Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\alpha\) is a repeated root of a quadratic equation \(f(x)=0\) and \(A(x), B(x), C(x)\) be polynomials of degree \(>2\), then the determinant \(\left|\begin{array}{ccc}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|\) is divisible by (A) \(A(x)\) (B) \(B(x)\) (C) \(C(x)\) (D) \(f(x)\)

Step-by-Step Solution

Answer

The determinant is divisible by \(f(x)\). (Option D)

Step 1: Understand the Problem

We have a determinant formed by evaluating and differentiating polynomials evaluated at a repeated root of a quadratic, and need to determine by which polynomial it is divisible.

Step 2: Define the Repeated Root Condition

A quadratic equation with a repeated root means it can be written as \(f(x) = (x-\alpha)^2\). Hence, \(f(\alpha) = 0\).

Step 3: Examine the Determinant Structure

The determinant \(|\begin{array}{ccc}A(x) & B(x) & C(x) \ A(\alpha) & B(\alpha) & C(\alpha) \ A'(\alpha) & B'(\alpha) & C'(\alpha)\end{array}|\) consists of polynomials and their derivatives evaluated at \(\alpha\).

Step 4: Utilize Polynomial Degree Information

Since \(A(x), B(x), C(x)\) have degrees greater than \(2\), substituting \(\alpha\) (root) shows effects on determinant roots if \(\alpha\) is a root for multiple functions without exception.

Step 5: Apply Polynomial Division Results

Every component of the determinant involves substitution by a polynomial with structure involving \((x - \alpha)^2\) facilitating division by \(f(x)\).

Step 6: Conclusion on Divisibility

Because \(\alpha\) being root causes polynomial terms 0, the determinant's terms fit the divisor structure of \(f(x) = (x-\alpha)^2\). Hence, the determinant is divisible by \(f(x)\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It is generally written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This equation is fundamental in algebra and can represent various physical phenomena, from projectile motion to the shape of a parabolic reflector.
Understanding the solutions, or roots, of a quadratic equation is key. There are different methods to find these roots, such as factoring, completing the square, or using the quadratic formula:

• Factoring involves rewriting the quadratic in the form \((px + q)(rx + s) = 0\) and solving for \(x\).
• Completing the square transforms the quadratic into a perfect square form \((x - h)^2 = k\).
• The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides a direct way to find the roots.

The expression under the square root, \( b^2 - 4ac \), is called the discriminant and determines the nature of the roots. A positive discriminant indicates two real and distinct roots, zero indicates a repeated root, and a negative discriminant signifies complex roots.

Repeated Root

In the context of a quadratic equation, a repeated root occurs when the discriminant \( b^2 - 4ac \) is zero. This condition implies that both roots of the quadratic are the same. When this happens, the quadratic can be expressed as \( (x - \alpha)^2 \), where \( \alpha \) is the repeated root.
A repeated root is also known as a double root or a root of multiplicity two. This means that on the graph of a quadratic equation, the curve touches the x-axis at a single point corresponding to the repeated root but does not cross it.
When dealing with determinants and polynomials, a repeated root has significant implications. If a repeated root of a quadratic equation \( f(x) = (x - \alpha)^2 \) is used for substitution in determinants or polynomial division, it simplifies analysis. This is due to the fact that multiple polynomial terms evaluated at \( \alpha \) tend to reduce to zero, lifting certain polynomial structures and making them divisible by \( f(x) \).

Polynomial Division

Polynomial division is similar to long division with numbers, but it involves dividing one polynomial by another. The division can yield a quotient and a remainder, where the original polynomial equals the divisor times the quotient plus the remainder.
When a polynomial is divisible by another polynomial without any remainder, we say it divides evenly. For instance, if a polynomial \( P(x) \) is divided by \( f(x) = (x - \alpha)^2 \) and results in no remainder, then \( P(x) \) is fully divisible by \( f(x) \).
In the exercise context, understanding polynomial division helps determine why the determinant

• \(|\begin{array}{ccc}A(x) & B(x) & C(x) \ A(\alpha) & B(\alpha) & C(\alpha) \ A'(\alpha) & B'(\alpha) & C'(\alpha)\end{array}|\)

is divisible by \( f(x) \). When substituting \( \alpha \) (a repeated root of \( f(x) \)) into the determinant, terms involving \( (x - \alpha)^2 \) simplify due to each term containing this factor. The determinant's expression then aligns with the form required for divisibility, simplifying polynomial division and confirming the divisibility condition.