JEE Advanced 2024 — Paper 1 — MATH-II solution

JEE Advanced2024Paper 1MATH-II

Let $S = {a + b 2 : a, b \in Z}$ , $T_{1} = {(- 1 + 2)^{n} : n \in N}$ and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is (are) TRUE?

A
$Z \cup T_{1} \cup T_{2} \subset S$
B
$T_{1} \cap (0, \frac{1}{2024}) = ϕ$ , where $ϕ$ denotes the empty set
C
$T_{2} \cap (2024, \infty) \neq = ϕ$
D
For any given $a, b \in Z$ , $cos (π (a + b 2)) + i sin (π (a + b 2)) \in Z$ if and only if $b = 0$ , where $i = - 1$

Solution

(A) Any integer $a$ can be written as $a + 0 \cdot 2 \in S$ . By the binomial expansion, $(\pm 1 + 2)^{n}$ has the form $p + q 2$ with integer $p, q$ , so $T_{1}, T_{2} \subset S$ . Hence $Z \cup T_{1} \cup T_{2} \subset S$ . True.

(B) Since $0 < - 1 + 2 < 1$ , the sequence $(- 1 + 2)^{n}$ tends to $0$ . For sufficiently large $n$ , it falls inside $(0, \frac{1}{2024})$ , so the intersection is non-empty. False.

(C) Since $1 + 2 > 1$ , $(1 + 2)^{n} \to \infty$ . For large enough $n$ it exceeds $2024$ . True.

(D) The given complex number equals $e^{iπ (a + b 2)}$ . This belongs to $Z$ (in particular equals $\pm 1$ ) iff $a + b 2$ is an integer. Since $2$ is irrational and $a \in Z$ , this forces $b = 0$ . True.

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