Project ForgeEngineering Mathematics
Calculus
GATE 2022
Medium
A polynomial φ(s) = aₙsⁿ + aₙ₋₁sⁿ⁻¹ + ⋯ + a₁s + a₀ of degree n>3 with constant real coefficients aₙ, aₙ₋₁, …a₀ has triple roots at s=−σ. Which one of the following conditions must be satisfied?
- Aφ(s)=0 at all the three values of s satisfying s³+σ³=0
- φ(s)=0, dφ(s)/ds=0, and d²φ(s)/ds²=0 at s=−σ
- Cφ(s)=0, d²φ(s)/ds²=0, and d⁴φ(s)/ds⁴=0 at s=−σ
- Dφ(s)=0, and d³φ(s)/ds³=0 at s=−σ
Explanation
A root of multiplicity 3 at s=−σ means φ(s) can be written as (s+σ)³g(s) with g(−σ)≠0. Differentiating, φ, φ′, and φ″ all vanish at s=−σ, while φ‴(−σ)=6g(−σ)≠0 in general. So exactly the first two derivatives (along with φ itself) must vanish — option (B).
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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