec{c}] = 90[\vec{a}\,\vec{b}\,\vec{c}]$. Thus the sum is $24[\vec{a}\,\vec{b}\,\vec{c}] + 90[\vec{a}\,\vec{b}\,\vec{c}] = 114[\vec{a}\,\vec{b}\,\vec{c}]$. Since the vectors are coplanar, $[\vec{a}\,\vec{b}\,\vec{c}] = 0$, so the entire expression equals $0$.Answer:$0$ (Option C).