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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Sequences and Series

If the set R=\ (a, b): a+5 b=42, a, b ∈ ℕ\ has m elements and ∑_...

If the set R={(a,b):a+5b=42,a,b∈N}R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\} has mm elements and ∑n=1m(1−in!)=x+iy\sum_{n=1}^m\left(1-i^{n !}\right)=x+i y, where i=−1i=\sqrt{-1}, then the value of m+x+ym+x+y is

Options

  1. A12
  2. B4
  3. C8
  4. D5

Correct answer

A. 12

Step-by-step solution

a+5b=42,a,b∈Na=42−5b,b=1,a=37b=2,a=32b=3,a=27⋮b=8,a=2\begin{aligned} & a+5 b=42, a, b \in N \\ & a=42-5 b, b=1, a=37 \\ & b=2, a=32 \\ & b=3, a=27 \\ & \vdots \\ & b=8, a=2\end{aligned} \mathrm{R} has " 8 " elements ⇒\Rightarrow\mathrm{m}=88∑n=18(1−in!)=x+iy for n≥4,in!=1⇒(1−i)+(1−i2!)+(1−i3!)=1−I+2+1+1=5−I=x+iym+x+y=8+5−1=12\begin{aligned} & \sum_{n=1}^8 \left(1-i^{n !}\right)=x+i y \\ & \text{ for } n \geq 4, i^{n !}=1 \\ & \Rightarrow(1-i)+\left(1-i^{2 !}\right)+\left(1-i^{3 !}\right) \\ & =1-I+2+1+1 \\ & =5-I=x+i y \\ & m+x+y=8+5-1=12\end{aligned}$$
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