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Consider the equation 3*cot²(x) + lambda*cot(x) + 3 = 0 for x in (0, 2*pi) excluding x = pi. For which value(s) of lambda does the sum of all solutions equal 3*pi or 5*pi as stated below? (A) Sum of solutions = 3*pi when lambda = 7 (B) Sum of solutions = 3*pi when lambda = -8 (C) Sum of solutions = 5*pi when lambda = 9 (D) Sum of solutions = 5*pi when lambda = -10

1. Sum of solutions = 3*pi when lambda = 7
2. Sum of solutions = 3*pi when lambda = -8
3. Sum of solutions = 5*pi when lambda = 9
4. Sum of solutions = 5*pi when lambda = -10

Correct answer: Sum of solutions = 3*pi when lambda = -8

Solution

The quadratic 3t²+lambda*t+3=0 has product of roots=1, so roots are either both positive or both negative. Both positive when -lambda/3>0, i.e. lambda<0, giving solution sum=3*pi; both negative when lambda>0, giving sum=5*pi. Option B (lambda=-8, both roots positive, sum=3*pi) and option C (lambda=9, both roots negative, sum=5*pi) are correct.

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