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k^(sin^2 x) + k^(cos^2 x) = c (k = 16 or 81): solutions in an interval

The same question, asked 3 times: 3 May 2023 S1*, 9 May 2024 S1*, 22 Apr 2025 S1. Every wording below, oldest first.

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  1. Question 1

    3 May 2023 S1

    122
    The number of roots of the equation (81)sin⁡2x+(81)cos⁡2x=30(81)^{\sin^2 x}+(81)^{\cos^2 x}=30 in the interval [0,π][0,\pi] is equal to

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  2. Question 2

    9 May 2024 S1

    130
    The number of solutions in [0,2π][0,2\pi] of the equation 16sin⁡2x+16cos⁡2x=1016^{\sin^2 x}+16^{\cos^2 x}=10 is

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  3. Question 3

    22 Apr 2025 S1

    149
    The number of solutions of 16sin⁡2x+16cos⁡2x=1016^{\sin^{2}x}+16^{\cos^{2}x}= 10 in 0⩽x⩽2π0 \leqslant x\leqslant 2\pi are

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