Let p,q,r be non-zero real numbers that are, respectively, the 10th ,100th and 1000th 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 qr=10, then the system of linear equations has P x=0,y=109,z=-19 as a solution II If pr≠100, then the system of linear equations has Q x=109,y=-19,z=0 as a solution III If pq≠10, then the system of linear equations has R infinitely many solutions IV If pq=10, then the system of linear equations has S no solution T at least one solution The correct option is:

(B) I→Q;II→S;III→S;IV→R

(A) I→T;II→R;III→S;IV→T

(B) I→Q;II→S;III→S;IV→R

(C) I→Q;II→R;III→P;IV→R

(D) I→T;II→S;III→P;IV→T

JEE Advanced · Mathematics · 19. Determinants

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 $\frac{q}{r} = 10$ , 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:

A $(I) \to (T); (II) \to (R); (III) \to (S); (IV) \to (T)$
B $(I) \to (Q); (II) \to (S); (III) \to (S); (IV) \to (R)$
C $(I) \to (Q); (II) \to (R); (III) \to (P); (IV) \to (R)$
D $(I) \to (T); (II) \to (S); (III) \to (P); (IV) \to (T)$

Verified Solution

Answer & Solution

Correct Answer

(B) $(I) \to (Q); (II) \to (S); (III) \to (S); (IV) \to (R)$

Step-by-step Solution

Detailed explanation

Given, $x + y + z = 1 \dots (1)$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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	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	$(Q)$	$x = \frac{10}{9}, y = - \frac{1}{9}, z = 0$ 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