If $\log_e \left( \frac{a + b}{2} \right) = \frac{1}{2}(\log_e a + \log_e b)$,then the relation between $a$ and $b$ is:

$\cos 36^{\circ} - \cos 72^{\circ}$ is equal to

If $200 \text{ MeV}$ of energy is released in the fission of one nucleus of ${}_{92}^{236}U$,the number of nuclei that must undergo fission to release an energy of $1000 \text{ J}$ is

In the isoelectronic series of metal carbonyls,the $CO$ bond strength is expected to increase in the order:

$A$ nucleus $_Z{X^A}$ emits an $\alpha$-particle with velocity $v$. The recoil speed of the daughter nucleus is

If the function $f$ defined on $\left( \frac{\pi }{6}, \frac{\pi }{3} \right)$ by $f(x) = \begin{cases} \frac{\sqrt{2} \cos x - 1}{\cot x - 1}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$ is continuous,then $k$ is equal to