JEE Advanced 2013 · Paper 2 · Mathematics Q48

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MathematicsMultiple correct+3 / -1Question 48 of 60

In a triangle $PQR$ , $P$ is the largest angle and $\cos P=\frac13$ . Further the incircle of the triangle touches the sides $PQ$ , $QR$ and $RP$ at $N$ , $L$ and $M$ respectively, such that the lengths of $PN$ , $QL$ and $RM$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

Answer and worked solution

Official answer: B, D

Tangent lengths. With $PN=x$ , $QL=y$ , $RM=z$ : $QR=y+z$ , $RP=z+x$ , $PQ=x+y$ . $P$ is the largest angle, so $QR$ is the largest side and $x$ is the smallest tangent length: $x=2k$ , $\{y,z\}=\{2k+2,2k+4\}$ .

Sides. With $m=2k+2$ : $QR=2(m+1)$ and the others $2(m-1)$ , $2m$ .

Cosine rule.
$\cos P=\frac{(m-1)^2+m^2-(m+1)^2}{2m(m-1)}=\frac{m-4}{2(m-1)}=\frac13\implies m=10.$

Sides 18, 20, 22.

(B), (D)

Topics:Properties of triangles

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JEE Advanced 2013 · Paper 2, Mathematics question 48