Multiple choice

Let the values of $a$ for which the roots of the equation $(x -a) (x - a - 1) = 0$ lie between the roots of the equation $\left ( x+a\right )\left ( x+a^{2}-2 \right )=0$ be $a\in \left ( -\infty ,p \right )\cup \left ( q,\dfrac{-1+\sqrt{5}}{2} \right )$, then the value of $q - p$ is

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

The roots of (x-a)(x-a-1)=0 are a and a+1. The roots of (x+a)(x+a^2-2)=0 are -a and 2-a^2. For the first roots to lie between the second, we need -a < a < a+1 < 2-a^2 or -a < a < a+1 < 2-a^2. Solving these inequalities leads to the specified interval. The resulting values for p and q yield q-p=2.

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