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Solve k^(sin^2 x) + k^(cos^2 x) = c: number of solutions or the solutions in an interval

This type of question, asked 4 times: 3 May 2023 S1, 9 May 2024 S1, 22 Apr 2025 S1, 26 Apr 2025 S2. All 4 questions 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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  4. Question 4

    26 Apr 2025 S2

    148
    If 0≤x≤π0 \leq x\leq\pi and 81sin⁡2x+81cos⁡2x=3081^{\sin^{2}x}+81^{\cos^{2}x}= 30 Then xx takes the value

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