$$\int_{\frac{\pi}{18}}^{\frac{4 \pi}{9}} \frac{2 \sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\ldots \ldots$$

The maximum value of $Z=5 x+4 y$, Subject to $y \leq 2 x, x \leq 2 y, x+y \leq 3, x \geq 0, y \geq 0$ is ........

In amplitude modulation, the amplitude of carrier wave is $A_c$ and that of the modulating signal is $A_m$. In practice, the ratio of $A_m$ to $A_c$ is kept less than or equal to one, to avoid

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their $P^{\text {th }}$ overtone is