Let $p, q, r$ be non-zero real numbers that are respectively the $10^{t h}, 100^{t h} a n d 1000^{t h}$ terms of a harmonic progression. Consider the system of linear equations:
$x + y + z = 1$ , $10 x + 100 y + 1000 z = 0$ , $(q r) x + (p r) y + (p q) z = 0$
Column-I Column-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) $x = \frac{10}{9}, y = - \frac{1}{9}, z = 0$ as a 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
The correct option is:

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$(I) - (T); (I I) - (S); (I I I) - (P); (I V) - (T)$

$(I) - (Q); (I I) - (R); (I I I) - (P); (I V) - (R)$

$(I) - (T); (I I) - (R); (I I I) - (S); (I V) - (T)$

$(I) - (Q); (I I) - (S); (I I I) - (S); (I V) - (R)$

answer is B.

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Detailed Solution

(I) $\frac{q}{r} = 10 \Rightarrow D = D_{x} = D_{y} = D_{z} = 0$
So, there will be infinitely many solutions and they will be of the type $(x, y, z) = (\frac{10}{9} + 10 λ, - \frac{1}{9} - 11 λ, λ)$ which gives (I)-(Q),(R),(T)
(II) $\frac{p}{r} \neq 100 \Rightarrow D_{y} \neq 0$
So, no solution which gives (II)-(S)
(III) $\frac{p}{q} \neq 10 \Rightarrow D_{z} \neq 0$
So, no solution which gives (III)-(S)
(IV) $\frac{p}{q} = 10 \Rightarrow D_{x} = D_{y} = D_{z} = D = 0$
So, infinitely many solutions which give (IV)-(Q),(R),(T)

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	Column-I		Column-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)	$x = \frac{10}{9}, y = - \frac{1}{9}, z = 0$ as a 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