Find the values of m for which the expression `2x^(2)+mxy +3y^(2) -5y-2` can be resolved into two rational linear factors.

Prove that the integral values of m for which the roots of the equation mx^(2)+ (2m-1)x+(m-2)=0 are rational are given by the expression n(n+1) , where n is integer.

Find the value of p so that the three lines 3 x+y-2=0, p x+2 y-3=0 . and 2 x-y-3=0 may intersect at one point.

Find 'the values of |x-y| such that |x-5|=|x-9|,|y-2|=5 .

Find the distance of the point on y = x^(4) + 3x^(2) + 2x which is nearest to the line y = 2x - 1.