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Two limits are the roots of a quadratic: find its coefficients

This type of question, asked 2 times: 2021 Paper 23, 8 Apr 2024. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    68
    If α=lim⁡x→π/4tan⁡3x−tan⁡xcos⁡(x+π4)\alpha=\lim_{x\rightarrow\pi/4} \frac{\tan^{3}x- \tan x}{\cos\left( x+\frac{\pi}{4} \right)} and β=lim⁡x→0(cos⁡x)cot⁡x\beta=\lim_{x\rightarrow 0} (\cos x)^{\cot x} are the roots of the equation, ax2+bx−4=0ax^{2}+bx- 4 = 0, then the ordered pair (a,b)(a,b) is:

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  2. Question 2

    8 Apr 2024

    172
    If α=lim⁡x→0+(etan⁡x−extan⁡x−x)\alpha =\lim_{x \rightarrow0^{+}} \left( \frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}} \right) and β=lim⁡x→0(1+sin⁡x)12cot⁡x\beta =\lim_{x \rightarrow 0} (1 + \sin x)^{\frac{1}{2}\cot x} are the roots of the quadratic equation ax2+bx−e=0ax^{2}+ bx -\sqrt{e}= 0, then 12 log⁡e(a+b)\log_{e}(a + b) is equal to