System of Linear Equations MCQ Quiz - Objective Question with Answer for System of Linear Equations - Download Free PDF
Last updated on Apr 5, 2025
Latest System of Linear Equations MCQ Objective Questions
System of Linear Equations Question 1:
Consider the following system of equations in three real variables x, y, z.
2x – 3y + 7z = 5
3x + y – 3z = 13
2x + 19y – 47z = 32
The system of equation has
A. No Solution
B. A Unique Solution
C. More than one but finite number of solutions
D. More than two but finite number of solutions
Answer (Detailed Solution Below)
System of Linear Equations Question 1 Detailed Solution
Explanation:
Augmented matrix will be
R3 → R3 – R1
R2 → 2R2 – 3R1
R3 → R3 – 2R2
Rank of A ≠ Rank of Augmented matrix.
Hence given system of equations has no solution.
Hence Option(5) is the correct answer.
System of Linear Equations Question 2:
Consider the following system of linear equations
x + y + 5z = 3, x + 2y + mz = 5 and x + 2y + 4z = k.
The system is consistent if
Answer (Detailed Solution Below)
System of Linear Equations Question 2 Detailed Solution
Explanation:
Consider the augmented matrix
Applying row operations:
The system is consistent if and only if
If
Hence, option (1) is true
If k = 5 , then
Hence, option (4) is true.
If
and the system will be inconsistent.
Hence, options (1) and (4) are correct.
System of Linear Equations Question 3:
What will be the conditions on p, q, r so that the linear system of equations
x + 2y – 3z = p
2x + 6y – 11z = q
x – 2y + 7z = r
is consistent
Answer (Detailed Solution Below)
System of Linear Equations Question 3 Detailed Solution
Concept:
Consistency means, the system of equations should have either infinite solutions or unique solution
Explanation:
For Consistent solution, the system of equations should have either infinite solutions or unique solution
R2 → R2 - 2R1
R3 → R3 – R1
R3 → R3 + 2R2
For consistency ρ [A/b] = ρ [A]
⇒ -5p + 2q + r = 0
Hence Option(3) is the correct answer.
System of Linear Equations Question 4:
Consider the following system of equations in three real variables x, y, z.
2x – 3y + 7z = 5
3x + y – 3z = 13
2x + 19y – 47z = 32
The system of equation has
A. No Solution
B. A Unique Solution
C. More than one but finite number of solutions
D. More than two but finite number of solutions
Answer (Detailed Solution Below)
System of Linear Equations Question 4 Detailed Solution
Explanation:
Augmented matrix will be
R3 → R3 – R1
R2 → 2R2 – 3R1
R3 → R3 – 2R2
Rank of A ≠ Rank of Augmented matrix.
Hence given system of equations has no solution.
Hence Option(1) is the correct answer.
System of Linear Equations Question 5:
For what value of k, the equations
x + 4y + z = a,
2x - 3y – 1z = – 3, and
5x – 13y + k z = 9
will not have a unique solution?
Answer (Detailed Solution Below)
System of Linear Equations Question 5 Detailed Solution
Given:
x + 4y + z = a,
2x - 3y – z = – 3, and
5x – 13y + k z = 9
Concept Used:
The given system of equation will not have unique solution if determinant of the coefficient of x, y, and z is equal to zero.
Solution:
D =
⇒ (-3k - 13) - (4) (2k + 5) + (- 26 + 15) = 0
⇒ -3k - 13 - 8k - 20 - 11 = 0
⇒ -11k - 44 = 0
⇒ k = -4
Top System of Linear Equations MCQ Objective Questions
System of Linear Equations Question 6:
For what value of k, the equations
x + 4y + z = a,
2x - 3y – 1z = – 3, and
5x – 13y + k z = 9
will not have a unique solution?
Answer (Detailed Solution Below)
System of Linear Equations Question 6 Detailed Solution
Given:
x + 4y + z = a,
2x - 3y – z = – 3, and
5x – 13y + k z = 9
Concept Used:
The given system of equation will not have unique solution if determinant of the coefficient of x, y, and z is equal to zero.
