If the set R = {(a,b) : a + 5b = 42, a,b ∈ N} has m elements and where then the value of m + x + y is

Engineering

Mathematics

Introduction to Determinants

Question

If the set R = {(a,b) : a + 5b = 42, a,b ∈ 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

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$a + 5 b = 2 + 40 \Rightarrow (a - 2) = - 5 b + 40 \Rightarrow \frac{a - 2}{5} = \frac{b - 8}{- 1} = λ \Rightarrow a = 2 + 5 λ, b = 8 - λ$

λ = 0 ⇒ a = 2, b = 8; ……. λ = 7 ⇒ a = 37, b = 1

R = {(37,1),(32,2),(27,3),(22,4),(17,5),(12,6),(7,7),(2,8)}

so m = 8

Now $N \sum_{n = 1}^{8} (1 - (i)^{n!}$

=8 – {i¹ +i^2! i^3! + i^4! + …………. i^8!}

= 8 –[i – 1 – i + 1 + 1 + 1 + 1 + 1]

= 8 – 4 = 4 = 4 + 0i

so m + x + y = 8 + 4 + 0 = 12

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