JEE-Advanced 2015 (18/40)

Geometry Level 3

Let \(\Delta PQR\) be a triangle. Let \(\vec{a}=\vec{QR}, \vec{b}=\vec{RP}\) and \(\vec{c}=\vec{PQ}\). If \(|\vec{a}|=12,.|\vec{b}|=4\sqrt{3}\) and \(\vec{b} \cdot \vec{c}=24\), then which of the following is(are) true \[\begin{array}{} (1) \, \frac{|\vec{c}|^2}{2}-|\vec{a}|=12 \quad \quad \quad \quad & (2) \, \frac{|\vec{c}|^2}{2}+|\vec{a}|=12 \\ (3) \, |\vec{a} \times \vec{b}+\vec{c} \times \vec{a}|=48\sqrt{3}& (4) \, \vec{a} \cdot \vec{b}=-72 \end{array}\]
Note :

  • Submit your answer as the increasing order of the serial numbers of all the correct options.

  • For eg, if your answer is \((1),(2)\), then submit 12 as the correct answer, if your answer is \((2),(3),(4)\), then submit 234 as the correct answer.

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