NIMCET 2013 — Mathematics PYQ

Step 1: Recall the properties of cube roots of unity

For a complex cube root of unity $\omega$:

• $1+\omega +{\omega }^2=0$

• ${\omega }^3=1$

From the first property, we can derive:

• $1+\omega =-{\omega }^2$

• $1+{\omega }^2=-\omega$

• ${\omega }^2+\omega =-1$

Step 2: Substitute these values into the determinant

Let the determinant be $\Delta$. Substituting the derived values into the first column:

Step 3: Simplify using row operations

Observe that Row 2 ($R_2$) and Row 3 ($R_3$) have identical entries in the second and third columns. We can perform $R_2\to R_2-R_3$:

Step 4: Expand along the second row

Expanding along $R_2$:

Step 5: Final simplification using ${\omega }^3=1$

Since ${\omega }^4={\omega }^3\cdot \omega =1\cdot \omega =\omega$:

Multiply the terms:

Substitute ${\omega }^3=1$:

Since $1+\omega =-{\omega }^2$:

Correct Option: 4. $-3{\omega }^2$