KCET 2025 Math Paper

Let's check each statement in turn:Statement (I): "If either $\left|\overset{\rightarrow}{a}\right|=0$ or $\left|\overset{\rightarrow}{b}\right|=0$, then $\overset{\rightarrow}{a}\cdot \overset{\rightarrow}{b}=0$."- If one of the vectors is the zero vector, say $\overset{\rightarrow}{a}=\overset{\rightarrow}{0}$, then for any $\overset{\rightarrow}{b}$$\overset{\rightarrow}{0}\cdot \overset{\rightarrow}{b}=0$- Hence $(I)$ is true. Statement (II): "If $\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}=\overset{\rightarrow}{0}$, then $\overset{\rightarrow}{a}$ is perpendicular to $\overset{\rightarrow}{b}$."- Recall $\left|\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}\right|=\left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right| \sin \theta$, where $\theta$ is the angle between them.$-\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}=\overset{\rightarrow}{0}$ means $\sin \theta=0$, so $\theta=0$ or $\pi$ : the vectors are parallel (or one is zero), not perpendicular. - Hence (II) is false.Conclusion:Statement (I) is true, Statement (II) is false.