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(1 + |sin x|)^(a/|sin x|) on one side and e^(cot 4x / cot 2x) on the other is continuous: find the constants

The same question, asked 2 times: 2021 Paper 16*, 25 Jan 2023. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 16

    46
    If f(x)={(1+∣sin⁡x∣)3a∣sin⁡x∣,x∈(−π4,0)b,x=0ecot⁡4xcot⁡2x,x>0f(x) =\left\{ \begin{matrix} (1 + |\sin x|)^{\frac{3a}{|\sin x|}}, & x\in\left( \frac{-\pi}{4},0 \right) \\ b, & x= 0 \\ e^{\frac{\cot4x}{\cot2x}}, & x> 0 \end{matrix} \right. is continuous at x=0x = 0, then 6a+b26a+b^{2} is equal to

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  2. Question 2

    25 Jan 2023

    163
    If the function f(x)={(1+∣cos⁡x∣)λ∣cos⁡x∣,0<x<π2μ,x=π2ecot⁡6xcot⁡4xπ2<x<πf(x) =\left\{ \begin{matrix} (1 + |\cos x|)^{\frac{\lambda}{|\cos x|}} & ,0 < x <\frac{\pi}{2} \\ \mu & ,x =\frac{\pi}{2} \\ e^{\frac{\cot6x}{\cot4x}} & \frac{\pi}{2}< x < \pi \end{matrix} \right. is continuous at x=π2x =\frac{\pi}{2}, then 9λ+6log⁡eμ+μ6−e6λ9\lambda + 6\log_{e}\mu +\mu^{6}-e^{6\lambda} is equal to

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