Consider the system of linear equations - x + y + 2 z = 0 3 x - a y + 5 z = 1 2 x - 2 y - a z = 7 Let S 1 be the set of all a ∈ R for which the system is inconsistent and S 2 be the set of all a ∈ R for which the system has infinitely many solutions. If n S 1 and n S 2 denote the number of elements in S 1 and S 2 respectively, then

Step-by-step solution

For system to be inconsistent D = - 1 1 2 3 - a 5 2 - 2 - a = 0 ⇒ a 2 - 7 a + 12 = 0 ⇒ a = 3 ,   4 D x = 0 1 2 1 - a 5 7 - 2 - a = 15 a + 31 D x ≠ 0  for  a = 3 , 4 Similarly we can show D y ≠ 0 ,   D z ≠ 0  for  a = 3 , 4 ⇒ n S 1 = 2 Now for infinitely many solutions D = 0   also D x = D y = D z = 0 which is not possible value of any real of a , since D = 0   &   D x = 0 have different solutions for a ⇒ n S 2 = 0 Hence, n S