Solution:
D =
⇒ (-3k - 13) - (4) (2k + 5) + (- 26 + 15) = 0
⇒ -3k - 13 - 8k - 20 - 11 = 0
⇒ -11k - 44 = 0
⇒ k = -4
System of Linear Equations Question 7:
What will be the conditions on p, q, r so that the linear system of equations
x + 2y – 3z = p
2x + 6y – 11z = q
x – 2y + 7z = r
is consistent
Answer (Detailed Solution Below)
System of Linear Equations Question 7 Detailed Solution
Concept:
Consistency means, the system of equations should have either infinite solutions or unique solution
Explanation:
For Consistent solution, the system of equations should have either infinite solutions or unique solution
R2 → R2 - 2R1
R3 → R3 – R1
R3 → R3 + 2R2
For consistency ρ [A/b] = ρ [A]
⇒ -5p + 2q + r = 0
Hence Option(3) is the correct answer.
System of Linear Equations Question 8:
Consider the following system of equations in three real variables x, y, z.
2x – 3y + 7z = 5
3x + y – 3z = 13
2x + 19y – 47z = 32
The system of equation has
A. No Solution
B. A Unique Solution
C. More than one but finite number of solutions
D. More than two but finite number of solutions
Answer (Detailed Solution Below)
System of Linear Equations Question 8 Detailed Solution
Explanation:
Augmented matrix will be
R3 → R3 – R1
R2 → 2R2 – 3R1
R3 → R3 – 2R2
Rank of A ≠ Rank of Augmented matrix.
Hence given system of equations has no solution.
Hence Option(5) is the correct answer.
System of Linear Equations Question 9:
Consider the following system of linear equations
x + y + 5z = 3, x + 2y + mz = 5 and x + 2y + 4z = k.
The system is consistent if
Answer (Detailed Solution Below)
System of Linear Equations Question 9 Detailed Solution
Explanation:
Consider the augmented matrix
Applying row operations:
The system is consistent if and only if
If
Hence, option (1) is true
If k = 5 , then
Hence, option (4) is true.
If
and the system will be inconsistent.
Hence, options (1) and (4) are correct.
System of Linear Equations Question 10:
Consider the following system of equations in three real variables x, y, z.
2x – 3y + 7z = 5
3x + y – 3z = 13
2x + 19y – 47z = 32
The system of equation has
A. No Solution
B. A Unique Solution
C. More than one but finite number of solutions
D. More than two but finite number of solutions
Answer (Detailed Solution Below)
System of Linear Equations Question 10 Detailed Solution
Explanation:
Augmented matrix will be
R3 → R3 – R1
R2 → 2R2 – 3R1
R3 → R3 – 2R2
Rank of A ≠ Rank of Augmented matrix.
Hence given system of equations has no solution.
Hence Option(1) is the correct answer.
System of Linear Equations Question 11:
Consider the following system of linear equations x + y + 5z = 3, x + 2y + mz = 5 and x + 2y + 4z = k. The system is consistent if
(A). m ≠ 4
(B). k ≠ 5, m = 4
(C). m = 4, k = 1
(D). m= 4, k = 5
Choose the correct answer from the options given below:
Answer (Detailed Solution Below)
System of Linear Equations Question 11 Detailed Solution
Explanation:
A system of linear equations is considered consistent if it has at least one solution.
Augmented Matrix:
Row Operation:
This gives us:
For the system to be consistent, the last row must not represent an inconsistent equation
If -m + 4 ≠ 0 (i.e., m ≠ 4), then we can solve for z in the last row and back-substitute to find x and y
If -m + 4 = 0 and k - 5 = 0 (i.e., m = 4 and k = 5), then the last row becomes 0 = 0, which is always true
The system has infinitely many solutions
If -m + 4 = 0 and k - 5 ≠ 0 (i.e., m = 4 and k ≠ 5),
then the last row represents an inconsistent equation (0 = a non-zero value), and the system has no solutions
Therefore, the system is consistent if:
1. m ≠ 4
2. m = 4 and k = 5
⇒ the correct options are (A) and (D)
Hence Option(2) is the correct answer.