Kerala KEAM 5 June 2024 Solved Paper

Let $z$ be a complex number of unit modulus, which means that $$| z | = 1.$$We need to evaluate: $$\left | \frac{1 + z}{1 + \overline{z}} \right | .$$Step 1: Express $\overline{z}$ in terms of $z$The complex conjugate of $z$ is denoted by $\overline{z}$, and for any complex number $z = x + iy$, its conjugate is given by $\overline{z} = x - iy$. Since $z$ has unit modulus, we know that:$$| z | ^2 = 1 \Rightarrow z \overline{z} = 1.$$Step 2: Simplify the expressionWe can simplify the expression as follows: $$\left | \frac{1+z}{1+\overline{z}} \right | = \frac{ | 1+z | }{ | 1+\overline{z} | }.$$Step 3: Evaluate $| 1+z |$ and $| 1+\overline{z} |$Using the fact that $| z | = 1$, we compute both the modulus of $1+z$ and $1+\overline{z}$. Since the modulus of a complex number is the distance from the origin, and $| z | = 1$, the expressions for $| 1+z |$ and $| 1+\overline{z} |$ are equal.Thus: $$\left | \frac{1+z}{1+\overline{z}} \right | = 1.$$Therefore, the correct answer is option (B), which is 1.Quick Tip: When simplifying expressions involving complex numbers, remember that the modulus of a complex number is its distance from the origin. If a complex number has unit modulus, its conjugate will also have unit modulus, which can simplify calculations.