JEE Main - Test of consistency and solution of linear systems using matrices One Line Questions & Answers

1. Which of the following represents the augmented matrix for the system: x + y = 5, 2x - z = 1? — [ [1, 1, 0, 5], [2, 0, -1, 1] ]

2. Which matrix represents the augmented matrix for the system of equations 2x - y + 3z = 9, x + y - z = 2, 3x - y + 2z = 7? — [ [2, -1, 3, 9], [1, 1, -1, 2], [3, -1, 2, 7] ]

3. The system x = 1, y = 2, x + y = 3 can be represented in matrix form as: — [[1, 0], [0, 1], [1, 1]] [ [x], [y] ] = [ [1], [2], [3] ]

4. For the system x - y = 1, 2x - 2y = 2, the rank of the coefficient matrix A is: — 1

5. For a system of 3 linear equations in 3 variables, if det(A) = 0 and the system is consistent, how many free variables are there? — 1

6. Consider the system: x + 2y = 3, 2x + 4y = 5. The rank of the coefficient matrix A is: — 1

7. For the system x + y + z = 1, x - y + z = 1, x + y - z = 1. The determinant of the coefficient matrix is: — 2

8. For the system of equations: x + y = 3, 2x + 2y = 6. The determinant of the coefficient matrix is: — 0

9. Consider the system: x + y + z = 1, 2x + 2y + 2z = 2, 3x + 3y + 3z = 3. The rank of the augmented matrix [A|b] is: — 1

10. If det(A) is non-zero for a homogeneous system Ax = 0, then the system has: — Only the trivial solution (x=0, y=0, ...)

11. If the rank of matrix A is r, and the rank of the augmented matrix [A|b] is r+1, then the system of linear equations Ax = b has: — No solution

12. If rank(A) = rank([A|b]) = r, and r < n (where n is the number of variables), then the system Ax = b has: — Infinitely many solutions

13. Consider the system: x + y + z = 1, x + 2y + 3z = 4, 3x + 5y + 7z = 10. If rank(A) = 2 and rank([A|b]) = 3, the system has: — No solution

14. For a system of n linear equations in n variables, represented by Ax = b, if det(A) = 0 and the system is consistent, then it has: — Infinitely many solutions

15. If rank(A) = 1, rank([A|b]) = 1, and there are 3 variables, the system Ax = b has: — Infinitely many solutions

16. For a system of m linear equations in n variables, if rank(A) = m and rank([A|b]) = m, and m < n, the system has: — Infinitely many solutions

17. If rank(A) = r, rank([A|b]) = r, and r < number of variables, the system Ax = b has: — Infinitely many solutions

18. Consider the system: x + y + z = 6, x + 2y + 3z = 10, 3x + 5y + 7z = 26. If rank(A) = 2 and rank([A|b]) = 3, the system has: — No solution

19. For a system of n linear equations in n variables, if det(A) = 0, the system is guaranteed to have: — At least one solution

20. If rank(A) = r, rank([A|b]) = r, and r < n (number of variables), the system Ax = b has: — Infinitely many solutions

21. For a system of n equations in n unknowns, if the rank of the coefficient matrix is n-1 and the rank of the augmented matrix is also n-1, the system has: — Infinitely many solutions

22. In the context of linear systems, what does it mean for a system to be homogeneous? — All constants on the right-hand side are zero

23. If a system of linear equations has no solution, it is called: — Inconsistent

24. The system x = 0, y = 0 is an example of a: — Homogeneous system with only trivial solution

25. Consider the system: x + y = 2, x + y = 3. This system is: — Inconsistent

26. Which method is commonly used to determine the rank of a matrix for consistency checks? — Gaussian Elimination

27. The condition for a system of n linear equations in n variables to have a non-trivial solution is: — det(A) = 0

28. Which condition guarantees a unique solution for a system of n linear equations in n variables Ax = b? — det(A) != 0

29. Cramer's rule can be used to solve a system of linear equations Ax = b if: — det(A) != 0

30. For a homogeneous system of linear equations Ax = 0, where A is a square matrix, a non-trivial solution exists if and only if: — det(A) = 0

31. If det(A) = 0 and the system Ax = b is inconsistent, then rank(A) must be: — Less than rank([A|b])

32. If a system Ax = b has infinitely many solutions, then rank(A) is: — Less than the number of variables

33. A system of linear equations is said to be consistent if it has: — At least one solution

34. A system of linear equations Ax = b is consistent if rank(A) is: — Equal to rank([A|b])

35. The system of equations x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 20 is: — Consistent with infinitely many solutions

36. If rank(A) = rank([A|b]) = r, and r is the number of variables, the system Ax = b is: — Consistent with a unique solution

37. If a system of linear equations has infinitely many solutions, it is: — Consistent and dependent

38. If a system of linear equations Ax = b has a unique solution, then rank(A) must be: — Equal to the number of variables

39. The number of free variables in a consistent system of n linear equations in m variables, where rank(A) = r, is given by: — m - r

40. For a system of linear equations Ax = b, if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|b] and this rank is equal to the number of variables, then the system has: — A unique solution

41. Consider the system of linear equations: x + 2y = 5 and 2x + 4y = 10. The system has: — Infinitely many solutions

42. If rank(A) = rank([A|b]) = r and r = n (number of variables), the system Ax = b has: — A unique solution

43. If rank(A) = n (number of variables) and rank([A|b]) = n, the system Ax = b has: — A unique solution

44. If the system Ax = b is consistent and has a unique solution, then rank(A) must be equal to: — Number of variables

45. For a homogeneous system Ax = 0 where A is an n x n matrix, if rank(A) = n, then the system has: — Only the trivial solution

46. The system x + y + z = 0, 2x + 2y + 2z = 0, 3x + 3y + 3z = 0 has: — Infinitely many non-trivial solutions

47. The number of non-trivial solutions for a homogeneous system Ax = 0 with n variables and rank(A) = r is: — n - r

48. A system of linear equations is inconsistent if and only if: — rank(A) < rank([A|b])

49. If a system of linear equations Ax = b is consistent, what is the relationship between rank(A) and rank([A|b])? — rank(A) = rank([A|b])

50. Which of the following systems represents a homogeneous system of linear equations? — x + y = 0

Test of consistency and solution of linear systems using matrices - One Line Questions