Topic-wise Practice

If zeros are α and -α, their sum = α + (-α) = 0. By Vieta's formulas, sum of zeros = 8k/4 = 2k. So 2k = 0, therefore k = 0.

LHS = sin²A + 2·sinA·cosecA + cosec²A + cos²A + 2·cosA·secA + sec²A = 1 + 2 + cosec²A + 2 + sec²A = 5 + (1 + cot²A) + (1 + tan²A) = 7 + tan²A + cot²A = RHS ∎

f'(x) = 3x² - 3 = 0 → x = ±1. Critical points + endpoints: x ∈ {-2, -1, 1, 3}. f(-2) = -8+6+2 = 0, f(-1) = -1+3+2 = 4, f(1) = 1-3+2 = 0, f(3) = 27-9+2 = 20. Absolute max = 20 at x=3. Absolute min = 0 at x=-2 and x=1.

z⁵ = r⁵·e^(5iθ). For 5th roots of unity: z⁵ = 1 (r=1, θ=2πk/5 for k=0,1,2,3,4). These are: z_k = e^(2πik/5) = cos(2πk/5) + i·sin(2πk/5). The five roots are equally spaced around the unit circle at 72° intervals: 1, e^(2πi/5), e^(4πi/5), e^(6πi/5), e^(8πi/5).