DETERMINANTS — Maths STD 12 Science — Question

1. $\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}_{\ =\ \left(\cos\theta\right)\left(\cos\theta\right)\ -\ \left(-\sin\theta\right)\left(\sin\theta\right)\ =\ \cos^2\theta\ +\ \sin^2\theta\ =\ 1}$

2. $\begin{vmatrix}x^2-x+1&x-1\\x+1&x+1\end{vmatrix}$

$=\left(x^2-x+1\right)\left(x+1\right)-\left(x-1\right)\left(x+1\right)$

$=x^3-x^2+x+x^2-x+1-\left(x^2-1\right)$

$=x^3+1-x^2+1$

$=x^3-x^2+2$