JEE (Main) Mathematics: Test of Consistency and Solution of Simultaneous Linear Equations

Below are multiple-choice questions (MCQs) to help you practice and master the topic. The correct answers are indicated for each question.

If the determinant of the coefficient matrix is zero and the augmented matrix also has a determinant of zero, the system of equations is:
- a) Consistent and has a unique solution
- b) Inconsistent
- c) Consistent and has infinitely many solutions (Correct Answer)
- d) None of the above
The determinant of the coefficient matrix of a system of linear equations is non-zero. What can be inferred?
- a) The system has no solution
- b) The system is inconsistent
- c) The system has a unique solution (Correct Answer)
- d) The system has infinitely many solutions
For the system of equations \(x + y + z = 6\), \(2x + 3y + z = 10\), \(x + 2y + 3z = 14\), what is the rank of the coefficient matrix?
- a) 1
- b) 2
- c) 3 (Correct Answer)
- d) 0
A system of linear equations is consistent if:
- a) The rank of the coefficient matrix equals the rank of the augmented matrix (Correct Answer)
- b) The rank of the coefficient matrix is greater than the rank of the augmented matrix
- c) The determinant of the coefficient matrix is zero
- d) The equations are dependent
For a 3x3 system of equations, if the rank of the coefficient matrix is 2 and the rank of the augmented matrix is also 2, the system:
- a) Is inconsistent
- b) Has a unique solution
- c) Has infinitely many solutions (Correct Answer)
- d) Cannot be solved
If the determinant of the coefficient matrix is zero and the system of equations is consistent, the equations are:
- a) Dependent (Correct Answer)
- b) Independent
- c) Contradictory
- d) None of the above
In solving simultaneous linear equations using matrices, if \(AX = B\) and \(A^{-1}\) exists, the solution is given by:
- a) \(X = AB\)
- b) \(X = A^{-1}B\) (Correct Answer)
- c) \(X = B^{-1}A\)
- d) \(X = BA^{-1}\)
A system of equations has an infinite number of solutions when:
- a) The determinant of the coefficient matrix is non-zero
- b) The rank of the coefficient matrix equals the rank of the augmented matrix but is less than the number of unknowns (Correct Answer)
- c) The equations are independent
- d) None of the above
For the system \(2x + y + z = 3\), \(x - y + z = 2\), and \(3x + 2y + 2z = 5\), the augmented matrix is:
- a) \(\begin{bmatrix} 2 & 1 & 1 \\ 1 & -1 & 1 \\ 3 & 2 & 2 \end{bmatrix}\)
- b) \(\begin{bmatrix} 2 & 1 & 1 & 3 \\ 1 & -1 & 1 & 2 \\ 3 & 2 & 2 & 5 \end{bmatrix}\) (Correct Answer)
- c) \(\begin{bmatrix} 3 & 2 & 1 & 5 \\ 1 & 1 & -1 & 2 \\ 2 & 3 & 1 & 5 \end{bmatrix}\)
- d) None of the above
If a system of equations is inconsistent, what is true about the rank of the coefficient matrix and augmented matrix?
- a) They are equal
- b) They are unequal (Correct Answer)
- c) Both are zero
- d) None of the above
For a consistent system of equations, if there are more equations than variables, what can be said about the solutions?
- a) No solution
- b) Infinite solutions (Correct Answer)
- c) Unique solution
- d) None of the above
The condition for consistency of a system of linear equations \(AX = B\) is:
- a) Rank(A) = Rank(A|B) (Correct Answer)
- b) Determinant of A is zero
- c) Rank(A) < Rank(A|B)
- d) None of the above
What happens if all equations in a system are multiples of each other?
- a) The system has a unique solution
- b) The system is inconsistent
- c) The system has infinitely many solutions (Correct Answer)
- d) None of the above
For the system \(x + y = 2\) and \(2x + 2y = 4\), the rank of the coefficient matrix is:
- a) 0
- b) 1 (Correct Answer)
- c) 2
- d) 3
If the determinant of a 2x2 coefficient matrix is zero, the system of equations is:
- a) Consistent with a unique solution
- b) Inconsistent
- c) Consistent with infinitely many solutions (Correct Answer)
- d) None of the above

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JEE (Main) Mathematics: Test of Consistency and Solution of Simultaneous Linear Equations

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