CUET PG 2023 — Mathematics PYQ

Step 1: Split the Determinant

We can split the given determinant into two separate determinants based on the third column:

Step 2: Simplify the second Determinant

In the second determinant, we can take $x$ common from Row 1 ($R_1$), $y$ from $R_2$, and $z$ from $R_3$:

Step 3: Align the Determinants

To make both determinants look identical, we rearrange the columns of the first one.

1. Swap $C_2$ and $C_3$ (changes sign to negative).

2. Swap $C_1$ and $C_2$ (changes sign back to positive).

Now the equation becomes:

Step 4: Factorize

Taking the determinant common:

The determinant $\left|\begin{array}{ccc}1& amp;x& amp;x^2\\ 1& amp;y& amp;y^2\\ 1& amp;z& amp;z^2\end{array}\right|$ is a standard Vandermonde determinant, which equals $(x-y)(y-z)(z-x)$.

Since $x,y,z$ are all distinct, we know that:

Therefore, the determinant itself cannot be zero. This leaves us with: