= Solution
The <power function of a statistical test> for Test 1 is
$$
\boxed{
\pi_1(\theta)=
\begin{cases}
0,&\theta\leq-0.05,\\
\theta+0.05,&-0.05<\theta<0.95,\\
1,&\theta\geq0.95.
\end{cases}}
$$
Put $a=1/\sqrt{10}$. Test 2 rejects when
$$
U_1+U_2>2-a-2\theta.
$$
Using the two branches of the triangular distribution gives
$$
\boxed{
\pi_2(\theta)=
\begin{cases}
0,&\theta\leq-a/2,\\
\frac12(a+2\theta)^2,
&-a/2<\theta<(1-a)/2,\\
1-\frac12(2-a-2\theta)^2,
&(1-a)/2\leq\theta<(2-a)/2,\\
1,&\theta\geq(2-a)/2.
\end{cases}}
$$
Both <functions> equal $0.05$ at the least favourable null value $\theta=0$.
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