JEE Advanced2021MathematicsProperties of TrianglesActual
Consider a triangle P Q R having sides of lengths p , q and r opposite to the angles P , Q and R , respectively. Then which of the following statements is (are) TRUE?
Options
- Acos P ≥ 1 - p 2 2 q r
- Bcos R ≥ q - r p + q cos P + p - r p + q cos Q
- Cq + r p < 2 sin Q   sin R sin P
- DIf p < q and p < r , then cos Q > p r and cos R > p q
Correct answer
A. cos P ≥ 1 - p 2 2 q r
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 triang