If $\mathrm{p}$ is a real number and if the middle term in the expansion of ${\left(\frac{\mathrm{p}}{2}+2\right)}^8$ is $1120$, find $\mathrm{p}$ .

In the expansion of $(\mathrm{x}+\mathrm{a}{)}^{\mathrm{n}}$ if the sum of odd terms is denoted by $\mathrm{O}$ and the sum of even terms by $\mathrm{E}$ then prove that 

${\mathrm{O}}^2-{\mathrm{E}}^2={\left({\mathrm{x}}^2-{\mathrm{a}}^2\right)}^{\mathrm{n}}$.

In the expansion of $(\mathrm{x}+\mathrm{a}{)}^{\mathrm{n}}$ if the sum of odd terms is denoted by $\mathrm{O}$ and the sum of even terms by $\mathrm{E}$ then prove that

$4\mathrm{OE}=(\mathrm{x}+\mathrm{a}{)}^{2\mathrm{n}}-(\mathrm{x}-\mathrm{a}{)}^{2\mathrm{n}}$

Find the term independent of $\mathrm{x}$ in the expansion of $\left(1+\mathrm{x}+2{\mathrm{x}}^3\right){\left(\frac{3}{2}{\mathrm{x}}^2-\frac{1}{3\mathrm{x}}\right)}^9$