If set R={(a, b) ; a+5 b=42, a, b ∈ mathbb{N}} has m elements and Σ

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4-1.Complex numbers

hard

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

$8$

$12$

$4$

$5$

(JEE MAIN-2024)

Solution

$ 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 $

$ R \text { has "8" elements } \Rightarrow \mathrm{m}=8 $

$ \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$

Standard 11

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Solution

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