The matrix $\begin{bmatrix} 1 & a \\ 8 & 3 \end{bmatrix}$ (where $a > 0$) has a negative eigenvalue if $a$ is greater than

If the value of the double integral

$\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$

is $\log_e(\frac{a}{24})$, then $a$ is __________ (answer in integer).

If $x(t)$ satisfies the differential equation

$t \frac{dx}{dt} + (t - x) = 0$

subject to the condition $x(1) = 0$, then the value of $x(2)$ is __________ (rounded off to 2 decimal places).

Let $X$ be a continuous random variable defined on $[0, 1]$ such that its probability density function $f(x) = 1$ for $0 \leq x \leq 1$ and $0$ otherwise. Let $Y = \log_e (X + 1)$. Then the expected value of $Y$ is _____ . (rounded off to 2 decimal places)