Advanced Triangle Properties & JEE Problems

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1.
In triangle ABC, a cos A + b cos B + c cos C equals:
2.
In a triangle, if r_1 = 2r_2 = 3r_3, then a : b : c is:
3.
In triangle ABC, if a = 5, b = 4, cos(A - B) = 31/32, find c.
4.
In triangle ABC, prove that a(b cos C - c cos B) equals:
5.
In triangle ABC, if (a^2 + b^2) sin(A - B) = (a^2 - b^2) sin(A + B), then the triangle is:

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