The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is

The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is

In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the probability, of getting exactly 4 successes, is

The function to be maximized is given by $Z=3 x+2 y$. The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by