Let $p$, $q$, $r$ be nonzero real numbers that are, respectively, the $10^{\text{th}}$, $100^{\text{th}}$ and $1000^{\text{th}}$ terms of a harmonic progression. Consider the system of linear equations $x + y + z = 1; 10x + 100y + 1000z = 0$; $qrx + pry + pqz = 0$. | List-I | List-II | | --- | --- | | (I) If $\frac{q}{r} = 10$, then the system of linear equations has | (P) $x = 0, y = \frac{10}{9}, z = -\frac{1}{9}$ as a solution | | (II) If $\frac{p}{r} \neq 100$, then the system of linear equations has | (Q) $x = \frac{10}{9}, y = -\frac{1}{9}, z = 0$ solution | | (III) If $\frac{p}{q} \neq 10$, then the system of linear equations has | (R) infinitely many solutions | | (IV) If $\frac{p}{q} = 10$, then the system of linear equations has | (S) no solution | | | (T) at least one solution |

$\text{Given: } p, q, r = 10^{\text{th}}, 100^{\text{th}}, 1000^{\text{th}} \text{ terms of an H.P.}$

$x + y + z = 1;\quad 10x + 100y + 1000z = 0;\quad qrx + pry + pqz = 0$

$\text{To find: match List-I with List-II; choose the correct option}$

$\Rightarrow \frac{1}{p}, \frac{1}{q}, \frac{1}{r} \text{ are in A.P.}$

$\frac{1}{q} - \frac{1}{p} = 90d, \qquad \frac{1}{r} - \frac{1}{q} = 900d$

$\Rightarrow \frac{1}{r} - \frac{1}{q} = 10\left(\frac{1}{q} - \frac{1}{p}\right)$

$u = \frac{1}{p},\quad v = \frac{1}{q},\quad w = \frac{1}{r} \Rightarrow w = 11v - 10u$

$10x + 100y + 1000z = 0 \Rightarrow x + 10y + 100z = 0$

qrx + pry + pqz = 0 ⇒ x/p + y/q + z/r = 0

$\text{System: } x + y + z = 1,\quad x + 10y + 100z = 0,\quad ux + vy + wz = 0$

Δ = 1 1 1 1 10 100 u v w = 90(w - 11v + 10u) = 0

$\text{Consistent} \iff (u,v,w) \parallel (1,10,100) \iff v = 10u$

$v = 10u \iff w = 10v \iff w = 100u$

$\frac{p}{q} = 10 \iff \frac{q}{r} = 10 \iff \frac{p}{r} = 100$

(P) (0, 10/9, -1/9): qrx+pry+pqz = p/9(10r - q) = 0 iff q/r = 10 (Q) (10/9, -1/9, 0): qrx+pry+pqz = r/9(10q - p) = 0 iff p/q = 10

$(I)\ \tfrac{q}{r}=10 \Rightarrow v=10u \Rightarrow \text{(Q) is a solution}$

$(II)\ \tfrac{p}{r}\ne100 \Rightarrow v\ne10u \Rightarrow \text{(S) no solution}$

(III) p/q≠10 ⇒ v≠10u ⇒ (S) no solution

$(IV)\ \tfrac{p}{q}=10 \Rightarrow v=10u \Rightarrow \text{(R) infinitely many}$

$\text{I}\to\text{Q},\quad \text{II}\to\text{S},\quad \text{III}\to\text{S},\quad \text{IV}\to\text{R}$

$\boxed{\text{option (2)}}$