Problem 31 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{10}\) be in G.P. with \(a_{i}>0\) for \(i=1,2, \ldots, 10\) and \(S\) be the set of pairs \((r, k), r, k \in N\) (the set of natural numbers) for which $$ \left|\begin{array}{lll} \log _{e} a_{1}^{r} a_{2}^{k} & \log _{e} a_{2}^{r} a_{3}^{k} & \log _{e} a_{3}^{r} & a_{4}^{k} \\ \log _{e} a_{4}^{r} a_{5}^{k} & \log _{e} a_{5}^{r} a_{6}^{k} & \log _{e} a_{6}^{r} & a_{7}^{k} \\ \log _{e} a_{7}^{r} a_{8}^{k} & \log _{e} a_{8}^{r} a_{9}^{k} & \log _{e} a_{9}^{r} a_{10}^{k} \end{array}\right|=0 $$ Then the number of elements in \(\mathrm{S}\), is : [Jan. \(\mathbf{1 0}, \mathbf{2 0 1 9}\) (II)] (a) 4 (b) infinitely many (c) 2 (d) 10

Step-by-Step Solution

Answer

The number of elements in S is 2.

Step 1: Understand the Problem

The problem statement involves a determinant of a 3x3 matrix where each element is expressed in terms of logarithms of terms from a geometric progression (G.P.). We need to find the values of \(r\) and \(k\) such that the determinant is zero.

Step 2: Express the Terms

Each term like \( \log_{e} (a_{i}^{r} a_{j}^{k}) \) can be expanded using logarithmic properties as \( r \log_{e} a_{i} + k \log_{e} a_{j} \). The terms \( a_{i} \) in G.P. can be written in terms of the first term \( a_{1} \) and common ratio \( q \), i.e., \( a_{i} = a_{1} q^{i-1} \). Use this to express each element in the determinant in terms of \( a_{1} \) and \( q \).

Step 3: Substitute into the Matrix

Substitute the expressions derived in Step 2 into the original determinant. It will involve replacing these terms with their logarithmic expressions expressed in \( a_{1} \) and \( q \).

Step 4: Simplify the Determinant

Recognize that logarithms reduce the multiplicative property of the G.P., simplifying the matrix entries. With algebraic manipulation, the structure of the matrix simplifies due to the common ratio factor cancelling out due to the G.P. structure of terms.

Step 5: Determine When the Determinant is Zero

This involves finding when the linear algebraic expressions in terms of \( r \) and \( k \) lead to a zero determinant. The determinant will be zero if the matrix is linearly dependent, which reduces in this case to patterns or constraints on \( r \) and \( k \).

Step 6: Solve for r and k

Conclude on the pairs \((r, k)\) that satisfy the determinant condition. This can involve setting constraints that arise from simplifying and checking common factors or relationships derived from symmetry or structure in a G.P. when the determinant becomes zero.

Key Concepts

Understanding Geometric Progression (G.P.)

In a geometric progression, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Suppose we have a sequence \(a_1, a_2, a_3, \ldots, a_n\). This sequence is in G.P. if and only if \(a_{i+1} = a_i \cdot q\) for some common ratio \(q\).

• The first term is denoted by \(a_1\).
• The \(i\)-th term is \(a_i = a_1 \cdot q^{i-1}\).
• A crucial property of G.P. is that the ratio \(\frac{a_{i+1}}{a_i}\) is constant and equal to \(q\).

This structure is essential for solving problems involving G.P., as it provides a simple way to express any term using the first term and the common ratio. This method allows for easier manipulation and application of additional concepts like logarithms or determinants.

Exploring Logarithm Properties in Matrices

Logarithms transform multiplicative processes into additive ones, thanks to properties such as \(\log_{b}(xy) = \log_{b}x + \log_{b}y\). This feature simplifies handling terms from a geometric progression within matrix elements.

• When terms like \(\log_e(a_i^r a_j^k)\) appear, they can be rewritten using the property: \(r \cdot \log_e a_i + k \cdot \log_e a_j\).
• This transformation turns the originally multiplicative G.P. factor into a simpler, additive form.

Applying these logarithmic transformations helps simplify expressions within matrices, particularly when calculating determinants. The use of logarithms is particularly effective in linear algebra, where operations are often performed on sums, allowing one to reduce complex problems to simpler forms.

Solving Linear Algebra Problems with Determinants

Linear algebra often involves determinants, particularly in solving systems of linear equations or analyzing properties of matrices. The determinant of a matrix provides key information about the matrix, such as whether it is invertible.

• If a determinant is zero, it indicates that the matrix might be rank-deficient.
• A rank-deficient matrix implies linear dependence among rows or columns.

In our exercise, we consider a determinant set to zero, which suggests examining the linear relations among the matrix's rows or columns. This step involves:

• Finding combinations of \(r\) and \(k\) in which the matrix reduces to a form showcasing these dependencies.
• Solving algebraic equations derived from the matrix expressions to establish the pairs \((r, k)\) that satisfy the zero determinant condition.

Understanding these determinants' properties allows one to explore configurations leading to specific solutions, essential in many areas of mathematics and its applications.