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Question:

Let α∈ℝ and the three vectors a→=αi^+j^+3k^, b→=2i^+j^−αk^ and c→=αi^−2j^+3k^. Then the set S={α: a→, b→ and c→ are coplanar}.

Contains exactly two numbers only one of which is positive

Is singleton

Contains exactly two positive numbers

Is empty

Solution:

Correct option is D. Is empty
|α 1 3|
|2 1 -α|=0 ⇒ 3α²+18=0 ⇒ α²=-6
|α -2 3|
Since α is a real number, α² cannot be negative. Therefore, there is no real value of α for which the vectors are coplanar. Hence the set S is empty.