GATE CS 2025 Set 1 | Question 23

Consider the given system of linear equations for variables \(x\) and \(y\), where \(k\) is a real-valued constant. Which of the following option(s) is/are CORRECT?

\(x + ky = 1\)
\(kx + y = -1\)

Option A: There is exactly one value of \(k\) for which the above system of equations has no solution. (correct)
Option B: There exist an infinite number of values of \(k\) for which the system of equations has no solution.
Option C: There exists exactly one value of \(k\) for which the system of equations has exactly one solution.
Option D: There exists exactly one value of \(k\) for which the system of equations has an infinite number of solutions. (correct)

Answer

(A), (D)

Explanation

The determinant of the coefficient matrix is \(1 \cdot 1 - k \cdot k = 1 - k^{2}\). For every \(k \ne \pm 1\) it is non-zero, so the system has exactly one solution.

For \(k = 1\): \(x + y = 1\) and \(x + y = -1\) contradict each other, so there is no solution. For \(k = -1\): \(x - y = 1\) and \(-x + y = -1\) are the same equation, so there are infinitely many solutions.

(A) True: only \(k = 1\) gives no solution.
(B) False: no solution happens for \(k = 1\) alone.
(C) False: infinitely many values (every \(k \ne \pm 1\)) give exactly one solution.
(D) True: only \(k = -1\) gives infinitely many solutions.

Answer: A, D