Question

Let \(z\) be a complex number such that the imaginary part of \(z\) is nonzero and \(a=z^{2}+z+1\) is real. Then \(a\) cannot take the value (A) \(-1\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{2}\) (D) \(\frac{3}{4}\)

Short answer

Given that the imaginary part of z is nonzero, and a is real, a can take any real value except those greater than 2. Therefore, a cannot take the value of (D) 3/4.

Step-by-step solution

Express the complex number

Let the complex number be expressed as z = x + yi, where x and y are real numbers and y ≠ 0 since the imaginary part of z is nonzero.

Square the complex number

Calculate the square of z: z^2 = (x + yi)^2 = x^2 + 2xyi - y^2. The real part of z^2 is x^2 - y^2 and the imaginary part is 2xyi.

Add the terms

Add up all the terms to get a: a = z^2 + z + 1 = (x^2 - y^2) + (x + yi) + 1 = (x^2 + x - y^2 + 1) + (2xy)i.

Set the imaginary part to zero

Since a is said to be real, the imaginary part must be zero: 2xy = 0. Given that y ≠ 0, it must be that x = 0.

Substitute x to find a

With x = 0, we can rewrite a as a = 0^2 + 0 + 1 - y^2 + 1 = 1 - y^2 + 1 = 2 - y^2.

Exclude the possible value of a

Since y ≠ 0, y^2 is positive, so 2 - y^2 < 2. Therefore, a cannot equal any value greater than or equal to 2.

Determine the impossible value

Review the given options and determine which cannot be the value of a. Since only -1 is less than 2 and the others are greater, a cannot take the value of (D) 3/4, as 2 - y^2 will never be 3/4 for any non-zero y.

Key concepts

Imaginary Part of Complex Numbers

Understanding the imaginary part of a complex number is crucial for anyone delving into complex number problems, especially in competitive exams like JEE Advanced. A complex number, denoted as \( z \), can be expressed in the form \( z = x + yi \), where \( x \) is the real part, and \( yi \) is the imaginary part with \( y \) being a real number and \( i \) representing \( \sqrt{-1} \).

Now, a key aspect to recognize when solving complex number problems is to distinguish properly between these two parts. In the exercise given, the condition that the imaginary part is non-zero tells us that \( y \) is not equal to zero. This requirement is vital when deducing the characteristics of the complex number and, subsequently, solving for the real number \( a \).

In the JEE Advanced, problems focusing on the imaginary part of complex numbers often involve operations like addition, subtraction, multiplication, or even complex conjugation. Remember, it is the manipulation of this imaginary part that will affect the nature of the complex number when considering if the final result is real or still has an imaginary component.

Real Part of Complex Numbers

Parallel to understanding the imaginary part, grasping the concept of the real part of complex numbers is equally essential. The real part, denoted as \( x \) in \( z = x + yi \), is what remains when you subtract the imaginary part from the complex number. It is a real number, meaning it can be found on the conventional number line.

When dealing with JEE Advanced mathematics problems, focusing on the real part can guide you towards simplifying complex equations or understanding the behavior of a complex function. If a problem specifies that the result of certain operations involving complex numbers is real, as in our exercise, then the equation's imaginary parts must cancel out.

Students should practice isolating the real part and setting the imaginary part to zero, which is a technique used in the textbook exercise. This approach led to identifying that the value of \( x \) should be zero, an insight that is vital for solving this particular type of complex number problem.

Solving Complex Number Problems

Solving complex number problems can seem daunting, but with practice, it becomes a fascinating journey through a mathematical landscape full of imagination. It's all about breaking down the problem into manageable steps and understanding properties and operations involving complex numbers.

In the JEE Advanced context, problems are crafted to test your agility with concepts like multiplication of complex numbers, finding their modulus, or dealing with their real and imaginary parts separately. As shown in our exercise, the process started with expressing the complex number in standard form and proceeded with operations like squaring the complex number, setting imaginary parts to zero, and manipulating the resulting real parts.

Improvement in solving complex problems comes from practicing these operations in various scenarios, keeping in mind the ultimate goal: obtaining a real or imaginary result based on the question's conditions. The process might involve equating the imaginary part to zero (to find the real values) or other manipulations as needed by the problem.

JEE Advanced Mathematics

JEE Advanced Mathematics encompasses a wide array of topics and challenging problems, with complex numbers being a significant part of the syllabus. For success in this competitive exam, understanding every nuance of complex numbers is indispensable.

It involves more than rote learning; students must internalize concepts and develop strong problem-solving strategies. The question types often test not just knowledge, but also the application and combination of multiple concepts, such as the usage of complex numbers in coordinate geometry or calculus.

Tackling problems effectively requires a balanced approach to both theoretical understanding and practical application. Regular and rigorous practice, particularly of problems like the one discussed in this article, will sharpen the students' skills. Students should also review step-by-step solutions and practice with varied problems to develop the depth of understanding required for the JEE Advanced Mathematics section.