Quadratic Equation


  1. If a and b are the roots of the equation x2 - 6x + 6 = 0, find the value of 2(a2 + b2).











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    a and b are the roots of the equation
    x2 - 6x + 6 = 0
    ∴ a + b = -B/A = 6 and ab = C/A = 6

    Correct Option: C

    a and b are the roots of the equation
    x2 - 6x + 6 = 0
    ∴ a + b = -B/A = 6 and ab = C/A = 6
    We know that,
    a2 + b2 = (a + b)2 - 2ab
    = (6)2 - 2 x 6 = 36 - 12 = 24
    ∴ 2(a2 + b2) = 2 x 24 = 48


  1. If α and β be the roots of the equation ax2 + bx + c = 0, find the value of α2/β + β2/α .











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    Given, α and β are the roots of the equation ax2 + bx + c = 0
    ∴ Sum of two roots = -b/a
    α + β = -b/a
    and product of two roots = c/a
    α β = c/a
    ∴ α2/β + β2/α = ( α3 + β3) / α β
    = [(α + β) 3 - 3 α β (α + β)] / αβ
    [∵ (a + b)3 = a3 + b3 + 3ab(a + b) ]

    Correct Option: D

    Given, α and β are the roots of the equation ax2 + bx + c = 0
    ∴ Sum of two roots = -b/a
    α + β = -b/a
    and product of two roots = c/a
    α β = c/a
    ∴ α2/β + β2/α = ( α3 + β3) / α β
    = [(α + β) 3 - 3 α β (α + β)] / αβ
    [∵ (a + b)3 = a3 + b3 + 3ab(a + b) ]
    = (-b/a)3 - [3c/a(-b/a)/c/a = -b3/a3 + 3bc/a2]/(c/a)
    = 3abc - b3/a2c



  1. If α and β are the roots of the equation x2 - 11x + 24 = 0, find the equation having the roots α + 2 and β + 2











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    Given equation is
    x2 - 11x + 24 = 0
    Then, required equation is
    (x - 2)2 - 11(x - 2) + 24 = 0

    Correct Option: E

    Given equation is
    x2 - 11x + 24 = 0
    Then, required equation is
    (x - 2)2 - 11(x - 2) + 24 = 0
    ⇒ x2 - 4x + 4 - 11x + 22 + 24 = 0
    ⇒ x2 - 15x + 50 = 0


  1. If x2 = 6 + √6 + √6 + √6 + .... ∞ , then what is one of the value of x equal to ?









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    Here, x2 = 6 + √6 + √6 + √6 + .... ∞ ,
    So, x2 = 6 + √x2

    Correct Option: D

    Here, x2 = 6 + √6 + √6 + √6 + .... ∞ ,
    So, x2 = 6 + √x2
    ⇒ x2 = 6 + x
    ⇒ x2 - x - 6 = 0
    ⇒ x2 + 2x - 3x - 6 = 0
    ⇒ x(x + 2) - 3(x + 2) = 0
    ⇒ (x - 3) (x + 2) = 0
    ∴ x = 3



  1. If a = p / (p + q) and b = q / (p - q), then and ab/a + b is equal to









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    ab/(a + b) = [p/(p + q)] x [q/(p - q)] / [p/(p + q) + q/(p - q)]
    = pq/p2 - pq + pq + q2

    Correct Option: A

    ab/(a + b) = [p/(p + q)] x [q/(p - q)] / [p/(p + q) + q/(p - q)]
    = pq/p2 - pq + pq + q2
    = pq/(p2 + q2)