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Given the determinant g(y) = |[p-y, q, r], [r, p-y, q], [q, r, p-y]| where p > q > r > 0, what is the coefficient of y (the linear term) in g(y)?
- independent of only p
- independent of q as well as r
- independent of only r
- none of these
Correct answer: independent of only p
Solution
Let u = p - y. The determinant becomes |[u, q, r], [r, u, q], [q, r, u]|. Expanding: = u³ - 3qru + q³ + r³. Substituting back u = p - y: g(y) = (p-y)³ - 3qr(p-y) + q³ + r³. The linear term in y comes from expanding: -(p-y)³ gives terms in y, and -3qr(p-y) gives -3qrp + 3qr*y. The coefficient of y from the cubic term is 3p² (from -1*(3p²*(-y))) = 3p² and from -3qr(p-y) is 3qr. Total coefficient of y = 3p² - 3qr... wait, that depends on p too. Let me recheck: (p-y)³ = p³ - 3p²*y + 3p*y² - y³. Coefficient of y in (p-y)³ is -3p². Coefficient of y in -3qr(p-y) = +3qr. Total coefficient of y = -3p² + 3qr = 3(qr - p²). This depends on p, q, and r. So the coefficient is NOT independent of any single variable alone — 'none of these' seems correct. However the standard result for this type of question is that the coefficient of y is -(3p² - 3qr) which depends on p, q and r, so 'none of these' applies.
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