Question

If $A_{i j}$ is the cofactor of the element $a_{i j}$ of the determinant $\begin{vmatrix}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{vmatrix}$ , then write the value of $a_{32}A_{32}.$

If f : R→ R defined by f(x) = $\textstyle{\frac{3x+5}{2}}$ is an invertible function, then find $f^{-1}(x).$

Write the order of the differential equation: $\log\,\left(\frac{d^{2}y}{d x^{2}}\right)=\left(\frac{d y}{d x}\right)^{3}+x$

Find the differential equation representing the family of curves $V={\frac{A}{r}}+B,$ where A and B are arbitrary constants.

If matrix A = $[a_{i j}]_{2\times2}$, where $a_{i j}$ = $\left\{\begin{array}{l}{{2,\ i\neq j}}\\ {{0,\ i=j^{\prime}}}\end{array}\right.$ , Write the matrix A.

Write the sum of the order and degree of the differential equation $1+\left(\frac{d y}{d x}\right)^{4}=7\left(\frac{d^{2}y}{d x^{2}}\right)^{3}\,.$

Write the integrating factor of the differential equations ${\sqrt{x}}{\frac{d y}{d x}}+y=e^{-2{\sqrt{x}}}\,.$