1. The solution of the system of equations

x + y + z = 4

x – y + z = 0

2x + y + z = 5 is

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

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

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

Which one of the following is correct?

- i, ii and iii are true
- only ii and iii are true
- only iii is true
- none of them is true

3. The solution to the system of equations is

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

- 6, 2
- -6, 2
- -6, -2
- 6, -2

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

- 7a – b – c = 0
- 3a + b – c = 0
- 3a – b + c = 0
- 7a – b + c = 0

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

- r
- n
- n – r
- n + r

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

px + qy + rz = 0

qx + ry + pz = 0

rx + py + qz = 0

- p – q + r = 0 or p = q = -r
- p + q – r = 0 or p = -q = r
- p + q + r = 0 or p = q = r
- p – q + r = 0 or p = -q = -r

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
- 2
- 3
- 4

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

2x + 3y = 5

3x + py = 10

- 1
- 2.5
- 3.5
- 4.5

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
- 2
- 3
- 4

10. The system of linear equations

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

- a unique solution
- infinitely many solutions
- no solution
- exactly two solutions

11. Given a system of equations:

x + 2y + 2z = b_{1}

5x + y + 3z = b_{2}

Which of the following is true regarding its solution?

- The system has a unique solution for any given b
_{1}and b_{2} - The system will have infinitely many solutions for any given b
_{1}and b_{2} - Whether or not a solution exists depends on the given b
_{1}and b_{2} - The system would have no solution for any values of b
_{1}and b_{2}

12.

x + 2y + z = 4

2x + y + 2z = 5

x – y + z = 1

The system of algebraic given below has

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

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

- λ = 6, μ = 20
- λ = 6, μ ≠ 20

- λ ≠ 6, μ = 20

- λ ≠ 6, μ ≠ 20

14. Consider the following system of equations

2x_{1} + x_{2} + x_{3} = 0

x_{2} – x_{3} = 0

x_{1} + x_{2} = 0

This system has

- a unique solution
- no solution
- infinite number of solutions
- five solutions

15. For the set of equations

x_{1} + 2x_{2} + x_{3} + 4x_{4} = 2

3x_{1} + 6x_{2} + 3x_{3} + 12x_{4} = 6

the following statement is true

- only the trivial solution x
_{1}= x_{2}= x_{3}= x_{4}= 0 exists - there is no solution
- a unique non-trivial solution exists
- multiple non-trivial solutions exists