Multiple choice

All the values of m for which both the roots of the equation x2 - 2mx + m2 - 1 = 0 are greater than -2 but less than 4 lie in the interval

  1. 1 < m < 4

  2. -2 < m < 0

  3. m > 3

  4. -1 < m < 3
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D Correct answer
Explanation

The roots of x^2 - 2mx + m^2 - 1 = 0 are (2m +/- sqrt(4m^2 - 4(m^2 - 1)))/2 = m +/- 1. Roots are m-1 and m+1. We need -2 < m-1 and m+1 < 4. This gives m > -1 and m < 3. Thus, -1 < m < 3.

AI explanation

The roots of the given quadratic equation x^2 - 2mx + m^2 - 1 = 0 are found by factoring as (x - m)^2 - 1 = 0, giving x = m + 1 and x = m - 1. For both roots to lie strictly between -2 and 4, we set up the inequalities -2 < m - 1 and m + 1 < 4. Solving these gives m > -1 and m < 3, meaning the values of m lie in the interval -1 < m < 3.

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