Discrete Random Variables (DRV) - Determining an Unknown Binomial Success Probability

The random variable \(X\) is the number of successes in \(200\) independent trials of an experiment in which the probability of success at any one trial is \(p\). Given that \(\mathrm{E}\left( {{X}^{2}} \right)=10.6008\), find the exact value of \(p\), and show that \(\frac{\mathrm{P}\left( X=k+1 \right)}{\mathrm{P}\left( X=k \right)}=\frac{7\left( 200-k \right)}{493\left( k+1 \right)}\) , for \(k=0,1,2,3,...,199\). Hence find the value of \(k\) such that \(\mathrm{P}\left( X=k \right)\) is the maximum.