JEE Math Practice Question

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 \[ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} \] List-I (I) If $\frac{q}{r}=10$, then the system of linear equations has (II) If $\frac{p}{r} \neq 100$, then the system of linear equations has (III) If $\frac{p}{q} \neq 10$, then the system of linear equations has (IV) If $\frac{p}{q}=10$, then the system of linear equations has List-II (P) $x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution (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 (R) infinitely many solutions (IV) If $\frac{p}{q}=10$, then the system of linear (S) no solution (T) at least one solution The correct option is:

1. A.(I) $\rightarrow$ (T); (II) $\rightarrow$ (R); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (T)
2. B.(I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (S); (IV) $\rightarrow$ (R)
3. C.(I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (R)
4. D.(I) $\rightarrow$ (T); (II) $\rightarrow$ (S); (III) $\rightarrow$ (P); (IV) $\rightarrow$ (T)