Question

If $n$ is the degree of the polynomial, ${\left[\frac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]}^8+{\left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]}^8$ and $m$ is the coefficient of $x^n$ in it, then the ordered pair $(n,m)$ is equal to :

• A. $(24,(10{)}^8)$
• B. $(8,5(10{)}^4)$
• C. $(12,(20{)}^4)$
• D. $(12,8(10{)}^4)$

Answer 1

The correct option is C $(12,(20{)}^4)$

Rationalizing the expression we will get
${[\sqrt{5x^3+1}+\sqrt{5x^3-1}]}^8+{[\sqrt{5x^3+1}-\sqrt{5x^3-1}]}^8=2\left[\sum _{r=0}^4{}^8{C}_{2r}{\left(\sqrt{5x^3+1}\right)}^{2r}{\left(\sqrt{5x^3-1}\right)}^{8-2r}\right]$
The degree of the polynomial will be,
$n=12$
Coefficient of the highest power
$=2[5^4+28\cdot 5^3\cdot 5+70\cdot 5^4+28\cdot 5^3\cdot 5+5^4]={20}^4$