JEE Main 2025 Apr 3 Shift 1, Mathematics Q4: Binomial Theorem and Its Simple Applications

General term: $T_{r+1}=\left(\genfrac{}{}{0ex}{}{8}{r}\right)2^{8-r}(\sqrt{3}{)}^r=(\genfrac{}{}{0ex}{}{8}{r})2^{8-r}{3}^{r/2}.$

For a rational term, $r$ must be even. Take $r=0,2,4,6,8$.

$r=0:\left(\genfrac{}{}{0ex}{}{8}{0}\right)2^8=256.$

$r=2:\left(\genfrac{}{}{0ex}{}{8}{2}\right)2^6\cdot 3=28\cdot 64\cdot 3=5376.$

$r=4:\left(\genfrac{}{}{0ex}{}{8}{4}\right)2^4\cdot 9=70\cdot 16\cdot 9=10080.$

$r=6:\left(\genfrac{}{}{0ex}{}{8}{6}\right)2^2\cdot 27=28\cdot 4\cdot 27=3024.$

$r=8:\left(\genfrac{}{}{0ex}{}{8}{8}\right)\cdot 81=81.$

Total: $256+5376+10080+3024+81=18817.$