JEE Main 2019 · 12 Apr · Shift II
JEE Main 2019 Mathematics question on Vector Algebra
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}- A.
is empty
- B.
is empty
- C.
contains exactly two positive numbers
- D.
contains exactly two numbers only one of which is positive
Show answer and solution
Correct option: B
Solution
For a, b, c to be coplanar, the scalar triple product [a b c] = 0.
Computing the determinant:
|α 1 3|
|2 1 -α|
|α -2 3|
= α(1·3 - (-α)(-2)) - 1(2·3 - (-α)·α) + 3(2·(-2) - 1·α)
= α(3 - 2α) - 1(6 + α²) + 3(-4 - α)
= 3α - 2α² - 6 - α² - 12 - 3α
= -3α² - 18
= -3(α² + 6)
Since α² + 6 ≥ 6 > 0 for all real α, this expression is never zero. Therefore no real α makes the vectors coplanar, and S is empty.
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