Let ' z^' be a complex number such that z+(1)/(z)=1 and a=z^2007+(1)/(z^2007) and b is the last

step 1 In this question it is given that if A is equal to Z to the power 2007 plus 1 upon Z to the powe

Let ' z^' be a complex number such that z+(1)/(z)=1 and...

Let $' z^{\prime}$ be a complex number such that $z+\frac{1}{z}=1$ and $a=z^{2007}+\frac{1}{z^{2007}}$ and $b$ is the last digit of the number $2^{2^{n}}+1$ where $\mathrm{n}$ is an integer $>1$. Then the value of $\left(a^{2}+b^{2}\right)$ is
(a) 23
(b) 13
(c) 53
(d) 1

Key Concepts

Euler's Formula and Polar Form of Complex Numbers

This concept involves expressing complex numbers in the form z = e^(i?), allowing the conversion from algebraic to trigonometric expressions. It simplifies operations such as multiplication, division, and exponentiation of complex numbers by encapsulating magnitude and angle information in a compact, exponential form.

De Moivre's Theorem

De Moivre's Theorem states that for any real number ? and integer n, (cos ? + i sin ?)^n = cos(n?) + i sin(n?). This theorem is particularly useful for calculating powers and roots of complex numbers expressed in polar form, as it directly relates the exponentiation of a complex number to multiples of its angle.

Trigonometric Identities in Complex Number Analysis

Trigonometric identities, such as cos? = (e^(i?) + e^(-i?))/2, play a key role when handling expressions that involve both a complex number and its reciprocal. They enable the conversion of sums of complex conjugate pairs into simpler trigonometric forms, facilitating the computation of real values from complex equations.

Modular Arithmetic in Exponential Calculations

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value, the modulus. It is particularly useful for determining the last digit of large exponential numbers by reducing the problem to identifying repeating cycles or patterns in the powers modulo 10.

Periodicity in Exponential and Trigonometric Functions

The concept of periodicity highlights that functions like cosine and sequences of powers modulo a fixed number repeat their values after a certain period. Recognizing these cycles can greatly simplify computations in problems dealing with large exponents by reducing them to their equivalent within one complete cycle.

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