NIMCET 2012 — Mathematics PYQ

Step 1: Understand the nature of the system

The given equations are a system of homogeneous linear equations of the form $AX=0$:

1. $x+{\omega }^{2}y+\omega z=0$

2. $\omega x+y+{\omega }^{2}z=0$

3. ${\omega }^{2}x+\omega y+z=0$

A homogeneous system is always consistent because it always has at least the trivial solution $(0,0,0)$.

Step 2: Calculate the determinant of the coefficient matrix

To check for unique vs. infinitely many solutions, we find the determinant $\Delta$ of the coefficient matrix:

Step 3: Simplify the determinant

Apply the operation $R_1\to R_1+R_2+R_3$:

We know that for the cube roots of unity, $1+\omega +{\omega }^2=0$. Substituting this:

Since an entire row consists of zeros, the determinant is:

Step 4: Interpret the result

• If $\Delta \ne 0$, the system has only a unique (trivial) solution.

• If $\Delta =0$, the system has infinitely many solutions (non-trivial solutions).

Since $\Delta =0$, the system is consistent and has infinitely many solutions (more than one solution).

Conclusion:

The system is consistent and has more than one solution.

Correct Option: (b)