Question 63

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

Solution

Share

a+5b=42, a,b∈ℕ: b=1→a=37,...,b=8→a=2. m=8. Σ(1-i^{n!}) for n=1..8: for n≥4, n! is multiple of 4, so i^{n!}=1. For n=1: i¹=i. n=2: i²=-1. n=3: i⁶=-1. So sum = (1-i)+(1-(-1))+(1-(-1))+5(1-1) = (1-i)+2+2+0 = 5-i. x=5, y=-1. m+x+y = 8+5-1 = 12.

Option (1): 12.

Create a FREE account and get:

  • Free JEE Mains Previous Papers PDF
  • Take JEE Mains paper tests