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

Official 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: To analyze the system of linear equations: bmatrix 1 & k \\ k & 1 bmatrix bmatrix x \\ y bmatrix = bmatrix 1 \\ -1 bmatrix The determinant of the coefficient matrix A is: (A) = 1 - k2 = (1 - k)(1 + k) 1. Unique solution: If (A) 0, which means k 1 and k -1, the system has a unique solution. Since this holds for infinitely many real values of k, statement (C) is incorrect. 2. Case k = 1: The equations become: x + y = 1 x + y = -1 These two parallel lines never intersect, so there is no solution. Thus, k = 1 is the only value for which the system has no solution, making statement (A) correct, and statement (B) incorrect. 3. Case k = -1: The equations become: x - y = 1 -x + y = -1 x - y = 1 Both equations represent the exact same line, resulting in infinitely many solutions. Thus, k = -1 is the only value for which the system has infinitely many solutions, making statement (D) correct. Therefore, the correct options are A and D.

A) There is exactly one value of k for which the above system of equations has no solution.

B) There exist an infinite number of values of k for which the system of equations has no solution.

C) There exists exactly one value of k for which the system of equations has exactly one solution.

D) There exists exactly one value of k for which the system of equations has an infinite number of solutions.

Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 1 Q23 · MSQ · 1 mark

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

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 1 Q23 · MSQ · 1 mark

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

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

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

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 1

More on Linear Algebra

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

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 1

More on Linear Algebra

Consider the given system of linear equations for variables xxx and yyy, where kkk is a real-valued constant. Which of the following option(s) is/are CORRECT? x+ky=1x + ky = 1x+ky=1 kx+y=−1kx + y = -1kx+y=−1

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 1

More on Linear Algebra

Share this solution

Consider the given system of linear equations for variables xxx and yyy, where kkk is a real-valued constant. Which of the following option(s) is/are CORRECT? x+ky=1x + ky = 1x+ky=1 kx+y=−1kx + y = -1kx+y=−1

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2025 Set 1

More on Linear Algebra

Share this solution

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

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