Multiple choice

All the values of $\mathrm { m }$ for which both roots of the equation $\mathrm { x } ^ { 2 } - 2 \mathrm { m } \mathrm { x } + \mathrm { m } ^ { 2 } - 1 = 0$ are greater than $- 2$ but less than $4 ,$ lie in the interval-

  1. $- 1 < \mathrm { m } < 3$
  2. $1 < m < 4$
  3. $- 2 < \mathrm { m } < 0$
  4. $m > 3$
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A Correct answer
Explanation

The roots of x^2 - 2mx + m^2 - 1 = 0 are (m-1) and (m+1). Given -2 < m-1 < 4 and -2 < m+1 < 4, we solve the inequalities: -1 < m < 5 and -3 < m < 3. The intersection is -1 < m < 3.

AI explanation

The discriminant of the quadratic equation is 4m squared minus 4m squared plus 4, which equals 4, ensuring the roots are always real and distinct. The roots are m plus 1 and m minus 1, and for both to lie strictly between negative 2 and 4, we set up the inequalities m plus 1 less than 4 and m minus 1 greater than negative 2. Solving m less than 3 and m greater than negative 1 shows that m must lie in the open interval from negative 1 to 3. The result is negative 1 less than m less than 3.

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