If $$x \in\left(0, \frac{\pi}{2}\right)$$ and $$x$$ satisfies the equation $$\sin x \cos x=\frac{1}{4}$$, then the values of $$x$$ are

If $$x=a \cos \theta, y=b \sin \theta$$, then $$\left[\frac{d^2 y}{d x^2}\right]_{\theta=\frac{\pi}{4}}=$$

$$\int \frac{x+\sin x}{1+\cos x} d x=$$

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity of ice melts in 20 minutes, $$x_0$$ is the initial quantity of ice. If after 40 minutes the amount of ice left is $$\mathrm{Kx}_0$$, then $$\mathrm{K}=$$