Partial Derivatives

Q. The partial differential equation that can be formed from $z=ax+by+ab$ has the form $(\text{with}p=\frac{\partial z}{\partial x}\text{and}q=\frac{\partial z}{\partial y})$

1. $z=px+qy$
2. $z=px+pq$
3. $z=px+qy+pq$
4. $z=qy+pq$

Q. If $z=f(u,v),u=x^2-2xy-y^2,v=a$, then

1. $(x+y)\frac{\partial z}{\partial x}=(x-y)\frac{\partial z}{\partial y}$
2. $(x-y)\frac{\partial z}{\partial x}=(x+y)\frac{\partial z}{\partial y}$
3. $(x+y)\frac{\partial z}{\partial x}=(y-x)\frac{\partial z}{\partial y}$
4. $(y-x)\frac{\partial z}{\partial x}=(x-y)\frac{\partial z}{\partial y}$

Q. if z = xy ln(xy), then

1. $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0$
2. $y\frac{\partial z}{\partial x}=x\frac{\partial z}{\partial y}$
3. $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$
4. $y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}=0$

Q. If $u=ln\left(\frac{x^3+y^3}{x^2+y^2}\right)$, then the value of $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$ is

1. $e^u$
2. 1
3. u
4. eu