Multiple choice

The least integral value of $a$ for which the equation $x^2-2(a-1)x+2a+1=0$ has both the roots positive is-

  1. $3$
  2. $4$
  3. $1$
  4. $5$
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B Correct answer
Explanation

For roots to be positive: 1) Discriminant D >= 0: 4(a-1)^2 - 4(2a+1) >= 0 => a^2 - 2a + 1 - 2a - 1 >= 0 => a^2 - 4a >= 0 => a(a-4) >= 0. So a <= 0 or a >= 4. 2) Sum of roots > 0: 2(a-1) > 0 => a > 1. 3) Product of roots > 0: 2a+1 > 0 => a > -0.5. Combining these, a >= 4. The least integral value is 4.

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