Let $$A$$ be an $$n \times n$$ matrix with rank $$r\left( {0 < r < n} \right).$$ Then $$AX=0$$ has $$p$$ independent solutions, where $$p$$ is

The double integral $$\,\,\int_0^a {\int_0^y {f\left( {x,y} \right)\,dx\,dy\,\,\,} } $$ is equivalent to

The magnitude of the directional derivative of the function $$f\left( {x,y} \right) = {x^2} + 3{y^2}$$ in a direction normal to the circle $$\,{x^2} + {y^2} = 2,$$ at the point $$(1,1),$$ is

A coin is tossed thrice. Let $$X$$ be the event that head occurs in each of the first two tosses. Let $$Y$$ be the event that a tail occurs on the third toss. Let $$Z$$ be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE?