Hint: The answer will be $pa+(1-p)b$, where $a$ is the probability that an exponential with mean $1$ lies between $1$ and $2$, and $b$ is the probability that an exponential with mean $2$ lies between $1$ and $2$.
The calculation of $a$ and $b$ can be done by integration, or by recalling the formula for the cumulative distribution function of the exponential with parameter $\lambda$.