$\therefore \frac{{\partial u}}{{\partial x}} = \frac{1}{{{x^2} + {y^2}}}.2x$
$\frac{{{\partial ^2}u}}{{\partial {x^2}}} = \frac{{({x^2} + {y^2}).2 - 2x.2x}}{{{{({x^2} + {y^2})}^2}}}$$ = \frac{{2({y^2} - {x^2})}}{{{{({x^2} + {y^2})}^2}}}$’
$\frac{{\partial u}}{{\partial y}} = \frac{1}{{{x^2} + {y^2}}}.2y$
$\therefore \frac{{{\partial ^2}u}}{{\partial {y^2}}} = \frac{{({x^2} + {y^2})\,.\,2 - 2y.2y}}{{{{({x^2} + {y^2})}^2}}} = \frac{{2({x^2} - {y^2})}}{{{{({x^2} + {y^2})}^2}}}$
$\therefore $ $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = 0$.
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| List $I$ | List $II$ |
| $P.\quad$ Volume of parallelopiped determined by vectors $\vec{a}, \vec{b}$ and $\overrightarrow{ c }$ is $2$ . Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is | $1.\quad$ $100$ |
| $Q.\quad$ Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$ . Then the volume of the parallelepiped determined by vectors $3(\overrightarrow{ a }+\overrightarrow{ b }),(\overrightarrow{ b }+\overrightarrow{ c })$ and $2(\overrightarrow{ c }+\overrightarrow{ a })$ is | $2.\quad$ $30$ |
| $R.\quad$ Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$ . Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a}+3 \vec{b})$ and $(\vec{a}-\vec{b})$ is | $3.\quad$ $24$ |
| $S.\quad$ Area of a paralelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$ . Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is | $4.\quad$ $60$ |
Codes: $ \quad P \quad Q \quad R \quad S $