The number of distinct real values of , for which the vectors , and are coplanar, is
- A.0
- B.1
- C.2
- D.3
Show answer and solution
Correct option: C
Solution
For coplanarity, the scalar triple product (determinant) must be zero:
| -λ² 1 1 |
| 1 -λ² 1 | = 0
| 1 1 -λ² |
Expanding: -λ²(λ⁴ - 1) - 1(-λ² - 1) + 1(1 + λ²) = 0
= -λ⁶ + λ² + λ² + 1 + 1 + λ² = 0
= -λ⁶ + 3λ² + 2 = 0
→ λ⁶ - 3λ² - 2 = 0
Let u = λ²: u³ - 3u - 2 = 0
Factor: (u+1)²(u-2) = 0, giving u = -1 (rejected, not real) or u = 2.
So λ² = 2 → λ = ±√2, giving exactly 2 distinct real values.
| -λ² 1 1 |
| 1 -λ² 1 | = 0
| 1 1 -λ² |
Expanding: -λ²(λ⁴ - 1) - 1(-λ² - 1) + 1(1 + λ²) = 0
= -λ⁶ + λ² + λ² + 1 + 1 + λ² = 0
= -λ⁶ + 3λ² + 2 = 0
→ λ⁶ - 3λ² - 2 = 0
Let u = λ²: u³ - 3u - 2 = 0
Factor: (u+1)²(u-2) = 0, giving u = -1 (rejected, not real) or u = 2.
So λ² = 2 → λ = ±√2, giving exactly 2 distinct real values.
[AIEEE 2007]