माना \(\alpha \in R\) तथा तीन सदिश \(\overrightarrow{ a }=\alpha \hat{ i }+\hat{ j }+3 \hat{ k }\), \(\overrightarrow{ b }=2 \hat{ i }+\hat{ j }-\alpha \hat{ k }\) तथा \(\overrightarrow{ c }=\alpha \hat{ i }-2 \hat{ j }+3 \hat{ k }\) है। तब समुच्चय \(S =\{\alpha: \overrightarrow{ a }, \overrightarrow{ b }\) तथा \(\overrightarrow{ c }\) समतलीय है \(\}\) होगा

A में तथ्यतः दो संख्यायें हैं जिनमें से केवल एक धनात्मक है।
B रिक्त है।
C में तथ्यतः (exactly) दो धनात्मक संख्यायें है।
D एकल है।

Verified Solution

Answer & Solution

Correct Answer

(B) रिक्त है।

Step-by-step Solution

Detailed explanation

\([\vec{a} \cdot \vec{b} \cdot \vec{c}]=0\) \(\left| {\begin{array}{*{20}{c}} \alpha &3&1\\ 2&1&{ - \alpha }\\ \alpha &{ - 2}&3 \end{array}} \right| = 0\) \(\alpha(3-2 \alpha)+1\left(-\alpha^{2}-6\right)+3(-4-\alpha)=0\) \(3 \alpha-2 \alpha^{2}-\alpha^{2}-6-12-3 \alpha=0\)…

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.

Same subject

Explore more questions on app

From JEE Mains

Explore more questions on app