The GATE Grind

GATE 2025 CS (CS1) – Question 23

Engineering Mathematics · Linear Algebra · 1 mark · Multiple select

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?

$x + ky = 1$
$kx + y = -1$

  1. There is exactly one value of $k$ for which the above system of equations has no solution.
  2. There exist an infinite number of values of $k$ for which the system of equations has no solution.
  3. There exists exactly one value of $k$ for which the system of equations has exactly one solution.
  4. There exists exactly one value of $k$ for which the system of equations has an infinite number of solutions.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) There is exactly one value of $k$ for which the above system of equations has no solution.; (D) There exists exactly one value of $k$ for which the system of equations has an infinite number of solutions.

Explanation

The determinant is 1-k^2. For k=1 the equations are x+y=1 and x+y=-1, so there is no solution. For k=-1 they are x-y=1 and -x+y=-1, so there are infinitely many solutions. For all other k the solution is unique.