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Symmetric expression of the roots (sum of squares, cubes or fourth powers) from the coefficients

This type of question, asked 7 times: 2021 Paper 1, 14 Jun 2022, 26 Jun 2022, 26 Jul 2022, 25 Jan 2023, 9 Apr 2024, 22 Jan 2025. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    78
    Let pp and qq be two positive number such that p+q=2p+q= 2 and p4+q4=272p^{4}+q^{4}= 272. Then pp and qq are roots of the equation :

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  2. Question 2

    14 Jun 2022

    67
    If the sum of the squares of the reciprocals of the roots α\alpha and β\beta of the equation 3x2+λx−1=03x^{2}+\lambda x- 1 = 0 is 15 , then 6(α3+β3)26\left( \alpha^{3}+\beta^{3} \right)^{2} is equal to:

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  3. Question 3

    26 Jun 2022

    172
    Let pp and qq be two real numbers such that p+q=p + q = 3 and p4+q4=369p^{4}+q^{4}= 369. Then (1p+1q)−2\left( \frac{1}{p}+\frac{1}{q} \right)^{- 2} is equal to

  4. Question 4

    26 Jul 2022

    151
    The minimum value of the sum of the squares of the roots of x2+(3−a)x+1=2ax^{2}+ (3 -a)x+ 1 = 2a is:

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  5. Question 5

    25 Jan 2023

    173
    Let a∈Ra \in R and let α,β\alpha,\beta be the roots of the equation x2+6014x+a=0x^{2}+60^{\frac{1}{4}}x + a = 0 If α4+β4=−30\alpha^{4}+\beta^{4}= - 30, then the product of all possible values of a is

  6. Question 6

    9 Apr 2024

    79
    Let α,β\alpha,\beta be the roots of the equation x2+22x−1=0x^{2}+ 2\sqrt{2}x- 1 = 0. The quadratic equation, whose roots are α4+β4\alpha^{4}+\beta^{4} and 110(α6+β6)\frac{1}{10}\left( \alpha^{6}+\beta^{6} \right), is:

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  7. Question 7

    22 Jan 2025

    137
    Let αθ\alpha_{\theta} and βθ\beta_{\theta} be the distinct roots of 2x2+(cos⁡θ)x−1=0,θ∈(0,2π)2x^{2}+ (\cos\theta)x - 1 = 0,\theta \in (0,2\pi). If mm and MM are the minimum and the maximum values of αθ4+βθ4\alpha_{\theta}^{4}+\beta_{\theta}^{4}, then 16(M+m)16(M + m) equals :

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