Complex Numbers

Step 1: Solve the quadratic equation: The discriminant: Hence, the roots are:
Step 2: Write the roots in exponential form.
Step 3: Evaluate the powers using periodicity. Reduce exponents modulo (or powers modulo ):
Step 4: Add the two terms:
Step 5: Writing the answer up to two decimal places:

Step 1: Given which implies
Step 2: Square both sides:
Step 3: Cancel the common term from both sides:
Step 4: Rearranging,
Step 5: Since , we divide both sides by :
Step 6: Hence, is false, so the required answer is:

Step 1: Given for some positive real constant .
Step 2: Hence,
Step 3: Squaring,
Step 4: Substitute into the given expression:
Step 5: Using the identity the expression becomes:
Step 6: Simplifying:
Step 7: Therefore, the required value is:

The condition represents a circle with centre and radius in the Argand plane. Step 1: To make minimum, draw a tangent from the origin to the circle. Let the tangent make an angle with the positive real axis, so its equation is
Step 2: Distance of the centre from the line is equal to the radius: Thus, the tangent is .
Step 3: The point of contact lies on both the line and the circle: Solving,
Step 4: Hence,

Step 1: Let .
Step 2:
Step 3: Separate real and imaginary parts:
Step 4: Eliminate : This is ellipse. Hence → (C).