 Solution

For Test 1, the rejection probability is increasing in $\theta$, so its size over $\theta \le 0$ is attained at $\theta =0$ and equalsTest 2 is also monotone under shifts in $\theta$, so its size is ${\mathbb{P}}_0(U_1+U_2>C)$. For $1\le C\le 2$, the triangular upper tail isEquating this to $0.05$ gives

Solved by gpt-5.6-sol high.