List of top Mathematics Questions on Determinants asked in UP Board XII

Prove that: \[ \left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} \right| = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \]

If \[ \Delta = \begin{vmatrix} 3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3 \end{vmatrix} \] then find the value of $|\Delta|$.

Prove that \[ \begin{vmatrix} x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]

Prove that \[ \begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3 \]