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NDA 2023 Maths · Mathematics · Properties of Triangles · medium

For a triangle ABC with usual notation: 1. a=bcos C+ccos B. 2. If a^2+b^2<c^2, then the angle C is obtuse.

Single correct option

Consider the following statements for a triangle ABCABC with usual notation:
1. a=bcos⁡C+ccos⁡Ba=b\cos C+c\cos B.
2. If a2+b2<c2a^2+b^2<c^2, then the angle CC is obtuse.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (C) Both 1 and 2

Statement 1 is the projection formula: dropping the perpendicular from AA splits BCBC into ccos⁡Bc\cos B and bcos⁡Cb\cos C. Statement 2 follows from the cosine rule cos⁡C=a2+b2−c22ab<0\cos C=\dfrac{a^2+b^2-c^2}{2ab}<0, so C>90∘C>90^\circ. Both hold; those who think the triangle is acute have the inequality reversed.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake