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Three columns are given below. Column I lists trigonometric equations, Column II lists the number of solutions in [0, 2*pi], and Column III lists the sum of all solutions in [0, 2*pi]. Column I: (I) 2*sin²(x) - 3*sin(x) + 1 = 0, (II) |cos(x)| = 1/2, (III) sqrt(3)*tan²(x) - 4*tan(x) + sqrt(3) = 0, (IV) cos(3x) - cos(2x) + cos(x) - 1 = 0. Column II: (i) 0 solutions, (ii) 2 solutions, (iii) 3 solutions, (iv) 4 solutions. Column III: (P) sum = 3*pi/2, (Q) sum = 4*pi, (R) sum = 2*pi, (S) sum = 3*pi. Which of the following is the only INCORRECT combination?

1. (I) (iii) (P)
2. (II) (iv) (Q)
3. (III) (iv) (S)
4. (IV) (i) (P)

Correct answer: (IV) (i) (P)

Solution

Equation (I) has 3 solutions summing to 3*pi/2 (correct). Equation (II) has 4 solutions summing to 4*pi (correct). Equation (III) has 4 solutions summing to 3*pi (correct). Equation (IV) has multiple solutions (x = 0, pi/2, 2pi/3, 4pi/3, 3pi/2), so the claim of 0 solutions in option D is incorrect.

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