UPSC CDS 2 2020 Math Paper

Concept:If three ratios are equal, set each equal to a common constant $k$ and express $x$, $y$, $z$ in terms of $k$. Then test the given options.Explanation:Let $\frac{x}{b+c} = \frac{y}{c+a} = \frac{z}{b-a} = k$, where $k$ is a constant. Then $x = k(b+c) = bk + ck$. $y = k(c+a) = ck + ak$. $z = k(b-a) = bk - ak$. Now check each option: Option A: $x + y + z = (bk+ck) + (ck+ak) + (bk-ak) = 2bk + 2ck$. This is not zero in general. Option B: $x - y - z = (bk+ck) - (ck+ak) - (bk-ak) = bk+ck - ck - ak - bk + ak = 0$. This holds for all $k$. Option C: $x + y - z = (bk+ck) + (ck+ak) - (bk-ak) = bk+ck+ck+ak - bk + ak = 2ck + 2ak$. Not zero generally. Option D: $x + 2y + 3z = (bk+ck) + 2(ck+ak) + 3(bk-ak) = bk+ck+2ck+2ak+3bk-3ak = 4bk + 3ck - ak$. Not zero generally. Therefore, only option B is always correct.Answer:B. $x - y - z = 0$