The volume of the parallelepiped with its edges represented by the vectors $\hat{i}+\hat{j}, \hat{i}+2\hat{j}, \hat{i}+\hat{j}+\pi \hat{k}$ is 

Which of the following is/one correct?

$i)$ Adjoint of a symmetric matrix is also a symmetric matrix

$ii)$ Adjoint of a diagonal matrix is also a diagonal matrix.

$iii)$ If $A$ is square matrix of order $n$ and $\lambda$ is a scaler, then $\mathrm{adj}\left(\lambda A\right)={\lambda }^n\mathrm{adj}\left(A\right)$

$iv)$ $A\left(\mathrm{adj}A\right)=(adjA)A=\left|A\right|I$

If $\left(a{x}^2+bx+c\right)y+a^{\text{'}}{x}^2+b^{\text{'}}x+c^{\text{'}}=0,$ find the condition that $x$ may be a rational function of $y.$

Find the equation of the circle passing through the points  $(0,0),(-2,1)\text{ and }(-3,2)$.