PYQ Vault
EbooksTutor

Two quadratics share a common root: find the parameter or an expression in the roots

This type of question, asked 5 times: 2021 Paper 19, 25 Jun 2022, 27 Jun 2022, 29 Jan 2023, 30 Jan 2023. All 5 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 19

    87
    Let λ≠0\lambda \neq 0 be in RR. If α\alpha and β\beta are the roots of the equation x2−x+2λ=0x^{2}- x + 2\lambda = 0, and α\alpha and γ\gamma are the roots of equation 3x2−10x+27λ=03x^{2}- 10x + 27\lambda = 0, then βγλ\frac{\beta\gamma}{\lambda} is equal to

  2. Question 2

    25 Jun 2022

    152
    Let a,b∈Ra,b \in R be such that the equation ax2−2bx+15=0ax^{2}- 2bx + 15 = 0 has a repeated root α\alpha. If α\alpha and β\beta are the roots of the equation x2−2bx+21=0x^{2}- 2bx + 21 = 0, then α2+β2\alpha^{2}+\beta^{2} is equal to:

    Tap an option to check your answer.

  3. Question 3

    27 Jun 2022

    172
    Let α,β\alpha,\beta be the roots of the equation x2−4λx+5=0x^{2}- 4\lambda x + 5 = 0 and α,γ\alpha,\gamma be the roots of the equation x2−(32+23)x+7+3λ3=0x^{2}- (3\sqrt{2}+ 2\sqrt{3})x + 7 + 3\lambda\sqrt{3}= 0. If β+γ=32\beta + \gamma = 3\sqrt{2}, then (α+2β+γ)2(\alpha + 2\beta + \gamma)^{2} is equal to:

  4. Question 4

    29 Jan 2023

    76
    Let λ≠0\lambda \neq 0 be a real number. Let α,β\alpha,\beta be the roots of the equation 14x2−31x+3λ=014x^{2}- 31x + 3\lambda = 0 and α,γ\alpha,\gamma be the roots of the equation 35x2−53x+4λ=035x^{2}- 53x + 4\lambda = 0. Then 3αβ\frac{3\alpha}{\beta} and 4αγ\frac{4\alpha}{\gamma} are the roots of the equation

    Tap an option to check your answer.

  5. Question 5

    30 Jan 2023

    178
    If the value of real number a>0a > 0 for which x2−5ax+1=0x^{2}- 5ax + 1 = 0 and x2−ax−5=0x^{2}- ax - 5 = 0 have a common real root is 32β\frac{3}{\sqrt{2\beta}} then β\beta is equal to