Solution: We say that a quadratic form $Q$ on $\mathbb{R} ^2$ is positive definite if $Q(x,y)>0$ for all $x,y\neq 0$. We will examine the same for all the given problems.

• $Q(x,y) = xy$. This is not positive definite as \[ Q(1,-1) = -1 \lt 0. \]

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• $Q(x,y) = x^2 - xy + y^2$. Note that from AM-GM inequality on $x^2$ and $y^2$, we have \[ x^{2} + y^2 \geq \frac{x^2 + y^2}{2} \geq \sqrt{x^2 y^2} = xy. \] Therefore, for $x,y$ nonzero, we have \[ x^2 + y^2 > xy \implies x^2 - xy + y^2 \gt 0. \]

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• $Q(x,y) = x^2 + 2xy + y^2$. This is not positive definite as \[ Q(1,-1) = 0. \]

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• $Q(x,y) = x^2 + xy$. Similarly, $Q(x,y) = x^2 + xy$ is not positive definite, as \[ Q(1,-2) = -1 \lt 0. \]