JEE Advanced Mathematics Question (2021) — Solution

Step-by-step solution

(A) Applying cosine rule cos P = q 2 + r 2 − p 2 2 q r = 1 2 q r + r q − p 2 2 q r We know, using  A . M . ≥ G . M . q r + r q ≥ 2 ⇒     1 2 q r + r q ≥ 1 ⇒ 1 2 q r + r q − p 2 2 q r ≥ 1 − p 2 2 q r ⇒ cos P ≥ 1 − p 2 2 q r (B)  p + q cos R ≥ q − r cos P + p − r cos Q p cos R + r cos P + q cos R + r cos Q ≥ q cos P + p cos Q using projection formula q + p ≥ r Which is true as sum of two sides of triangles is greater than third side. (C)  q + r p < 2 sin Q sin R sin P q + r p < 2 sin Q sin R sin 2 P Applying sine rule ⇒ q + r p < 2 q r p 2 ⇒ q + r 2 p < q r p 2 Applying  A . M . ≥ G . M . q p + r p 2 ≥ q r p 2 So, (C) is not true ⇒ q + r 2 p ≥ q r p 2 (D)  p < q  &  p < r ⇒   p  is smallest side Let  cos Q > p r  and  cos R > p q  is true ⇒ p < r cos Q p < q cos R adding both inequalities ⇒ 2 p < r cos Q + q cos R using projection formula ⇒ 2 p < p 2 < 1 Which is not correct.