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Sum of squares, cubes or fourth powers of the roots from sum and product

This type of question, asked 6 times: CDS 1 2017, CDS 2 2021, CDS 2 2022, CDS 1 2023, CDS 2 2024, CDS 2 2026. All 6 questions below, oldest first.

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  1. Question 1

    CDS 1 2017

    17
    If α\alpha and β\beta are the roots of the equation x2+px+q=0x^2 + px + q = 0, then what is α2+β2\alpha^2 + \beta^2 equal to ?

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  2. Question 2

    CDS 2 2021

    31
    If one of the roots of the equation ax2−4ax+15=0ax^2-4ax+15=0 is 32\frac{3}{2}, then what is the sum of the squares of the roots ?

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  3. Question 3

    CDS 2 2022

    6
    If the sum of the squares of the roots of the equation x2−14x+k=0x^2 - 14x + k = 0 is 100, then what is the value of k ?

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  4. Question 4

    CDS 1 2023

    21
    If α\alpha and β\beta are the roots of the equation x2−7x+1=0x^2 - 7x + 1 = 0, then what is the value of α4+β4\alpha^4 + \beta^4?

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  5. Question 5

    CDS 2 2024

    20
    If pp and qq are the roots of the equation x2−sin⁡2θ x−cos⁡2θ=0x^2 - \sin^2\theta\,x - \cos^2\theta = 0, then what is the minimum value of p2+q2p^2 + q^2 ?

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  6. Question 6

    CDS 2 2026

    16
    If α\alpha and β\beta are the roots of the equation 2x2−2(n+1)x+(n2+n+1)=02x^2 - 2(n + 1)x + (n^2 + n + 1) = 0, then what is (2α3+2β3)(2\alpha^3 + 2\beta^3) equal to ?

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