Q.33
Let f(x) = (x-1)(x-2)(x-3)(x-4) and let f(3/2) = α, f(5/2) = β and f(-7/2) = γ.
Which of the following is/are CORRECT?
- (A) α and β have the same sign
- (B) α and γ have the same sign
- (C) β and γ have the same sign
- (D) αβ and γβ have the same sign
f(x) = (x-1)(x-2)(x-3)(x-4) is a quartic polynomial with roots at x=1,2,3,4. Evaluating signs at x=3/2=1.5 (α), x=5/2=2.5 (β), and x=-7/2=-3.5 (γ) determines which sign relationships hold.
Sign Analysis
At x=1.5 (between roots 1 and 2): factors are (+)(-)(-)(-) → one negative → α < 0.
At x=2.5 (between roots 2 and 3): factors are (+)(+)(-)(-) → two negatives → β > 0.
At x=-3.5 (left of all roots): factors are (-)(-)(-)(-) → four negatives → γ > 0.
Option Evaluation
Option (A): α (negative) and β (positive) have opposite signs → Incorrect.
Option (B): α (negative) and γ (positive) have opposite signs → Incorrect.
Option (C): β (positive) and γ (positive) have the same sign → Correct.
Option (D): αβ (negative × positive = negative) and γβ (positive × positive = positive) have opposite signs → Incorrect.
Only option (C) is correct.
The polynomial f(x)=(x-1)(x-2)(x-3)(x-4) features four consecutive roots, making sign analysis essential for competitive exams like CSIR NET. This guide details evaluation at x=3/2 (α), x=5/2 (β), and x=-7/2 (γ), verifying options on same-sign pairs and products.
Step-by-Step Sign Chart
Critical points divide the real line: (-∞,1), (1,2), (2,3), (3,4), (4,∞).
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Test points confirm: negative in (1,2) and (3,4); positive elsewhere.
| Point | Interval | Factors Signs | Product Sign | Value |
|---|---|---|---|---|
| 1.5 (α) | (1,2) | (+)(-)(-)(-) | Negative | -0.9375 |
| 2.5 (β) | (2,3) | (+)(+)(-)(-) | Positive | 0.5625 |
| -3.5 (γ) | (-∞,1) | (-)(-)(-)(-) | Positive | 1206.5625 |
Detailed Option Verification
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(A) α and β same sign? Negative α, positive β → False.
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(B) α and γ same sign? Negative α, positive γ → False.
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(C) β and γ same sign? Both positive → True.
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(D) αβ and γβ same sign? αβ (negative), γβ (positive) → False.
CSIR NET Exam Tips
Master polynomial sign analysis by counting negative factors between roots. Exact computation unnecessary—signs suffice for multiple-choice. Practice similar quartics for quick interval testing.
