A polynomial ψ(s) = a n s n + a n-1 s n-1 + ......+ a 1 s + a 0 of degree n > 3 with constant real coefficients a n , a n-1 , ... a 0 has triple roots at s = -σ. Which one of the following conditions must be satisfied?

Correct Answer is : ψ(s) = 0, dψ(s)ds=0 and d2ψ(s)ds2=0 at s = -σ

ψ(s) = 0 at all the three values of s satisfying s 3 + σ 3 = 0

ψ(s) = 0, \frac{d\psi (s)}{ds} = 0 and {d^{2}\psi (s)}/{ds^{2}} = 0 at s = -σ

ψ(s) = 0, {d^{2}\psi (s)}/{ds^{2}} = 0 and {d^{4}\psi (s)}/{ds^{4}} = 0 at s = -σ

ψ(s) = 0, {d^{3}\psi (s)}/{ds^{3}} = 0 at s = -σ

GATE Mechanical Engineering (ME) 2022 Shift 2 Solved Paper

Question : 15 of 65

Marks: +1, -0

A polynomial ψ(s) = a_nsⁿ + a_n-1s^n-1 + ......+ a₁s + a₀ of degree n > 3 with constant real coefficients a_n, a_n-1, ... a₀ has triple roots at s = -σ. Which one of the following conditions must be satisfied?

ψ(s) = 0 at all the three values of s satisfying s³ + σ³ = 0
ψ(s) = 0, $\;{dψ\;(s)}/{ds} = 0$ and $\;{d^{2}ψ\;(s)}/{ds^{2}} = 0$ at s = -σ
ψ(s) = 0, $\;{d^{2}ψ\;(s)}/{ds^{2}} = 0$ and $\;{d^{4}ψ\;(s)}/{ds^{4}} = 0$ at s = -σ
ψ(s) = 0, $\;{d^{3}ψ\;(s)}/{ds^{3}} = 0$ at s = -σ

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