Matrices & Determinants
System of linear equations
Grade Class 12
Question:
<p>Let α, β and γ be real numbers. consider the following system of linear equations</p><p>x + 2y + z = 7</p><p>x + αz = 11</p><p>2x - 3y + βz = γ</p><p>Match each entry in <b>List-I</b> to the correct entries in <b>List-II</b></p><table border="1"><tr><td><b>List-I</b></td><td><b>List-II</b></td></tr><tr><td>(P) If β = 1/2(7α - 3) and γ = 28, then the system has</td><td>(1) a unique solution</td></tr><tr><td>(Q) If β = 1/2(7α - 3) and γ ≠ 28, then the system has</td><td>(2) no solution</td></tr><tr><td>(R) If β ≠ 1/2(7α - 3) where α = 1 and γ ≠ 28, then the system has</td><td>(3) infinitely many solutions</td></tr><tr><td>(S) If β ≠ 1/2(7α - 3) where α = 1 and γ = 28, then the system has</td><td>(4) x = 11, y = -2 and z = 0 as a solution</td></tr><tr><td></td><td>(5) x = -15, y = 4 and z = 0 as a solution</td></tr></table>
(A) (P) → (3) (Q) → (2) (R) → (1) (S) → (4)
(B) (P) → (3) (Q) → (2) (R) → (5) (S) → (4)
(C) (P) → (2) (Q) → (1) (R) → (4) (S) → (5)
(D) (P) → (2) (Q) → (1) (R) → (1) (S) → (3)
Step-by-Step Solution
Key Concept: Analyze the determinant of the coefficient matrix and use Cramer's rule or Gaussian elimination to determine the nature of solutions based on the parameters \alpha, \beta, and \gamma.
The system is represented by the matrix equation AX = B. The determinant of the coefficient matrix is \Delta = |1 2 1; 1 0 \alpha; 2 -3 \beta| = 1(0 + 3\alpha) - 2(\beta - 2\alpha) + 1(-3 - 0) = 3\alpha - 2\beta + 4\alpha - 3 = 7\alpha - 2\beta - 3. For a unique solution, \Delta \neq 0, i.e., \beta \neq (7\alpha - 3)/2. If \Delta = 0, i.e., \beta = (7\alpha - 3)/2, the system has either no solution or infinitely many solutions depending on the consistency of the augmented matrix. By checking the conditions, (P) leads to infinitely many solutions (3), (Q) leads to no solution (2), (R) leads to a unique solution (1), and (S) leads to a unique solution (4).
Correct Answer: A