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The point (a², a+1) lies in the region between the lines 3x - y + 1 = 0 and x + 2y - 5 = 0 that contains the origin. Find the values of a satisfying this condition.
- a >= 1 or a <= -3
- a in (0, 1)
- a in (-3, 0) union (1/3, 1)
- None of these
Correct answer: a in (-3, 0) union (1/3, 1)
Solution
Condition 1: 3(a²) - (a+1) + 1 > 0 => 3a² - a > 0 => a(3a - 1) > 0 => a < 0 or a > 1/3. Condition 2: a² + 2(a+1) - 5 < 0 => a² + 2a - 3 < 0 => (a+3)(a-1) < 0 => -3 < a < 1. Intersection: (a < 0 or a > 1/3) AND (-3 < a < 1) = (-3, 0) union (1/3, 1).
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