Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Consider a system of linear equations <i>a</i><sub>i</sub><i>x</i> + <i>b</i><sub>i</sub><i>y</i> + <i>c</i><sub>i</sub><i>z</i> = <i>d</i><sub>i</sub> (where <i>a</i><sub>i</sub>, <i>b</i><sub>i</sub>, <i>c</i><sub>i</sub> ≠ 0 and <i>i</i> = 1,2,3 ) & (α,β,γ) is its unique solution, then match list-I with list-II<br><b>List-I</b><br>(I) If <i>a</i><sub>i</sub> = <i>d</i><sub>i</sub> = <i>k</i><sup>2</sup>, (<i>k</i> ≠ 0) and α + β + γ = 2, then <i>k</i> is<br>(II) If <i>a</i><sub>i</sub> = <i>d</i><sub>i</sub> = <i>k</i> ≠ 0, then α + β + γ is<br>(III) If <i>a</i><sub>i</sub> = <i>k</i> > 0, <i>d</i><sub>i</sub> = <i>k</i> + 1, then α + β + γ can be<br>(IV) If <i>a</i><sub>i</sub> = <i>k</i> < 0, <i>d</i><sub>i</sub> = <i>k</i> + 1, then α + β + γ can be<br><b>List-II</b><br>(P) 1<br>(Q) 2<br>(R) 0<br>(S) 3<br>(T) -1
(A) I → P,Q; II → R; III → Q, S; IV → T
(B) I → P; II → Q; III → R, S; IV → T
(C) I → Q; II → P; III → S; IV → T
(D) I → Q; II → P; III → Q, S; IV → R, T
Step-by-Step Solution
Key Concept: For a system of linear equations with a unique solution, if the system is homogeneous or has a specific structure, the solution can be determined by inspection or by using Cramer's rule properties.
For a system of linear equations <i>a</i><sub>i</sub><i>x</i> + <i>b</i><sub>i</sub><i>y</i> + <i>c</i><sub>i</sub><i>z</i> = <i>d</i><sub>i</sub>, if <i>a</i><sub>i</sub> = <i>d</i><sub>i</sub>, then (1, 1, 1) is a solution. Since the solution is unique, (\alpha, \beta, \gamma) = (1, 1, 1). Thus, \alpha + \beta + \gamma = 3. For (I), <i>k</i><sup>2</sup> = 1 implies <i>k</i> = 1 or -1. For (II), \alpha + \beta + \gamma = 3. For (III) and (IV), the system properties lead to the matched values.
Correct Answer: 3