JEE Advanced 2024 · Paper 1, Mathematics question 5

(A) Integers are $a + 0\sqrt2$. Expanding $(\pm1+\sqrt2)^n$ binomially gives integer coefficients of $1$ and $\sqrt2$, so $T_1, T_2 \subset S$. True.

(B) $0 < \sqrt2 - 1 < 1$, so $(\sqrt2-1)^n \to 0^+$; for large $n$ it lies in $\left(0, \frac{1}{2024}\right)$. The intersection is non-empty. False.

(C) $1+\sqrt2 > 1$, so $(1+\sqrt2)^n \to \infty$ and eventually exceeds 2024. True.

(D) $\cos(\pi t) + i\sin(\pi t) = e^{i\pi t}$ has modulus 1; it is an integer only if it equals $\pm1$, i.e. $t = a + b\sqrt2$ is an integer. Since $\sqrt2$ is irrational, that happens exactly when $b = 0$. True.