Sequences and Series · Question 2

JEE Advanced 2024 Paper 1Multiple CorrectMedium📈 Topic in 33% papers+4−1

Let S= \a+b 2 : a,b Z \, T1= \ (-1+ 2 )n : n N \, and T2= \ (1+ 2 )n : n N \. Then which of the following statements is (are) TRUE?

Z T1 T2 S
T1 (0, 12024 )= , where denotes the empty set.
T2 (2024, )
For any given a,b Z, ( (a+b 2 ) )+i ( (a+b 2 ) ) Z if and only if b=0, where i= -1.

Answer

(1) Z T1 T2 S, (3) T2 (2024, ), (4) For any given a,b Z, ( (a+b 2 ) )+i ( (a+b 2 ) ) Z if and only if b=0, where i= -1.

Solution

(A) a Z is a+0 2; by the binomial theorem ( 1+ 2)n=A+B 2 with A,B Z. True.

(B) 0< 2-1<1, so ( 2-1)n 0; for large n it lies in (0, 12024 ). False.

(D) ei (a+b 2) Z needs (a+b 2)=0, i.e. a+b 2 Z, i.e. b=0 ( 2 irrational); conversely b=0 gives a= 1. True.

Answer: (A), (C), (D)

JEE Advanced 2024 Paper 1

Keep practising — Sequences and Series

Let R denote the set of all real numbers. Let f: R R be a function such that f(x) > 0 for all x R, and f(x + y) = f(x)f(y) for all x, y R. Let the real numbers a1, a2, , a50 be in an arithmetic progression. If f(a31) = 64f(a25), and i=150 f(ai) = 3(225 + 1), then the value of i=630 f(ai) is .

JEE Advanced 2025 Paper 1

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