Question

If n is the degree of the polynomial, $\left[\frac{2}{\sqrt{5x^{2} + 1} - \sqrt{5x^{3} - 1 }}\right]^{8} + \left[ \frac{2}{\sqrt{5x^{3} + 1} + \sqrt{5x^{3} - 1}}\right]^{8}$ 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

Answer: (C)

$\left[\frac{2\left(\sqrt{5x^{3}\,\,\,1}\,\,\,\sqrt{5x^{3}\,\,\,1}\right)}{2}\right]^{8}+\left[\frac{2\left(\sqrt{5x^{3}+1}-\sqrt{5x^{3}-1}\right)}{2}\right]^{8}$
$= 2\left[^{8}C_{0}\left(\sqrt{5x^{3}+1}\right)^{8}+^{8}C_{2}\left(\sqrt{5x^{3}+1}\right)^{8}\left(5x^{3}-1\right)+^{8}C_{4}\left(\sqrt{5x^{3}+1}\right)^{4}\left(5x^{3}-1\right)^{2}+^{8}C_{8}\left(\sqrt{5x^{3}+1}\right)^{2}\left(5x^{3}-1\right)^{3}+^{8}C_{8}\left(5x^{3}-1\right)^{4}\right]$
$=2 \left[\left(5x^{3}-1\right)^{4}+28\left(5x^{3}-1\right)^{3}\left(5x^{3}-1\right)+70\left(5x^{3}-1\right)^{2}+28\left(5x^{3}-1\right)\left(5x^{3}-1\right)^{3}+\left(5x^{3}-1\right)^{4}\right]$
$h = 12 \quad\&\quad m = 2\left(5^{4} +140.5^{3} + 70.5^{4} +140.5^{3} + 5^{4}\right)$
$= 160000 = \left(20\right)^{4}$