NIMCET 2025 — Mathematics PYQ

The cube roots of 27 are the values that satisfy the equation $a^3=27$. Using the complex cube roots of unity ($1,\omega ,{\omega }^2$), these roots are:

• $x=3(1)=3$

• $y=3\omega$

• $z=3{\omega }^2$

2. Property of the Roots

A fundamental property of the cube roots of unity is that their sum is zero ($1+\omega +{\omega }^2=0$). Therefore:

3. Expansion of the Determinant

The determinant of a circulant matrix $\left[\begin{array}{ccc}x& amp;y& amp;z\\ y& amp;z& amp;x\\ z& amp;x& amp;y\end{array}\right]$ is given by the formula:

Alternatively, using row operations ($R_1\to R_1+R_2+R_3$):

Since we found that $x+y+z=0$, we substitute this value into the first row:

4. Final Value

If any row or column of a determinant consists entirely of zeros, the value of the determinant is $0$.

Correct Option: 2