Question
Let S=\ a+b 2 : a, b Z \ , T_1= \ (-1+ 2 )^n: n N \ , and T_2= \ (1+ 2 )^n: n N \ . Then which of the following statements is (are) TRUE?
JEE Advanced
Mathematics
Complex Number
2024
JEE Advanced 2024 (Paper 1)
JEE Advanced Mathematics Question (2024) — Solution
Options
- A. Z T_1 T_2 S
- B. T_1 (0, 1 2024 )= , where denotes the empty set.
- C. T_2 (2024, )
- D. For any given a, b Z , ( (a+b 2 ))+i ( (a+b 2 )) Z if and only if b=0, where i= -1 .
Answer
D. For any given a, b Z , ( (a+b 2 ))+i ( (a+b 2 )) Z if and only if b=0, where i= -1 .
Step-by-step solution
(1) aligned & (-1+ 2 )^ n = m + 2 n , m , where, n Z \\ & (1+ 2 )^ n = m _1+ 2 n _1, where, ~m _1, n _1 Z \\& Z T _1 T _2 S aligned but b 2 S for negative b Z . So Z T _1 T _2 S (2) aligned & ( 2 -1)^ n = 1 ( 2 +1)^ n 2024, where, n N \\ & T _2 (2024, ) aligned (4) ( (a+b 2 )=0) b=0, where, a Z Options (1), (3), (4) are Correct.
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