JEE Advanced 2017 · Paper 2 · Mathematics Q16

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MathematicsSingle correct+3Question 52 of 54

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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

$\cos(P + Q) + \cos(Q + R) + \cos(R + P)$

-\frac{5}{3}

-\frac{3}{2}

\frac{3}{2}

\frac{5}{3}

Answer and worked solution

Official answer: B

Rewrite. Since $P+Q=\pi-R$ , and similarly for the others,
$\cos(P+Q)+\cos(Q+R)+\cos(R+P)=-(\cos P+\cos Q+\cos R).$

Bound. In a triangle, $\cos P+\cos Q+\cos R=1+4\sin\frac P2\sin\frac Q2\sin\frac R2\le\frac32$ , with equality for an equilateral triangle.

Minimum. $-\dfrac32$ .

(B)

Topics:Trigonometric ratios and identities AM-GM-HM inequalities

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JEE Advanced 2017 · Paper 2, Mathematics question 16