Problem 7 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)=64 x^{3}+\frac{1}{x^{3}}\) and \(a, b\) are the roots of \(4 x+\frac{1}{x}=3\), (A) \(f(a)=12\) (B) \(f(b)=11\) (C) \(f(a)=f(b)\) (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these.

Step 1: Finding the Roots

We start by solving the equation \(4x + \frac{1}{x} = 3\) for \(x\). To do this, multiply every term by \(x\) to eliminate the fraction, obtaining \(4x^2 - 3x + 1 = 0\). Now, solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -3\), and \(c = 1\).

Step 2: Applying the Quadratic Formula

Plug the coefficients into the quadratic formula: \[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} = \frac{3 \pm \sqrt{9 - 16}}{8} = \frac{3 \pm \sqrt{-7}}{8}\]. Since the discriminant \(-7\) is negative, the roots are complex numbers, which cannot be substituted directly into \(f(x)\) as real numbers.

Step 3: Re-evaluating "Roots" and Their Properties

Given that \(4x + \frac{1}{x} = 3\) does not have real solutions, consider a possible misinterpretation involves looking for conditions these relationships might satisfy (or errors in checking properties), but typically we'd handle complex roots or assess separately given stated explicit value constraints. Here the goal is applying consistent standards or checking domain specifics, i.e., if on problem context allow complex evaluation authorization.

Step 4: Evaluating the Function f(x)

Given the speculated issues above, if considering complex numbers handling or substitute corrections for real roots but dealing explicitly through constraints, normally substitution would lead exploring solving or evaluate with \( f(x) = 64 x^3 + \frac{1}{x^3} \) at set 'roots', but typical functional methods would need proper approach or values to assert. Without real solutions and clarifications, choice answers become not viable set for defined direct roots solving without computations scalably managed towards function assessment optionality.

Key Concepts

Complex Numbers

In this problem, complex numbers play a critical role as solutions to the quadratic equation we encounter. You may not be familiar with complex numbers, so let's dive into their basics. A complex number is a number that comprises both a real part and an imaginary part. It is usually expressed in the form of \( a + bi \), where:

• \( a \) is the real part,
• \( b \) is the imaginary part, and
• \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

The roots of many equations, especially quadratics with a negative discriminant, are complex. In this exercise, the roots of the equation \( 4x + \frac{1}{x} = 3 \) involve complex numbers.
You can think of complex numbers as points or vectors in a two-dimensional space, with the real part determining the horizontal position, and the imaginary part determining the vertical position. This visualization helps you grasp how they behave differently than just plain real numbers, and why direct substitution into a function might not always be straightforward.
Understanding how to handle these numbers is crucial for comprehending many higher-level math concepts, including variables and functions with complex roots.

Discriminant

The discriminant is a key component in the quadratic formula and is essential in determining the nature of roots for any quadratic equation. The discriminant \( \Delta \) of a quadratic equation given by \( ax^2 + bx + c = 0 \) is calculated using the formula \( b^2 - 4ac \).

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the quadratic has exactly one real root, also known as a repeated or double root.
• When the discriminant is negative, the quadratic equation has no real roots, but instead, two complex roots.

In our example with the quadratic \( 4x^2 - 3x + 1 = 0 \), the discriminant is calculated as \( (-3)^2 - 4 \cdot 4 \cdot 1 = 9 - 16 = -7 \). Since \( -7 \) is negative, it confirms the presence of complex roots.
Mastering the concept of the discriminant allows you to quickly ascertain whether quadratic equations can be solved using real numbers and guides you on how to handle them, especially when dealing with functions or algebraic expressions requiring specific types of solutions.

Quadratic Formula

The quadratic formula is a well-known and fundamental tool in algebra for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides a direct method to derive both real and complex roots of a quadratic equation, utilizing the coefficients \( a \), \( b \), and \( c \) from the equation.
For \( 4x^2 - 3x + 1 = 0 \), we specify \( a = 4 \), \( b = -3 \), and \( c = 1 \). Substituting these into the quadratic formula yields complex solutions. The term under the square root, \( b^2 - 4ac \), is the discriminant, which we already discussed when negative, gives us complex solutions.

• The solution \( x = \frac{3 \pm \sqrt{-7}}{8} \) results in complex solutions clearly expressed within the formula.

Understanding how to use the quadratic formula is essential for anyone working with polynomials. It allows you to resolve equations efficiently and aids in recognizing the type of solutions you can expect based on the discriminant’s value. This problem, where the roots are complex, illustrates the formula's versatility.