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BITSAT2019MathematicsQuadratic EquationActual

In the real number system, the equation x + 3 - 4 x - 1 + x + 8 - 6 x - 1 = 1 has

Options

  1. ANo solution
  2. BExactly two distinct solution
  3. CExactly four distinct solution
  4. DInfinitely many solution

Correct answer

D. Infinitely many solution

Step-by-step solution

We have, x + 3 - 4 x - 1 + x + 8 - 6 x - 1 = 1 ( x - 1 ) 2 - 2 × 2 x - 1 + 4 + ( x - 1 ) 2 - 2 × 3 x - 1 + 9 = 1 ⇒   ( x - 1 - 2 ) 2 + ( x - 1 - 3 ) 2 = 1 ⇒   | x - 1 - 2 | + | x - 1 - 3 | = 1 ⇒   | x - 1 - 2 | + | x - 1 - 3 | = ( x - 1 - 2 ) - ( x - 1 - 3 ) We know that, If | x - a | + | x - b | = ( x - a ) - ( x - b ) then ( x - a ) ( x - b ) < 0 ∴   ( x - 1 - 2 ) ( x - 1 - 3 ) < 0 ⇒   2 < x - 1 < 3 ⇒ 5 < x < 10 ∴

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