The system of equations 2x+6y+11=0, 6x+20y−6z+3=0 and 6y−18z+1=0 will have:Consistent with unique solution.Consistent with infinitely many solution.Inconsistent.Data insufficient to give the answer.

To determine the nature of the solutions of the system of equations, we can use the determinant method. The determinant of the coefficient matrix of the system of equations is used to determine whether the system has a unique solution, infinitely many solutions, or no solutions.

The system of equations is:

2x + 6y + 0z - 11 = 0 6x + 20y - 6z - 3 = 0 0x + 6y - 18z - 1 = 0

The coefficient matrix is:

| 2 6 0 | | 6 20 -6 | | 0 6 -18 |

The determinant of the coefficient matrix is:

2*(20*-18 - -66) - 6(6*-18 - -60) + 0(6*-6 - 20*0)

= 2*(-360 - -36) - 6*(-108 - 0) + 0 = 2*-324 - 6*-108 = -648 + 648 = 0

Since the determinant of the coefficient matrix is 0, the system of equations does not have a unique solution.

Now, let's check for the condition of infinite solutions or no solution. For this, we form the augmented matrix and check its rank with the coefficient matrix.

The augmented matrix is:

| 2 6 0 -11 | | 6 20 -6 -3 | | 0 6 -18 -1 |

If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has infinitely many solutions. If not, the system is inconsistent and has no solution.

In this case, the rank of both the coefficient matrix and the augmented matrix is 2 (since the third row can be expressed as a linear combination of the first two rows), so the system has infinitely many solutions.

So, the correct answer is: Consistent with infinitely many solutions.