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Range of a sin x + b cos x, alone or inside another function

This type of question, asked 3 times: 2021 Paper 17, 9 Apr 2024, 23 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    73
    Let α=max⁡x∈R{82sin⁡3x⋅44cos⁡3x}\alpha=\max_{x\in R} \left\{ 8^{2\sin3x}\cdot4^{4\cos3x} \right\} and β=min⁡x∈R{82sin⁡3x⋅44cos⁡3x}\beta=\min_{x\in R} \left\{ 8^{2\sin3x}\cdot4^{4\cos3x} \right\}. If 8x2+bx+c=08x^{2}+bx+c= 0 is a quadratic equation whose roots are α1/5\alpha^{1/5} and β1/5\beta^{1/5}, then the value of c−bc-b is equal to:

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  2. Question 2

    9 Apr 2024

    158
    Let the range of the function f(x)=12+sin⁡3x+cos⁡3x,x∈Rf(x) =\frac{1}{2 + \sin3x + \cos3x},x \in R be [a,b]\lbrack a,b\rbrack. If α\alpha and β\beta are respectively the A.M. and the G.M. of aa and bb, then αβ\frac{\alpha}{\beta} is equal to:

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  3. Question 3

    23 Jan 2026 S2

    63
    The least value of (cos⁡2θ−6sin⁡θcos⁡θ+3sin⁡2θ+2)\left( \cos^{2}\theta - 6\sin\theta \cos\theta + 3\sin^{2}\theta + 2 \right) is

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