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If a R and the equation -3(x - [x])² + 2(x - [x]) + a² = 0 (where [x] denotes the greatest integer x) has no integral solution, then all possible values of a lie in the interval

1. (-2, -1)
2. (-, -2) (2,)
3. (-1, 0) (0, 1)
4. (1, 2) 2

Correct answer: (-1, 0) (0, 1)

Solution

The equation has no integral solutions when the quadratic expression in terms of the fractional part of x, which is bounded between 0 and 1, does not yield any real roots. The values of a must be such that the discriminant of the quadratic is negative, which occurs in the intervals (-1, 0) and (0, 1), ensuring that the equation cannot equal zero for any integer value.

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