Multiple choice

Let $ax^2+a^2x+2=0$ be a quadratic equation, $a \in R$ and $S$ be the set of values of $a$ for which roots of this equation are imaginary or equal, then answer the following questions: Number of integral values of $a$ in set $S$ are

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

For ax^2 + a^2x + 2 = 0, roots are imaginary or equal if discriminant D <= 0. D = (a^2)^2 - 4(a)(2) = a^4 - 8a <= 0. a(a^3 - 8) <= 0. a(a-2)(a^2+2a+4) <= 0. Since a^2+2a+4 is always positive, we need a(a-2) <= 0, so 0 <= a <= 2. The integral values are 0, 1, 2. However, if a=0, the equation is not quadratic (2=0), so a cannot be 0. Thus, integral values are 1 and 2.

AI explanation

For the roots of ax^2 + a^2x + 2 = 0 to be imaginary or equal, the discriminant must be less than or equal to zero, giving a^4 - 8a <= 0. Factoring out a, we get a(a^3 - 8) <= 0, which further factors to a(a - 2)(a^2 + 2a + 4) <= 0. Since the quadratic factor a^2 + 2a + 4 is always positive, the inequality holds when 0 <= a <= 2. The integers satisfying this range are 1 and 2, so there are exactly 2 integral values.

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