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MHT CET202519 Apr 2025Evening ShiftMathematicsQuadratic EquationActual

The value of ' a ' so that the sum of squares of the roots of the equation x^2-(a-2) x-a+1=0 assumes the least value is

Options

  1. A2
  2. B1
  3. C4
  4. D0

Correct answer

B. 1

Step-by-step solution

Given the quadratic equation x^2 - (a-2)x - a + 1 = 0 with roots α and β , Vieta's formulas yield: α + β = a - 2 αβ = 1 - a The sum of squares of the roots is S = α^2 + β^2 = (α + β)^2 - 2αβ . Substituting the expressions in terms of a : S = (a - 2)^2 - 2(1 - a) = a^2 - 4a + 4 - 2 + 2a = a^2 - 2a + 2 This quadratic function S(a) = a^2 - 2a + 2 has its minimum at the vertex a = -(-2)/(2 1) = 1 . The sum of squares of the roots assumes the least value when a = 1 .

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