Let $y_1$ and $y_2$ are two solutions of the differential equation $y'' + P(x)y' + Q(x)y = 0$ for all $x \in I$, where $P(x)$ and $Q(x)$ are continuous on $I$. Let $y_1$ and $y_2$ have common zero on $I$. Let $S_1 = \{\alpha \in \mathbb{R} \mid y_1(\alpha) = 0\}$, and $S_2 = \{\beta \in \mathbb{R} \mid y_2(\beta) = 0\}$, then-
(a) $S_1 \cap S_2 = \phi$
(b) $S_1 = S_2$
(c) $S_1 \subseteq S_2$ or $S_2 \subseteq S_1$
(d) $S_1 \subseteq S_2$ and $S_2 \subseteq S_1$
Explanation: (b) and (d) are correct.
As $y_1$ and $y_2$ have common zero in $I \Rightarrow y_1$ and $y_2$ are L.D. $\Rightarrow y_1 = Ky_2$ for $K \ne 0 \in \mathbb{R}$ Let $x_0 \in S_1 \Rightarrow y_1(x_0) = 0 \Rightarrow K y_2(x_0) = 0 \Rightarrow y_2(x_0) = 0$ \begin{align*} &\Rightarrow x_0 \in S_2 \\ &\Rightarrow S_1 \subseteq S_2 \end{align*} Similarly $\quad S_2 \subseteq S_1$ $\Rightarrow$ (b) and (d) are true.