= Solution
The two tests have the same size, but their powers cross. At $\theta=0.1$,
$$
\pi_1(0.1)=0.15,
\qquad
\pi_2(0.1)=\frac12\left(\frac1{\sqrt{10}}+0.2\right)^2<0.15.
$$
At $\theta=0.5$,
$$
\pi_1(0.5)=0.55,
\qquad
\pi_2(0.5)=1-\frac12\left(1-\frac1{\sqrt{10}}\right)^2>0.55.
$$
Thus Test 2 is a same-size test that beats Test 1 at one alternative, while Test 1 beats Test 2 at another. By the <crossing-power obstruction to a uniformly most powerful test>,
$$
\boxed{\text{neither Test 1 nor Test 2 is uniformly most powerful}.}
$$
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