$$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is

The eigen values of the matrix $$\left[ {\matrix{ a & 1 \cr a & 1 \cr } } \right]$$ are

The number of linearly independent solutions of the system of equations
$$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { - 1} & 0 \cr 2 & { - 2} & 0 \cr } } \right]\,\,\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right] = 0$$ is equal to

The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value