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Nature of the triangle or relation among sides from a trig condition

This type of question, asked 5 times: May 2021 S1, 22 Apr 2025 S2, 23 Apr 2025 S1, 17 Apr 2026 S1, 12 May 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    May 2021 S1

    101
    With usual notations, in any △ABC\triangle ABC, if acos⁡B=bcos⁡Aa\cos B = b\cos A, then the triangle is

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  2. Question 2

    22 Apr 2025 S2

    108
    In a triangle ABC with usual notations if, cot⁡A2=b+ca\cot\frac{A}{2}=\frac{b + c}{a}, then the triangle ABCABC is

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  3. Question 3

    23 Apr 2025 S1

    103
    In △ABC\bigtriangleup ABC, with usual notations, if cos⁡B2=c+a2a\cos\frac{B}{2}=\sqrt{\frac{c+a}{2a}}, then a2=a^{2}=

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  4. Question 4

    17 Apr 2026 S1

    63
    With usual notations, in △ABC\triangle ABC, if cos⁡C=sin⁡A2sin⁡B\cos C = \dfrac{\sin A}{2\sin B}, then which of the following is true ?

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  5. Question 5

    12 May 2026 S1

    54
    In ΔABC\Delta ABC, with usual notations, if sin⁡B−cos⁡C=cos⁡A\sin B - \cos C = \cos A, then

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