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JEE Advance - Mathematics (2017 - Paper 2 Offline - No. 15)

Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.
Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.
Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is
$$ - {3 \over 2}$$
$${3 \over 2}$$
$${5 \over 3}$$
$$ - {5 \over 3}$$

Explanation

cos(P + Q) + cos(Q + R) + cos(R + P)

= $$-$$ (cosR + cosP + cosQ)

Max. of cosP + cosQ + cosR = $${3 \over 2}$$

Min. of cos(P + Q) + cos(Q + R) + cos(R + P) is = $$ - {3 \over 2}$$

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