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Continuity of a piecewise function with (1 + |sin x|)^(1/|sin x|) or tan/cot ratio powers

This type of question, asked 3 times: 2021 Paper 16, 25 Jan 2023, 9 Apr 2024. All 3 questions 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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  3. Question 3

    9 Apr 2024

    86
    Let f:(0,π)→Rf:(0,\pi) \rightarrow R be a function given by
    f(x)={(87)tan⁡8xtan⁡7x,0<x<π2a−8,x=π2(1+∣cot⁡x∣)ba∣tan⁡x∣,π2<x<πf(x) =\left\{ \begin{matrix} \left( \frac{8}{7} \right)^{\frac{\tan8x}{\tan7x}}, & 0 <x<\frac{\pi}{2} \\ a- 8, & x=\frac{\pi}{2} \\ (1 + |\cot x|)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2}<x<\pi \end{matrix} \right.
    Where a,b∈Za,b \in Z. If ff is continuous at x=π2x =\frac{\pi}{2}, then a2+b2a^{2}+b^{2} is equal to