Problem 50 - A Complete Resource Book in Mathematics for JEE Main 2017

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

A. The domain is all real numbers except at \( x = -\frac{q}{2p} \).

Step 1: Identify the Condition

Given the condition, \( q^2 - 4pr = 0 \), this implies that the quadratic equation \( px^2 + qx + r = 0 \) has a double root, known as a repeated or coincident root. The root can be calculated as: \( x = \frac{-q}{2p} \). This also means the expression within the function can be factored as a perfect square.

Step 2: Rewrite the Function

The function \( f(x) = \log \left[ px^3 + (p+q)x^2 + (q+r)x + r \right] \) must be positive for the logarithm to be defined. Rewrite this expression as: \( p(x^3) + (p+q)x^2 + (q+r)x + r = (px^2 + qx + r)(x+1) \) since \( q^2 - 4pr = 0 \) has the root \( x = \frac{-q}{2p} \), indicating a perfect square \( px^2 + qx + r = (px + \frac{q}{2})^2 \).

Step 3: Analyze the Expression Sign

For \( f(x) \) to be defined, the expression inside the logarithm must be positive. It means \( (px^2 + qx + r)(x+1) > 0 \). The term \( px^2 + qx + r \) is zero at \( x = \frac{-q}{2p} \). This expression changes its sign at this root and at \( x = -1 \) if \( (x+1) \) is involved.

Step 4: Determine the Domain

Since \( px^2 + qx + r \) is zero at \( x = \frac{-q}{2p} \), the function is not defined at this point. The interval \( (-rac{q}{2p}, \infty) \) is where the expression is positive, excluding \( x = -1 \) when negative for \( x < -1 \). The domain includes all real numbers except \( x = \frac{-q}{2p} \) where \( f(x) \) is zero.

Key Concepts

Logarithmic Functions

Logarithms are powerful mathematical tools used to reverse the effects of exponentiation. In simpler terms, if you know the result of raising a number to a power, a logarithm helps you find the original power. The function for a logarithm is usually written as \( \log_b(a) = c \) meaning that \( b^c = a \).

In the context of our exercise, we refer to the domain of the logarithmic function\( f(x) = \log(px^3 + (p+q)x^2 + (q+r)x + r) \). The domain is crucial, as the expression inside a logarithm must always be positive.

• This ensures the function is well-defined.
• The argument cannot be zero or negative since the logarithm of non-positive numbers is not defined in real numbers.

Thus, analyzing conditions when the expression \( px^3 + (p+q)x^2 + (q+r)x + r \) equals zero helps us determine where the function is undefined.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). The solutions or roots of these equations can be found using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

When discussing the exercise, the condition \( q^2 - 4pr = 0 \) tells us the equation \( px^2 + qx + r = 0 \) has a special characteristic known as a repeated root, where both solutions of the equation are the same.

• This occurs when the discriminant, \( b^2 - 4ac \), is zero, indicating that there is one unique solution.
• In our specific case, this root is \( x = \frac{-q}{2p} \).

This impacts the problem because the value of the expression inside the logarithm at this root is zero, which affects the domain.

Repeated Roots

Repeated roots occur when a quadratic equation has a discriminant of zero, leading to both roots being equal. This causes the quadratic to factor into a perfect square.

• For the quadratic equation \( px^2 + qx + r = 0 \), observing a repeated root results in \( (px + \frac{q}{2})^2 \) being the factorization.
• The root \( x = \frac{-q}{2p} \) repeats, meaning the expression \( px^2 + qx + r \) changes sign at this point and affects the expression's positivity.

Understanding where roots repeat helps define where related functions are zero or where sign changes occur, which is essential when determining the domain, particularly when combined with other terms as in our exercise.