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The complete set of values of the real parameter 'a' for which the function f(x) = 2*sin²(x) - 3*cos²(x) - (a² + a - 7)*x + 5 is strictly increasing for all x in R is [p, q], where p and q are integers. Find |p + q|.

1. 1
2. 2
3. 3
4. 5

Correct answer: 1

Solution

f(x) = 2*sin²(x) - 3*cos²(x) - (a²+a-7)x + 5. f'(x) = 4*sin(x)*cos(x) + 6*cos(x)*sin(x) - (a²+a-7). Wait: d/dx(2sin² x) = 4 sin x cos x = 2 sin(2x). d/dx(-3cos² x) = 6 cos x sin x = 3 sin(2x). So f'(x) = 2sin(2x) + 3sin(2x) - (a²+a-7) = 5sin(2x) - (a²+a-7). For f to be strictly increasing, f'(x) >= 0 for all x. Since min of 5sin(2x) = -5: -5 - (a²+a-7) >= 0 => -(a²+a-7) >= 5 => a²+a-7 <= -5 => a²+a-2 <= 0 => (a+2)(a-1) <= 0 => -2 <= a <= 1. So [p,q] = [-2, 1]. |p+q| = |-2+1| = |-1| = 1.

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