Complex Numbers

To determine which statement is correct, let's analyse each option given and also carefully observe that where and are nonzero real numbers.Option A: "I is a root of , then must be even."The roots of are determined by setting . Since and when , the only root is . For a complex root like to be a root, must be 0 , which can only happen if , leading to and . However, since is nonzero, this condition cannot be satisfied. Thus, cannot be a root.Therefore, Option A is incorrect.Option B: "If is a root of , then must be odd."If is a real root, then which leads to or . This is true for any , not necessarily odd. Therefore, Option B is incorrect.Option C: "If is a root of , then can never be a root of ."As established earlier, only gives real roots. Thus, if were a root, would need to equal zero, and . This makes the question of whether can be a root irrelevant because no complex roots exist for .Therefore, Option C is irrelevant.Option D: "If is a root of , then must be an integer."If is a root, leads to . Since -1 is an integer, this statement holds true.Option D is correct.

Revision Note on : is a primitive cube root of unity with the following key properties: $ \omega^3 $$ $

Given the subset in the -plane: we need to determine the nature of this subset.Step-by-step Breakdown:Identify Coordinates: -coordinate: -coordinate: Let :Compute -coordinate as: Compute -coordinate as: Establish the Relationship:We have: Thus: The equation describes a parabola in the -plane, which confirms that the correct characterization of the subset is that it is a parabola.

We are given complex numbers satisfying We want to find Step 1. Simplify magnitudesFrom So lies on the unit circle.Also: Hence, lies on a circle of radius 2 centered at , lies on a circle of radius 2 centered at .Step 2. Introduce new variablesLet’s define Then by the given magnitudes, We can express the condition on reciprocals: But . So the condition becomes: Step 3. Note all have magnitude 2Let’s divide everything by 2 to normalize to unit circle magnitudes.Let’s write: Then , etc.The reciprocal condition becomes: Multiply both sides by 2 to simplify the numeric coefficient? Wait, better to keep as is.We have: Step 4. Magnitude constraint boundingEach term has magnitude 1, ½, ½ respectively, so the maximum possible magnitude is To achieve equality ,all three exponentials must have the same argument, i.e. Hence they must be all proportional with positive real coefficients.That means , up to integer multiple of .Step 5. So we can take them equal (since only magnitudes matter)Let .Then the condition is automatically satisfied: OK.Step 6. Return to Since all are equal, say Then: Step 7. Go back to From definitions: Step 8. Compute the required expressionWe need .Substitute: Simplify: Hence ✅Final Answer: ✅ OptionA. 8