Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(q^{2}-4 p r=0, p>0\), then the domain of the function \(f(x)=\log \left[p x^{3}+(p+q) x^{2}+(q+r) x+r\right]\) is (A) \(R-\left\\{-\frac{q}{2 p}\right\\}\) (B) \(R-\left[(-\infty,-1] \cup\left\\{-\frac{q}{2 p}\right\\}\right]\) (C) \(R-\left[(-\infty,-1) \cap\left\\{-\frac{q}{2 p}\right\\}\right]\) (D) None of these

Step-by-Step Solution

Answer

Option A is correct. Domain: \(R-\left\{-\frac{q}{2 p}\right\}\).

Step 1: Understanding the Function Inside the Logarithm

The function inside the logarithm is \(px^3 + (p+q)x^2 + (q+r)x + r\). For the logarithm function to be defined, this polynomial must be positive, i.e., \(px^3 + (p+q)x^2 + (q+r)x + r > 0\).

Step 2: Analyzing the Given Condition

From the given condition \(q^2-4pr=0\), we can identify that this is the discriminant of a quadratic equation. This implies that the quadratic \(prx^2 + qx + r\) corresponding to the discriminant has equal roots, specifically at \(x = -\frac{q}{2p}\) since \(p > 0\).

Step 3: Analyzing the Behavior Near the Root

The polynomial \(px^3 + (p+q)x^2 + (q+r)x + r\) can be rewritten by substituting the root: \(px^3 + (p+q)x^2 + (q+r)x + r = p(x + \frac{q}{2p})^2g(x)\), where \(g(x)\) is some polynomial function. The function inside the log becomes zero at \(x = -\frac{q}{2p}\).

Step 4: Domain of the Function

The function \(f(x) = \log(px^3 + (p+q)x^2 + (q+r)x + r)\) is not defined at \(x = -\frac{q}{2p}\) because the logarithm of zero is undefined. Other than this point, the function is defined everywhere else on \(\mathbb{R}\) since there are no other restrictions indicated in the problem. Thus, the domain of \(f(x)\) is \(\mathbb{R} - \{-\frac{q}{2p}\}\).

Step 5: Selection of the Correct Answer Option

Compare the obtained domain with the given answer options. The correct option is (A), which states the domain as \(\mathbb{R} - \{-\frac{q}{2p}\}\). This matches our derived domain.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients. They involve only non-negative integer exponents of variables. A polynomial function can be presented in the form \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \). Here, each \( a_i \) represents a coefficient and is a real number, while \( x \) is a variable raised to the power of \( i \), an integer. This function can have one or more terms depending on the degree of the polynomial, which is the highest exponent of the variable present in the expression.

Examples of polynomial functions include:

• Linear polynomials: like \( 2x + 3 \)
• Quadratic polynomials: like \( 4x^2 + x + 1 \)
• Cubic polynomials: like \( x^3 - 3x + 5 \)

Polynomials are essential in algebra because they form the building blocks for more complex functions and equations. Their roots or solutions represent points where the value of the function equals zero, which are pivotal in determining the behavior of functions and solving higher degree equations.

Discriminant of Quadratic Equations

The discriminant is a part of the quadratic formula used to find the roots of quadratic equations of the form \( ax^2 + bx + c = 0 \). The discriminant \( \Delta \) is given by the expression \( b^2 - 4ac \). It determines the nature and number of roots of the quadratic function:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), the quadratic equation has exactly one real root, also known as a repeated or double root.
• If \( \Delta < 0 \), the quadratic equation has no real roots, but two complex roots.

In the context of our problem, the condition \( q^2 - 4pr = 0 \) implies that the polynomial related to the discriminant has a repeated root. This critical insight helps identify the root \( x = -\frac{q}{2p} \) for this particular quadratic equation when combined with the trigger that \( p > 0 \). Understanding the discriminant is crucial for predicting and analyzing polynomial behavior.

Domain of a Function

The domain of a function consists of all the input values \( x \) for which the function is defined. A function's domain is integral to understanding its usability and range of operation. In mathematical terms, it includes all real numbers \( x \) such that the function \( f(x) \) takes on real values.

To determine the domain of a logarithmic function like \( f(x) = \log(px^3 + (p+q)x^2 + (q+r)x + r) \), it is essential to consider where the inside of the logarithm is positive. Since the logarithm of zero or a negative number is undefined, the polynomial must be strictly greater than zero for the logarithm to exist.

• From our problem, we identified that the expression becomes zero at \( x = -\frac{q}{2p} \). Hence, this point is excluded from the domain.
• Other than this excluded point, \( f(x) \) is defined across all real numbers causing the domain to be \( \mathbb{R} - \{-\frac{q}{2p}\} \).

Ensuring comprehension of domain helps resolve where functions behave correctly, thus offering a complete understanding of potential inputs and resultant computations.