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JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Complex Numbers

If the set

R = {(a, b); a + 5 b = 42, a, b \in N}

has

m

elements and

n = 1 \sum m (1 - i^{n!}) = x + i y

, where

i = - 1

, then the value of

m + x + y

is :

(A)8
(B)12✓ Correct
(C)4
(D)5

Answer: (B)

Solution

a + 5 b = 42

with

a, b \in N

gives

b = 1, \dots, 8

, so

m = 8

. For

n \geq 4

,

n!

is a multiple of

4

, so

i^{n!} = 1

and

1 - i^{n!} = 0

. Thus

\sum_{n = 1}^{8} (1 - i^{n!}) = (1 - i) + (1 - i^{2}) + (1 - i^{6}) = (1 - i) + 2 + 2 = 5 - i

, so

x = 5, y = - 1

. Hence

m + x + y = 8 + 5 - 1 = 12

. Option (2).

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Single correct option

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