STD 12 - 1. relation and function — Mathematics STD 12 Science — Question

$A) f(x, y)=x \sin y+y \sin x$

Now

$f(\lambda x, \lambda y)=\lambda x \sin (\lambda y)+\lambda y \sin (\lambda x)$

$=\lambda(x \sin \lambda y+y \sin (\lambda x))$

But

$x \sin \lambda y+y \sin (\lambda x) \neq f(x, y)$

Hence it is not a homogeneous function.

$B) f(x, y)=x e^{y / x}+y e^{x / y}$

$f(\lambda x, \lambda y)=\lambda x e^{\lambda y / \lambda x}+\lambda y e^{\lambda x / \lambda y}$

$=\lambda\left(e^{x / y}+e^{y / x}\right)$

$=\lambda f(x, y)$

Hence it is a homogeneous function. $C) f(x, y)=x^{2}-x y$

$f(\lambda x, \lambda y)=\lambda^{2} x^{2}-\lambda^{2}(x y)$

$=\lambda^{2}\left(x^{2}-x y\right)$

$=\lambda^{2} f(x, y)$

Hence it is a homogeneous function.