Question
Let p , q , r be non-zero real numbers that are, respectively, the 10 th   , 100 th   and 1000 th   terms of a harmonic progression. Consider the system of linear equations x + y + z = 1 10 x + 100 y + 1000 z = 0 q r x + p r y + p q z = 0 List- I List- II I If q r = 10 , then the system of linear equations has P x = 0 , y = 10 9 , z = - 1 9 as a solution II If p r ≠ 100 , then the system of linear equations has Q x = 10 9 , y = - 1 9 , z = 0 as a solution III If p q ≠ 10 , then the system of linear equations has R infinitely many solutions IV If p q = 10 , then the system of linear equations has S no solution T at least one solution The correct option is:
Step-by-step solution
Given, x + y + z = 1             ⋯ 1 10 x + 100 y + 1000 z = 0             ⋯ 2 q r x + p r y + p q z = 0               ⋯ 3 Now equation 3 can be re-written as x p + y q + z r = 0         ∵   p , q , r ≠ 0 Now given p ,   q   &   r  are 10 t h , 100 t h  & 1000 t h  term of an h.p , So, let p = 1 a + 9 d ,   q = 1 a + 99 d   &   r = 1 a + 999 d Now, equation 3 will be a + 9 d x + a + 99 d y + a + 999 d z = 0 Now from equation 1 ,   2   &   3 we get, Δ = 1 1 1 10 100 1000 a + 9 d a + 99 d a + 999 d = 0 Δ x = 1 1 1 0 100 1000 0 a + 99 d a + 999 d = 900 d - a Δ y = 1 1 1 10 0 1000 a + 9 d 0 a + 999 d = 990 a - d Δ z = 1 1 1 10 100 0 a + 9 d a + 99 d 0 = 90 d - a Option I: If q r = 10 ⇒ a = d Δ = Δ x = Δ y = Δ z = 0 And eq. 1 and eq. 2 represents non-parallel planes eq. 2 and eq. 3 represents same plane ⇒ Infinitely many solutions Now finding solution by taking z = λ so from equation 1   &   2 we get, x + y = 1 - λ and x + 10 y = - 100 λ ⇒ x = 10 9 + 10 λ ,   y = - 1 9 - 11 λ ⇒ x , y , z ∈ 10 9 + 10 λ ,   - 1 9 - 11 λ ,   λ So, P is not valid for any value of λ rest are valid. So, option (i) → Q , R , T Option II: p r ≠ 100 ⇒ a ≠ d Δ = 0   &   Δ x , Δ y , Δ z ≠ 0 So, no solution Option (ii) → S Option (iii): If p q ≠ 10 ⇒ a ≠ d then Δ z ≠ 0 So, no solution Option (iii) → S Now option (iv): If p q = 10 ⇒ a = d then Δ z = 0 ⇒ Δ x = Δ y = 0 So, infinitely many solutions Option (iv) → Q , R , T similar to option (i)