Identity which one of the following is an eigen vectors of the matrix $$A = \left[ {\matrix{ 1 & 0 \cr { - 1} & { - 2} \cr } } \right]$$

Let $$A$$ be $$3 \times 3$$ matrix with rank $$2.$$ Then $$AX=O$$ has

$$f = {a_0}\,{x^n} + {a_1}\,{x^{n - 1}}\,y + - - + \,{a_{n - 1}}\,x\,{y^{n - 1}} + {a_n}\,{y^n}$$
where $$\,\,{a_i}\,\,$$ ($$i=0$$ to $$n$$) are constants then $$x{{\partial f} \over {\partial x}} + y{{\partial f} \over {\partial y}}\,\,\,$$ is

The value of the integral $$\int\limits_{ - 1}^1 {{1 \over {{x^2}}}dx} \,\,\,$$ is

A scalar field is given by $$f = {x^{2/3}} + {y^{2/3}},$$ where $$x$$ and $$y$$ are the Cartesian coordinates. The derivative of $$'f'$$ along the line $$y=x$$ directed away from the origin at the point $$(8, 8)$$ is

If a vector $$\overrightarrow R \left( t \right)$$ has a constant magnitude then

The probability that there are $$53$$ Sundays in a randomly chosen leap year is

The general solution of the differential equation $$\left( {{D^2} - 4D + 4} \right)y = 0$$ is of the form (given $$D = {d \over {dx}}$$ and $${C_1},{C_2}$$ are constants)

Let $${z^3}\, = \,\overline z $$, where z is a complex number not equal to zero. Then z is a solution of

Consider the circle $$\left| {z\, - 5\, - 5i} \right|\, = \,2$$ in the complex number plane (x, y) with z = x + iy. The minimum distance from the origin to the circle is