System of Linear Equations Matrix Method MCQ

1. The solution of the system of equations

x + y + z = 4
x – y + z = 0
2x + y + z = 5 is

  1. x = 2, y = 2, z = 0
  2. x = 1, y = 4, z = 1
  3. x = 2, y = 4, z = 3
  4. x = 1, y = 2, z = 1

Answer
Answer. d

2. Consider the systems, each consisting of m inear equations in n variables.

  1. if m < n, then all such systems have a solution.
  2. if m > n, then none of these systems has a solution.
  3. if m = n, then there exists a system which has a solution

Which one of the following is correct?

  1. i, ii and iii are true
  2. only ii and iii are true
  3. only iii is true
  4. none of them is true

Answer
Answer. c

3. The solution to the system of equations is

    \[\begin{bmatrix} 2 &5 \\ -4 &3 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}=\begin{bmatrix} 2\\ -30 \end{bmatrix}\]

  1. 6, 2
  2. -6, 2
  3. -6, -2
  4. 6, -2

Answer
Answer. d

4. Consider the following linear system.

x + 2y – 3z = a
2x + 3y + 3z = b
5x + 9y – 6z = c

This system is consistent if a, b and c satisfy the equation

  1. 7a – b – c = 0
  2. 3a + b – c = 0
  3. 3a – b + c = 0
  4. 7a – b + c = 0

Answer
Answer. b

5. Let A be an n x n matrix with rank r (o < r < n). Then AX = 0 has p independent solutions, where p is

  1. r
  2. n
  3. n – r
  4. n + r

Answer
Answer. c

6. If the following system has non-trivial solution,

px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0

  1. p – q + r = 0 or p = q = -r
  2. p + q – r = 0 or p = -q = r
  3. p + q + r = 0 or p = q = r
  4. p – q + r = 0 or p = -q = -r

Answer
Answer. c

7. Consider a system of linear equations:

x – 2y + 3z = -1
x – 3y + 4z = 1
-2x + 4y – 6z = k

The value of k for which the system has infinitely many solution is

  1. 1
  2. 2
  3. 3
  4. 4

Answer
Answer. b

8. For what value of p the following set of equations will have no solution?

2x + 3y = 5
3x + py = 10

  1. 1
  2. 2.5
  3. 3.5
  4. 4.5

Answer
Answer. d

9. Consider the following system of equations

3x + 2y = 1
4x + 7z = 1
x + y + z =3
x – 2y + 7z = 0

The number of solutions for this system is

  1. 1
  2. 2
  3. 3
  4. 4

Answer
Answer. a

10. The system of linear equations

    \[\begin{bmatrix} 2 & 1 & 3\\ 3 & 0 & 1\\ 1 & 2 & 5 \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix}=\begin{bmatrix} 5\\ -4\\ 14 \end{bmatrix}\]

  1. a unique solution
  2. infinitely many solutions
  3. no solution
  4. exactly two solutions

Answer
Answer. b

11. Given a system of equations:

x + 2y + 2z = b1
5x + y + 3z = b2

Which of the following is true regarding its solution?

  1. The system has a unique solution for any given b1 and b2
  2. The system will have infinitely many solutions for any given b1 and b2
  3. Whether or not a solution exists depends on the given b1 and b2
  4. The system would have no solution for any values of b1 and b2

Answer
Answer. b

12. 

x + 2y + z = 4
2x + y + 2z = 5
x – y + z = 1

The system of algebraic given below has

  1. a unique solution of x = 1, y = 1 and z = 1
  2. only the two solutions of ( x = 1, y = 1,  z = 1) and ( x = 2, y = 1,  z = 0)
  3. infinite number of solutions
  4. no feasible solution

Answer
Answer. c

13. The system of equations

x + y + z = 6
x + 4y + 6z = 20
x + 4y + λz = μ

has no solution for values of λ and μ given by 

  1. λ = 6, μ = 20
  2. λ = 6, μ ≠ 20
  3. λ ≠ 6, μ = 20
  4. λ ≠ 6, μ ≠ 20

Answer
Answer. b

14. Consider the following system of equations

2x1 + x2 + x3 = 0
x2 – x3 = 0
x1 + x2 = 0

This system has

  1. a unique solution
  2. no solution
  3. infinite number of solutions
  4. five solutions

Answer
Answer. c

15. For the set of equations

x1 + 2x2 + x3 + 4x4 = 2
3x1 + 6x2 + 3x3 + 12x4 = 6

the following statement is true

  1. only the trivial solution x1 = x2 = x3 = x4 = 0 exists
  2. there is no solution
  3. a unique non-trivial solution exists
  4. multiple non-trivial solutions exists

Answer
Answer. d

error: Content is protected !!