Multiple choice

Consider the two quadratic polynomials: $\displaystyle C_{a}:y=\frac{x^{2}}{4}-ax+a^{2}+a-2$ and $\displaystyle C:y=2-\frac{x^{2}}{4}$ If the origin lies between the zeroes of the polynomial $ C_{a},$ then the number of integral value(s) of $'a'$ is

  1. $1$
  2. $2$
  3. $3$
  4. more than $3$
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B Correct answer
Explanation

For the origin to lie between the roots of Ca, the product of the roots must be negative, or f(0) < 0. f(0) = a^2 + a - 2 < 0. (a+2)(a-1) < 0. This holds for -2 < a < 1. The integers are -1 and 0. There are 2 such values.

AI explanation

For the origin to lie between the zeroes of the polynomial x^2/4 - ax + a^2 + a - 2, the quadratic must have real roots with a negative y-intercept, meaning the constant term is less than zero. The constant term is a^2 + a - 2, so setting a^2 + a - 2 < 0 gives the interval -2 < a < 1. The integers in this range are -1 and 0. There are exactly 2 integral values of a.

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