If the vectors ( x 2 − 1 ) i ^ + 2 ( x 2 − 1 ) j ^ − 3 ( x 2 − 1 ) k ^ , ( 2 x 2 − 1 ) i ^ + ( 2 x 2 + 1 ) j ^ + x 2 k ^ and ( 3 x 2 + 2 ) i ^ + ( x 2 + 4 ) j ^ + ( x 2 + 1 ) k ^ are non-coplanar, then the number of real values x cannot take is

Step-by-step solution

For non-coplanar vectors x 2 - 1 2 x 2 - 1 - 3 x 2 - 1 2 x 2 - 1 2 x 2 + 1 x 2 3 x 2 + 2 x 2 + 4 x 2 + 1 ≠ 0 x 2 - 1 1 2 - 3 2 x 2 - 1 2 x 2 + 1 x 2 3 x 2 + 2 x 2 + 4 x 2 + 1 ≠ 0 x 2 - 1 1 2 x 4 + 3 x 2 + 1 - x 4 - 4 x 2 - 2 2 x 4 + x 2 - 1 - 3 x 4 - 2 x 2 - 3 2 x 4 + 7 x 2 - 4 - 6 x 4 - 7 x 2 - 2 ≠ 0 x 2 - 1 x 4 - x 2 + 1 + 2 x 4 + 2 x 2 + 2 + 12 x 4 + 18 ≠ 0 x 2 - 1 15 x 4 + x 2 + 21 ≠ 0 ⇒ x ≠ + 1 , - 1