निम्नलिखित दो कथनों के बीच: कथन \(-I\) : माना \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\) और \(\vec{b}=2 \hat{i}+\hat{j}-\hat{k}\) है। तब सदिश \(\vec{r}\) जो \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\) और \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{r}}=0\) को संतुष्ट करता है, का परिमाण \(\sqrt{10}\) है। कथन \(-II\) : एक त्रिभुज \(A B C\) में, \(\cos 2 A+\cos 2 B\) \(+\cos 2 \mathrm{C} \geq-\frac{3}{2}\)

A दोनों कथन \(-I\) और कथन \(-II\) असत्य हैं
B कथन \(-I\) असत्य है किंतु कथन \(-II\) सत्य है
C दोनों कथन \(-I\) और कथन \(-II\) सत्य हैं
D कथन \(-I\) सत्य है किंतु कथन \(-II\) असत्य है

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Correct Answer

(B) कथन \(-I\) असत्य है किंतु कथन \(-II\) सत्य है

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Detailed explanation

\( \bar{a}=\hat{i}+2 \hat{j}-3 \hat{k} \) \( \bar{a}=2 \hat{i}+\hat{j}-\hat{k} \) \( \bar{a} \times \bar{r}=\bar{a} \times \bar{b} ; \bar{a} \cdot \bar{r}=0 \) \( \Rightarrow \bar{a} \times(\bar{r}-\bar{b})=\overline{0} \) \( \Rightarrow \bar{a}=\lambda(\bar{r}-\bar{b}) \)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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