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Quadratic Equation and Inequalities question

2019 · 9 Apr · Shift 2 · Q34
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Quadratic Equation and Inequalities question

2019 · 9 Apr · Shift 2 · Q34

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-
  1. A
    434\sqrt 343​
  2. B
    838\sqrt 383​
  3. C
    858\sqrt 585​
  4. D
    10510\sqrt 5105​
View written solutionFree

Correct answer: C

  1. Write the quadratic in standard form

    The equation is (m2+1)x2−3x+(m2+1)2=0.(m^2+1)x^2-3x+(m^2+1)^2=0.(m2+1)x2−3x+(m2+1)2=0.

    Comparing with ax2+bx+c=0,ax^2+bx+c=0,ax2+bx+c=0, we have a=m2+1, b=−3, c=(m2+1)2.a=m^2+1,\, b=-3,\, c=(m^2+1)^2.a=m2+1,b=−3,c=(m2+1)2.

  2. Sum of roots

    If the roots are α,β\alpha,\betaα,β, then α+β=−ba=3m2+1.\alpha+\beta=-\frac{b}{a}=\frac{3}{m^2+1}. α+β=−ab​=m2+13​.

    We need this sum to be greatest.

    Since m2≥0m^2\ge 0m2≥0, we have m2+1≥1,m^2+1\ge 1,m2+1≥1, so 3m2+1≤3.\frac{3}{m^2+1}\le 3.m2+13​≤3.

    Hence the maximum value of the sum occurs when m2=0  ⟹  m=0.m^2=0 \implies m=0.m2=0⟹m=0.

  3. Substitute m=0m=0m=0 into the quadratic

    Then the equation becomes x2−3x+1=0.x^2-3x+1=0.x2−3x+1=0.

    Let its roots be α,β\alpha,\betaα,β. Then α+β=3,αβ=1.\alpha+\beta=3,\qquad \alpha\beta=1.α+β=3,αβ=1.

  4. Find ∣α3−β3∣|\alpha^3-\beta^3|∣α3−β3∣

    Use α3−β3=(α−β)(α2+αβ+β2).\alpha^3-\beta^3=(\alpha-\beta)(\alpha^2+\alpha\beta+\beta^2).α3−β3=(α−β)(α2+αβ+β2).

    First, α2+β2=(α+β)2−2αβ=32−2(1)=9−2=7.\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=3^2-2(1)=9-2=7.α2+β2=(α+β)2−2αβ=32−2(1)=9−2=7.

    So, α2+αβ+β2=7+1=8.\alpha^2+\alpha\beta+\beta^2=7+1=8.α2+αβ+β2=7+1=8.

    Next, (α−β)2=(α+β)2−4αβ=9−4=5,(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=9-4=5,(α−β)2=(α+β)2−4αβ=9−4=5, hence ∣α−β∣=5.|\alpha-\beta|=\sqrt{5}. ∣α−β∣=5​.

    Therefore, ∣α3−β3∣=∣α−β∣ ∣α2+αβ+β2∣=5⋅8=85.|\alpha^3-\beta^3|=|\alpha-\beta|\,|\alpha^2+\alpha\beta+\beta^2|=\sqrt{5}\cdot 8=8\sqrt{5}. ∣α3−β3∣=∣α−β∣∣α2+αβ+β2∣=5​⋅8=85​.

  5. Match with options

    858\sqrt{5}85​ corresponds to Option C.

  6. Comparison with stored answer

    Stored correct answer: C

    Our derived answer: C

    So they agree.

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