Mathematics · Binomial Theorem

JEE Main 2026 — 28 January, Evening Shift — Question 1

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II :

Statement I : $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7 .

Statement II : The integral part of $(7+4 \sqrt{3})^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

Answer: B

Statement I : ∴ divisible by $7$

Statement II: $\mathrm{R}=(7+4 \sqrt{3})^{25}=\mathrm{I}+\mathrm{f}$

$R^{\prime}=(7-4 \sqrt{3})^{25}=f^{\prime}$

$\therefore \mathrm{R}+\mathrm{R}^{\prime}=2\left[{ }^{25} \mathrm{C}_{0} 7^{25}+{ }^{25} \mathrm{C}_{2} 7^{23}(4 \sqrt{3})^{2}+\ldots.\right]$

$+\mathrm{f}+\mathrm{f}^{\prime}=$ even integer

$\therefore \mathrm{I}=$ odd integer

$\because 0<\mathrm{f}+\mathrm{f}^{\prime}<2 \Rightarrow \mathrm{f}+\mathrm{f}^{\prime}=1$

⇒ Both the statements are correct

Answer key and solution verified before publishing.

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