JEE Advanced 2014 · Paper 1, Mathematics question 60

Dot products. $\vec a\cdot\vec b=\vec b\cdot\vec c=\vec c\cdot\vec a=\frac12$. Let $V=[\vec a\,\vec b\,\vec c]\ne0$.

Dot the equation with $\vec a$, $\vec b$ and $\vec c$ in turn:
$$\begin{aligned}V&=p+\tfrac q2+\tfrac r2\\0&=\tfrac p2+q+\tfrac r2\\V&=\tfrac p2+\tfrac q2+r\end{aligned}$$
(using $\vec a\cdot(\vec b\times\vec c)=V$, $\vec c\cdot(\vec a\times\vec b)=V$, and that the other triple products vanish).

Solve. The first minus the third gives $p=r$. The second then gives $q=-p$, and the first gives $p=V$. So $p=r=V$, $q=-V$.

Value.
$$\frac{p^2+2q^2+r^2}{q^2}=\frac{1+2+1}1=4.$$

Answer: 4