The GATE Grind

GATE 2022 ME (ME2) – Question 15

Engineering Mathematics · Calculus: Limit, continuity, differentiability, mean value theorems and indeterminate forms · 1 mark · Multiple choice

A polynomial $\varphi(s)=a_ns^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0$ of degree $n>3$ with constant real coefficients has triple roots at $s=-\sigma$. Which one of the following conditions must be satisfied?

  1. $\varphi(s)=0$ at all three values satisfying $s^3+\sigma^3=0$
  2. $\varphi=\varphi^\prime=\varphi^{\prime\prime}=0$ at $s=-\sigma$
  3. $\varphi=\varphi^{\prime\prime}=\varphi^{(4)}=0$ at $s=-\sigma$
  4. $\varphi=\varphi^{(3)}=0$ at $s=-\sigma$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $\varphi=\varphi^\prime=\varphi^{\prime\prime}=0$ at $s=-\sigma$

Explanation

If $s=-\sigma$ is a root of multiplicity 3, then
$$\varphi(s)=(s+\sigma)^3h(s),\qquad h(-\sigma)\neq0 .$$

**Differentiate.**
- $\varphi^\prime(s)=3(s+\sigma)^2h+(s+\sigma)^3h^\prime$, which is zero at $s=-\sigma$.
- $\varphi^{\prime\prime}(s)=6(s+\sigma)h+\cdots$, which is also zero at $s=-\sigma$.
- $\varphi^{(3)}(-\sigma)=6h(-\sigma)\neq0$.

So the condition is $\varphi=\varphi^\prime=\varphi^{\prime\prime}=0$ at $s=-\sigma$ (option **B**). The third derivative does not vanish, so C and D are wrong. Option A is about the roots of $s^3+\sigma^3$, which is unrelated.