Let `S=sqrt3x+2y:x,yinZ}.` Prove that the operation `**` on S defined by `(sqrt3x_(1)+2y)**(sqrt3x_(2)+2y_(2))=sqrt3(x_(1)+x_(2))+2(y_(1)+y_(2))` for all `x_(1),x_(2),y_(1),y_(2)inZ` is closed under `**`.

Let A={3x+sqrt5y:x,yinZZ}. Show that an operation ** on A defined by, (3x_(1)+sqrt5y_(1))**(3x_(2)+sqrt5y_(2))=3(x_(1)+x_(2))+sqrt5(y_(1)+y_(2)) for all x_(1),x_(2),y_(1),y_(2)inZZ is binary operation on A.

y = sqrt(1 + x^(2)) : y' = (xy)/(1 + x^(2))

The maximum value of y = sqrt((x-3)^(2)+(x^(2)-2)^(2))-sqrt(x^(2)-(x^(2)-1)^(2)) is

The locus of the point P(x,y) satisfying the relation sqrt((x-3) ^(2) +(y-1)^(2))+ sqrt((x+3) ^(2) +(y-1)^(2))=6 is-

IF y=f(x^(2)) and f'(x)=sqrt(3x^(2)+1) , find [(dy)/(dx)]_(x=2) .

If y = sqrt(x) + (1)/(sqrt(x)) prove that 2x(dy)/(dx) + y = 2sqrt(x)

If y="tan"^(-1) (sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2))) show that, (dy)/(dx)=(x)/(sqrt(1-x^(4)))

If sqrt(1-x^2)+sqrt(1-y^2)=a(x-y) ,show that dy/dx=sqrt((1-y^2)/(1-x^2))

If x>y prove that sqrt(y+sqrt(2xy-x^2))+sqrt(y-sqrt(2xy-x^2)) = sqrt(2x) .

The solution of (x d x+y dy)/(x dy-y dx)=sqrt((1-x^2-y^2)/(x^2+y^2)) is