Let S 1 and S 2 be respectively the sets of all a ∈ R - 0 for which the system of linear equations a x + 2 a y - 3 a z = 1 2 a + 1 x + 2 a + 3 y + a + 1 z = 2 3 a + 5 x + a + 5 y + a + 2 z = 3 has unique solution and infinitely many solutions. Then

Step-by-step solution

Given, S 1 and S 2 be respectively the sets of all a ∈ R - 0 for which the system of linear equations a x + 2 a y - 3 a z = 1   . . . . . . . . . 1 2 a + 1   x + 2 a + 3   y + a + 1 z = 2   . . . . 2 3 a + 5   x + a + 5   y + a + 2   z = 3   . . . . . 3 Now from above equations finding Δ = a 2 a - 3 a 2 a + 1 2 a + 3 a + 1 3 a + 5 a + 5 a + 2 = a 15 a 2 + 31 a + 36 = 0 ⇒ a = 0 As 15 a 2 + 31 a + 36 cannot be zero as 31 2 - 4 × 15 × 36 < 0 So, &#