Allied & ICAR Exam Bank • ALLIED_1019MCQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Systems of linear equations
Question Statement:
The system of equations $x + y + z = 6$, $x + 2y + 3z = 10$, $x + 2y + kz = \lambda$ has infinitely many solutions if:
Options:
(A) k = 3 \text{ and } \lambda = 10
(B) k = 3 \text{ and } \lambda \ne 10
(C) k \ne 3 \text{ and } \lambda = 10
(D) k \ne 3 \text{ and } \lambda \ne 10
Answer Key:
A
Step-by-Step Solution & Explanation
For infinitely many solutions, $\text{rank}(A) = \text{rank}(A|B) < 3$. Augmented matrix row reduction: $R_3 - R_2$ gives $(0, 0, k-3 | \lambda - 10)$. For the third row to vanish completely, we must have $k = 3$ and $\lambda = 10$.