Complex Numbers And Quadratic Equations

Concept: Complex Numbers - Algebraic Operations, Modulus, and Conjugate.

Step 1: Simplify the numerator of the complex number . Expand . Since , this becomes . Thus, .

Step 2: Rationalize the denominator to express in standard form. Multiply the numerator and denominator by the conjugate of the denominator, : .

Step 3: Calculate the modulus and solve for . The modulus is given by . .

Step 4: Equate to the given magnitude and find . We are given that . So, . This implies , or . Since the problem states , we must choose .

Step 5: Substitute back into and find the conjugate . Substitute into our standard form expression from
Step 2: . The question asks for the complex conjugate , which is found by simply changing the sign of the imaginary part. Therefore, . $ $

Concept: Calculus - Application of Derivatives (Monotonicity) and Intermediate Value Theorem.

Step 1: Define the function to analyze. Let . Finding the roots of the equation means finding the values of for which .

Step 2: Find the first derivative of the function. Differentiate with respect to to determine the behavior of the function. .

Step 3: Analyze the sign of the derivative. For any real number , the square of that number is non-negative ( ). Multiplying by 3 maintains this non-negativity ( ).
Adding 1 guarantees that the expression is strictly positive: . Thus, for all .

Step 4: Determine the monotonicity of the function. Because for all real numbers, the function is strictly increasing over its entire domain.
A strictly increasing continuous function can cross the x-axis at most one time. Therefore, it has at most one real root.

Step 5: Confirm the existence of a root using the Intermediate Value Theorem. Let's test some simple values. (which is negative). (which is positive). Because is a polynomial (and therefore continuous), and it changes sign between and , the Intermediate Value Theorem guarantees there is at least one root in the interval . Combined with Step 4, there is exactly one real root. $ $

We are given the complex number , and we need to find its modulus. Step 1: Recall the formula for the modulus of a complex number The modulus of a complex number is given by: Here, and . Step 2: Calculate the modulus Substitute and into the formula: Answer: The modulus of is , so the correct answer is option (1).

Step 1: Use the quadratic formula The quadratic formula to solve the equation is: For the given equation , we have: - , - , - . Step 2: Substitute the values into the quadratic formula Substitute , , and into the quadratic formula: Step 3: Solve for the two roots The two possible values for are: and Answer: Therefore, the roots of the quadratic equation are . So, the correct answer is option (1).