The p.d.f of a random variable $x$ is given by

$$\begin{aligned} & f(x)=\frac{1}{4 a}, \quad 00) \\ & =0 \text {, otherwise } \end{aligned}$$

and $P\left(x<\frac{3 a}{2}\right)=k P\left(x>\frac{5 a}{2}\right)$ then $k=$ ..............

If the function $f(x)=\frac{\left(e^{k x}-1\right) \tan k x}{4 x^2}, x \neq 0$

$$\qquad \qquad=16 \qquad x=0$$

is continuous at $x=0$, then $k=\ldots \ldots$

The solution of the differential equation $y d x-x d y=x y d x$ is ......

The maximum value of $z=6 x+8 y$ subject to $x-y \geq 0, x+3 y \leq 12, x \geq 0, y \geq 0$ is $\ldots \ldots$.