Problem 33 - Algebra for JEE ADVANCED

Question

If \(a, b, c\) are positive and are the \(p^{\text {ll }}, q^{\text {Ht }}\) and \(r^{\text {th }}\) terms, respectively, of a G.P., then \(\Delta=\left|\begin{array}{lll}\log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1\end{array}\right|\) is a. 0 b. \(\log (a b c)\) c. \(-(p+q+r)\) d. none of these

Step-by-Step Solution

Answer

The value of \(\Delta\) is 0, so the answer is (a) 0.

Step 1: Understand the Given Problem

Identify the terms in the problem. We have that \(a, b, c\) are the \(p^{\text{th}}, q^{\text{th}}\), and \(r^{\text{th}}\) terms of a geometric progression (G.P.) respectively.

Step 2: Express Terms in Mathematical Form

In a G.P., the \(n^{\text{th}}\) term is given by \(ar^{n-1}\). Hence, \(a = ar^{p-1}, b = ar^{q-1}, c = ar^{r-1}\). Simplifying gives \(a = a, b = ar^{q-p}, c = ar^{r-p}\).

Step 3: Construct the Determinant

The determinant \(\Delta = \left| \begin{array}{ccc} \log a & p & 1 \ \log b & q & 1 \ \log c & r & 1 \end{array} \right|\). To compute this, use the cofactor expansion across the third column.

Step 4: Compute the Cofactors

Cofactor expansion gives \(\Delta = \log a \cdot (q - r) - \log b \cdot (p - r) + \log c \cdot (p - q)\).

Step 5: Substitute Logarithmic Expressions

Substitute the expressions \(\log a = \log a\), \(\log b = \log(ar^{q-p}) = \log a + (q-p)\log r\), \(\log c = \log(ar^{r-p}) = \log a + (r-p)\log r\) into the cofactor expansion.

Step 6: Simplify the Expression

Using the expressions from Step 5, substitute them back into the determinant formula and simplify. It will result in terms cancelling out, yielding \(\Delta = 0\).

Step 7: Conclusion

After substituting and simplifying the determinant using properties of logarithms and the terms of G.P., we find that \(\Delta = 0\). This matches option a.

Key Concepts

Geometric Progression

Geometric Progression (G.P.) is a fascinating concept in mathematics, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Let's delve a bit deeper:

• The common ratio can be any real number, which may be greater than 1 (causing the sequence to grow) or between 0 and 1 (causing the sequence to shrink).
• The formula for the nth term of a G.P. is given as \( a_n = ar^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio.
• In the exercise, \( a \), \( b \), and \( c \) are terms of a G.P., meaning each term is derived from multiplying the previous term by the same common ratio.

Understanding these basics aids in finding values or simplifying expressions in problems involving geometric sequences.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinant of a matrix. The process involves breaking down a larger determinant into smaller, more manageable minors, making it a fundamental technique in linear algebra:

• Start by selecting a row or column from the matrix for expansion; typically, a row or column with the most zeros is chosen for simplicity.
• Each element of the selected row or column is then multiplied by the determinant of the minor matrix, which is formed by removing the row and column of the element in question.
• The elements are summed, with their respective signs determined by their position (alternating signs starting with a positive).

The exercise demonstrates this by using cofactor expansion across the third column of the given matrix, simplifying the computation of the determinant.

Logarithmic Expressions

Logarithmic expressions provide a powerful tool for simplifying complex algebraic calculations. A logarithm answers the question: "What power must a given base be raised to, to obtain this number?" Here's a closer look:

• The expression \( \log_b(x) \) reads as "the logarithm of \( x \) to base \( b \)," meaning the power to which \( b \) is raised to yield \( x \).
• Logarithms have various properties that transform products into sums and powers into products, thus simplifying multiplication and division.
• For example, \( \log(ar^{n}) = \log a + n \log r \), decomposing into more manageable parts.

In the provided exercise, logarithmic properties were used to transform geometrically progressive terms into simpler expressions in preparation for determinant evaluation.

Properties of Determinants

Determinants are mathematical objects that summarize certain properties of matrices, useful in solving systems of linear equations, among other applications. Here's why they are important:

• A determinant of a square matrix provides insight into the matrix's characteristics, such as solvability of corresponding linear systems and matrix invertibility.
• Properties like linearity with respect to rows and columns, and the effect of row swapping or multiplication, facilitate easier computation.
• An important property used in the exercise was that when a determinant has two identical rows (or columns), its value is zero.

Understanding and applying properties of determinants simplifies problem-solving processes, like confirming that the given determinant in the problem is zero.