In a non-right-angled triangle △PQR, Let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p=3, q=1, and the radius of the circumcircle of the △PQR equals 1, then which of the following options is/are correct?

Correct option is C. Radius of incircle of △PQR = 3/2(2−√3)
p sin P = q sin Q = 2(1) ⇒ sin P = 3/2, sin Q = 1/2 ⇒ ∠P = 60° or 120° and ∠Q = 30° or 150° because ∠P + ∠Q must be less than 180° but not equal to 90°
∠P = 120° and ∠Q = 30° and ∠R = 30°
r sin R = 2 ⇒ r = 1
Now length of median RS = 1/2√2p² + 2q² − r² = 1/2√6 + 2 − 1 = 7/2 ⇒ option (A) is correct
Inradius = Δ/s = 2Δ/p+q+r = pqr/4(1)p+q+r = 1/2(1×1×3/1+1+3) = 3/2(2−√3) ⇒ option (C) is correct
⇒ 1/2 × 3 × PE = pqr/4(1) (equal area of △)
⇒ PE = 1 × 1 × 3/4 × 2/3 = 1/2 ⇒ OE = 2(Area of △OQR)/QR = 2 × 1/3(1/2 × 1 × 3 sin 30°) /3 = 1/6.