Complex Numbers

The correct answer is(A): 3x²y + 6xy-y³ + c

The correct option is (C):Statement-I is correct but Statement-II is false

Step 1: Understanding the Concept:
The equation defines the roots of unity. The question asks for the product of terms of the form , where the are the roots of unity excluding 1 itself.
Step 2: Key Formula or Approach:
The polynomial has the roots of unity as its roots. Therefore, it can be factored as:
We can also use the formula for the sum of a geometric series to write:
By comparing these two factorizations, we can deduce an expression for the product involving the roots.
Step 3: Detailed Explanation:
From the two factorizations of , we can equate the parts that do not contain the factor:
We are asked to find the value of the expression .
This can be obtained by substituting into the polynomial identity above.
Let .
We need to find .
The sum on the right-hand side has terms, and each term is equal to 1.
Step 4: Final Answer:
The value of the expression is .

Step 1: Understanding the Concept:
For three complex numbers to be the vertices of an equilateral triangle, they must satisfy the condition . In this problem, the vertices are , , and the origin, so we take .
Step 2: Key Formula or Approach:
An alternative approach is using the geometric property of rotation. If 0, , and form an equilateral triangle, then can be obtained by rotating about the origin by an angle of ( radians).
Step 3: Detailed Explanation:
From the rotation property:
Let . We can form a quadratic equation with these roots. Sum of roots: .
Product of roots: .
The quadratic equation is .
Substitute back :
Multiply the entire equation by (since ):
Rearranging this gives:
Step 4: Final Answer:
The condition for the vertices 0, , and to form an equilateral triangle is .

Step 1: Understanding the Concept:
This question tests fundamental properties of the complex exponential function .
Step 2: Detailed Explanation:

(A) is never zero: Let . Then . The magnitude is . Since is always positive for any real , the magnitude is never zero. Therefore, can never be zero. This statement is True.
(B) if x is real: By Euler's formula, . The magnitude is . This statement is True.
(C) if z is an integral multiple of : Let for some integer . Then . This statement is True.
(D) ...: The condition is equivalent to . From statement (C), this means that must be an integral multiple of . The statement given is , which is incorrect due to the factor. This statement is False.
(E) for : This inequality is not well-defined because is generally a complex number, and inequalities (other than for magnitude) are not defined for complex numbers. Furthermore, if is a positive real number, , so the inequality is false. This statement is False.
Step 3: Final Answer:
Statements (A), (B), and (C) are true. Therefore, option (C) is the correct choice.

Step 1: Understanding the Concept:
This problem requires solving four different equations in the complex domain and matching them to the correct description or set of roots.

Step 2: Detailed Explanation:
(A) :
This implies . So, . Let this ratio be . Since , solving gives If , then simplifying leads to The roots are purely imaginary. (A) matches (II).

(B) :
This is the sum of a geometric series: Thus, roots are the 6th roots of unity excluding 1. Explicitly: In Cartesian form: . (B) matches (III).

(C) :
This gives So, Let . Simplification gives: Thus, (C) matches (IV).

(D) :
So, The roots are These correspond to . (D) matches (I).

Step 3: Final Answer:
The correct matching is: Correct Choice: Option (A).

Step 1: Understanding the Concept:
The imaginary cube roots of unity are and . They satisfy two fundamental properties: 1. 2. We can let the imaginary cube root 'a' be .

Step 2: Key Formula or Approach:
We will use the properties of the cube roots of unity to simplify the expressions inside the parentheses before raising them to the power of 5. From , we can derive: - -

Step 3: Detailed Explanation:
Let's substitute into the given expression. First term: Group the terms: Substitute : Since , we can simplify . So the first term is . Second term: Group the terms: Substitute : Simplify . So the second term is . Now, add the two simplified terms: From the property , we know that .
Step 4: Final Answer:
The value of the expression is 32.

Step 1: Understanding the Concept:
The argument of a complex number, , represents the angle it makes with the positive real axis. The condition has a geometric interpretation: the angle subtended by the line segment from to at the point is . The locus of points satisfying such a condition is an arc of a circle passing through the points -1 and 1. We can find the equation of this circle algebraically.

Step 2: Key Formula or Approach:
Let . We will substitute this into the expression, simplify the complex fraction into the form , and then use the property that .

Step 3: Detailed Explanation:
First, substitute : Multiply the numerator and denominator by the conjugate of the denominator: The argument of this complex number is . We are given that this is . Therefore, we have: Rearrange the equation to find the locus: Divide by to get a standard form for a circle: This is the equation of a circle. It matches option (D) exactly.

Step 4: Final Answer:
The locus of the point z is the circle given by the equation .