Kerala KEAM 5 June 2024 Solved Paper

We are asked to find the modulus of the complex number raised to a large power.Step 1: First, write the given complex number in polar form. The complex number $1 + i$ can be written as:$$1 + i = \sqrt{1^2 + 1^2} \, cis \, \theta = \sqrt{2} \, cis \, \left( \frac{\pi}{4} \right),$$where $cis \, \theta = \cos \theta + i \sin \theta$.Step 2: Now, divide by $\sqrt{2}$: $$\frac{1+i}{\sqrt{2}} = \frac{\sqrt{2} \, cis \, \frac{\pi}{4}}{\sqrt{2}} = cis \, \frac{\pi}{4}.$$Step 3: We are asked to find the modulus of $\left( \frac{1+i}{\sqrt{2}} \right)^{2024}$. Since the modulus of $cis \, \theta$ is 1, we have: $$\left | cis \, \frac{\pi}{4} \right | = 1.$$Step 4: Raising this to the power of 2024, we get: $$\left | \left( cis \, \frac{\pi}{4} \right)^{2024} \right | = 1.$$Thus, the value of the expression is $1$.Therefore, the correct answer is option (C).Quick Tip: When raising a complex number in polar form to a power, the modulus remains the same, and only the argument is multiplied by the exponent. In this case, the modulus of $cis \, \theta$ is 1, so the modulus remains 1 regardless of the power.