Binomial Theorem – Rational and Irrational Terms – JEE Main 03 April 2025 Shift 1

Solution:

$$(x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r$$

$$T_{r+1} = \binom{8}{r} (2)^{8-r} (\sqrt{3})^r$$

For rational terms, the power of $\sqrt{3}$ must be an even integer.

$$r \in \{0, 2, 4, 6, 8\}$$

$$\text{Sum} = \binom{8}{0}(2)^{8}(\sqrt{3})^{0} + \binom{8}{2}(2)^{6}(\sqrt{3})^{2} + \binom{8}{4}(2)^{4}(\sqrt{3})^{4} + \binom{8}{6}(2)^{2}(\sqrt{3})^{6} + \binom{8}{8}(2)^{0}(\sqrt{3})^{8}$$

$$= (1)(256)(1) + (28)(64)(3) + (70)(16)(9) + (28)(4)(27) + (1)(1)(81)$$

$$= 256 + 5376 + 10080 + 3024 + 81$$

$$= 18817$$

Ans. (4)