Show that `f(x,y) = (x^(2)-y^(2))/(y^(2)+1)` is continous at every, `(x,y) in R^(2)`.

Show that f (x,y) = (2x ^(2) -y ^(2))/sqrt(x ^(2) + y ^(2)) is a homogeneous function of degree 1.

Show that F(x,y)=(x^2+5xy-10y^2)/(3x+7y) is a homogeneous function of degree 1.

Show that y = (cos^(-1) x)^(2) is a solution of the differential equation . (1-x^(2))y '' - xy' - 2 = 0

Let g(x,y) = (x^(2)y)/(x^(4) + y^(2)) for (x,y) ne (0,0) and f(0,0)=0. (i) Show that lim_((x,y) to (0,0)) g(x,y)=0 along every line y=mx, m in R . (ii) Show that lim_((x,y) to (0,0)) g(x,y) = k/(1+k^(2)) , along every parabola y=kx^(2), k in R{0} .

Let F (x,y) = (2x - 3y +4)/(x ^(2) + y ^(2) +4) for all (x,y) in R ^(2), Show that f is continuous on R ^(2).

Show that |(1,1,1),(x,y,z),(x^(2),y^(2),z^(2))|=(x-y)(y-z)(z-x)

If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the vlaue of (1)/(y_(1))+(1)/(y_(2)) is

Let (x,y) = 2x y ^(2) + 3x ^(2) y calculate u _(x) (2,1) and u_(y) (1,-2).

Show that (x^2+y^2)^4=(x^4-6x^2y^2+y^4)^2+(4x^3y-4x y^3)^2dot