Consider a system equation

,

and

where and , then identify the correct statement(s).

	List-I	List-II
(I)	Number of ordered pairs for which system of equation has unique solution is	(P) 1
(II)	Number of ordered pairs for which system of equation has no solution is	(Q) 9
(III)	Number of ordered pairs for which system of equation has infinite solution is	(R) 91
(IV)	Number of ordered pairs for which system of equation has atleast one solution is	(S) 90

Calculation:

Given:

System of linear equations:

 and 

The coefficient matrix A is:

The augmented matrix [A|B] is:

Calculate the determinant of A, |A|:

⇒ For unique solution, :

For no solution or infinite solutions, , so .

⇒ If , the system becomes:

⇒ From the second and third equations, for a solution to exist, .

⇒ If and , infinite solutions exist.

⇒ If and , no solution exists.

(I) Unique solution: . can take 9 values (1 to 10 except 4). can take 10 values. Total pairs: 9 × 10 = 90.

(II) No solution: and . 9 values for . 1 pair for . Total pairs: 1 × 9 = 9.

(III) Infinite solutions: and . 1 pair only.

(IV) At least one solution: Total pairs - No solution pairs = 100 - 9 = 91.

∴ (I) - (S), (II) - (Q), (III) - (P), (IV) - (R)