Let w = i + j - 2 k, and u and v be two vectors, such that u v = w and v w = u. Let , , , and t be real numbers such that u = i + j + k, -t + + = 0, - t + = 0, and + - t = 0. Match each entry in List-I to the correct entry in List-II and choose the correct option.
- (P) → (2), (Q) → (1), (R) → (4), (S) → (5)
- (P) → (2), (Q) → (4), (R) → (3), (S) → (5)
- (P) → (2), (Q) → (1), (R) → (4), (S) → (3)
- (P) → (5), (Q) → (4), (R) → (1), (S) → (3)
Answer
(1) (P) → (2), (Q) → (1), (R) → (4), (S) → (5)
Solution
u v = w w u, v; v w = u u v. So the three are mutually perpendicular.
| u| = | v|| w| and | w| = | u|| v| = | v|2| w| | v|2 = 1 (P), and | u|2 = 6.
The system has coefficient matrix J - (t + 1)I (J = all-ones), eigenvalues 2 - t and -(t + 1); u 0 needs t = 2 or t = -1. Also u w = 0: + - 2 = 0.
t = 2: = = (satisfies u w = 0), 3 2 = 6 = 2.
t = -1: + + = 0 with + - 2 = 0 = 0, = - ; 2 2 = 6 = 3.
(Q) = 3: 2 = 0. (R) ( + )2 = 3. (S) = 2 t = 2, t + 3 = 5.
P → 2, Q → 1, R → 4, S → 5.
Answer: (A)