Complex Numbers And Quadratic Equations

...(i) One root is and ...(ii) E (ii) - E (i), we get Then, equation is Roots Sum of roots So, another root .

Let and . From the first equation , we get: Now, from the second equation , we have: Thus: Substituting into the equation: Thus, . Therefore, the correct answer is (D).

Step 1: Understanding the Concept:
. So .

Step 2: Detailed Explanation:
because . Thus .

Step 3: Final Answer:
Option (D).

Concept:

Step 1: Rationalize denominator.


Step 2: Expand numerator.


Step 3: Extract real and imaginary parts.


Step 4: Apply given condition.


Step 5: Solve equation.

Concept: Let , then:

Step 1: Find real and imaginary parts.


Step 2: Compute modulus squared.

Concept: Use and simplify complex numbers before exponentiation.

Step 1: Simplify given expression.


Step 2: Convert to polar form (optional shortcut).


Step 3: Apply De Moivre’s theorem.

Concept:

Step 1: Find modulus of base.


Step 2: Apply power property.

Concept:
• Powers of are cyclic with period 4:
• This cycle repeats for all higher powers

Step 1: Identify repeating pattern
Each group of 4 terms sums to:

Step 2: Count number of terms
From to :

Step 3: Divide into complete cycles


Step 4: Sum remaining terms
Remaining terms: Final Conclusion:

Step 1: Understanding the Concept:
Vertices of a triangle given as complex numbers can be mapped directly to coordinates on the Cartesian (Argand) plane.
The problem involves finding an unknown coordinate based on the properties of a right-angled isosceles triangle.

Step 2: Key Formula or Approach:
Convert the complex numbers to coordinate points: , , and .
Use geometric properties: The angle at is , meaning line segments and are perpendicular.
Also, since it is isosceles, the lengths of the legs adjacent to the right angle must be equal (i.e., ).

Step 3: Detailed Explanation:
Let the vertices in the coordinate plane be:



The triangle is right-angled at .
Notice that the x-coordinates of points and are both .
This implies that the line passing through and is a vertical line parallel to the y-axis.
For angle to be , the line passing through and must be perpendicular to .
Since is a vertical line, must be a completely horizontal line parallel to the x-axis.
A horizontal line has a constant y-coordinate for all its points.
Therefore, the y-coordinate of must equal the y-coordinate of .
This gives us: .
Let us verify if the triangle is isosceles with :
The length of segment is the distance between and :
units.
The length of segment is the distance between and :
units.
Since , the triangle is indeed an isosceles right-angled triangle.

Step 4: Final Answer:
The value of is .

Step 1: Rationalize the Denominator


Step 2: Simplify
Denominator: Numerator:

Step 3: Find Argument
Since lies in 4th quadrant:

Step 4: Final Answer