Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|\begin{array}{ccc}\frac{1}{a+x} & \frac{1}{b+x} & \frac{1}{c+x} \\\ \frac{1}{a+y} & \frac{1}{b+y} & \frac{1}{c+y} \\ \frac{1}{a+z} & \frac{1}{b+z} & \frac{1}{c+z}\end{array}\right|=\frac{P}{Q}\), where \(Q\) is the product of denominators, then \(P\) is equal to (A) \((a-b)(b-c)(c-a)\) (B) \((x-y)(y-z)(z-x)\) (C) \((a-b)(b-c)(c-a)(x-y)(y-z)(z-x)\) (D) None of these

Step-by-Step Solution

Answer

P is equal to (x-y)(y-z)(z-x), so the answer is (B).

Step 1: Understand the Matrix and Determinant

We have a 3x3 determinant with elements that are reciprocals of linear expressions. To find what \(P\) equals, we need to evaluate this determinant by considering common factorization patterns and properties specific to determinants.

Step 2: Express the Determinant

Express the determinant \[\det \begin{bmatrix}\frac{1}{a+x} & \frac{1}{b+x} & \frac{1}{c+x} \\frac{1}{a+y} & \frac{1}{b+y} & \frac{1}{c+y} \\frac{1}{a+z} & \frac{1}{b+z} & \frac{1}{c+z}\end{bmatrix}.\]This determinant is calculated by expansion as a sum involving the products of elements of diagonals.

Step 3: Hypothesize and Simplify

Recognize that differences like \((a-b), (b-c), (c-a)\) often factor determinants of this form. We could, for instance, suspect symmetry in the determinant and apply any available symmetrical identities that simplify evaluation.

Step 4: Relate Q and Apply Property

Since \(Q\) is the product of all the denominators, it would look like \[(a+x)(b+x)(c+x)(a+y)(b+y)(c+y)(a+z)(b+z)(c+z).\] Note that the determinant being zero (when differences vanish) gives clues as it should contain factors of such differences.

Step 5: Evaluate Determinants Directly and Isolate P

By factoring common expressions multiple times, note that the determinant's non-zero evaluations often will depend on particular differences in diagonals: \((x-y), (y-z), (z-x)\). Due to the known pairing and factoring of expanded determinant transposes, \(P = (x-y)(y-z)(z-x)\).

Step 6: Select the plausible answer

Considering each option and identifying the factors present, examine how the symmetry suggests repeating patterns like \((x-y)(y-z)(z-x)\). Hence, choice (B) \((x-y)(y-z)(z-x)\) is appropriate.

Key Concepts

Determinant Properties

Determinants are mathematical expressions that are useful in solving systems of linear equations, finding matrix inverses, and more. They have special properties that help simplify their evaluation. One helpful property is the idea of linearity, which lets you pull out constants from rows or columns. This is handy when factors repeat across rows, because you can factor them out, making the determinant simpler to calculate.

Another useful property is the ability to interchange rows or columns. Swapping two rows or two columns changes the sign of the determinant. This is useful if you need to rearrange elements for simplification purposes.

• Linearity: Constants can be factored out from rows/columns, which simplifies calculations.
• Row/Column Swap: Switching rows or columns reverses the sign of the determinant.

By understanding these properties, you can make smarter choices when solving determinant problems, leading to quicker and more accurate solutions.

Factorization

Factorization is a vital concept in simplifying mathematical expressions, including determinants. In our specific determinant problem, factorization helps us identify common patterns or differences, like \(a-b\), \(b-c\), \(c-a\), which are often found in polynomial expressions associated with determinants.

When evaluating determinants of reciprocal linear expressions, like the exercise presented, looking for these common factors across rows or columns can reduce complexity. If the determinant can be factored to extract such expressions, then it's possible to condense it significantly.

• Often involve differences like \(a-b, b-c, c-a\).
• Simplifies evaluation significantly by reducing crossing elements.

By applying factorization techniques, determinant evaluation becomes less daunting, making the solution both clear and manageable.

Symmetrical Identities

Symmetrical identities involve expressions that maintain certain properties even if their elements are rearranged. In our matrix with determinants, symmetrical identities help identify forms or patterns that simplify the evaluation. Recognizing symmetry can sometimes reveal intrinsic relationships between elements of the determinant.

Think of symmetric properties in terms of repeating patterns like \(x-y\), \(y-z\), \(z-x\) in a determinant. These patterns emerge in problems with three-variable dependencies, such as the presented exercise. Applying symmetrical identities lets you exploit inherent balance among matrix components.

• Recognizes patterns \(x-y, y-z, z-x\).
• Simplifies asymmetric matrices to derive relationships quickly.

Leveraging symmetry, we find solutions that otherwise might be concealed amid complex calculations, highlighting elegant expressions like \(x-y\) in the process.