Problem 83 - Algebra for JEE ADVANCED

Question

If \(a, b, c\) are in G.P. with common ratio \(r_{1}\) and \(\alpha, \beta, \gamma\) are in G.P. with common ratio \(r_{2}\), and equations \(a x+\alpha y+z=0\), \(b x+\beta y+z=0, c x+y y+z=0\) have only zero solution, then which of the following is not true? a. \(r_{1} \neq 1\) b. \(r_{2} \neq 1\) c. \(r_{1} \neq r_{2}\) d. none of these

Step-by-Step Solution

Answer

None of the given statements are always true due to the zero determinant condition.

Step 1: Define G.P.

A geometric progression (G.P.) has the property that each term after the first is the previous term multiplied by a fixed, non-zero number called the common ratio. Here, for the sequence \(a, b, c\), the common ratio is \(r_1\) such that \(b = a \cdot r_1\) and \(c = b \cdot r_1 = a \cdot r_1^2\). Similarly, for the sequence \(\alpha, \beta, \gamma\), the common ratio is \(r_2\) such that \(\beta = \alpha \cdot r_2\) and \(\gamma = \beta \cdot r_2 = \alpha \cdot r_2^2\).

Step 2: Interpret Conditions for Zero Solutions

For the system of equations \(ax + \alpha y + z = 0\), \(bx + \beta y + z = 0\), \(cx + \gamma y + z = 0\) to have only the trivial (zero) solution, the determinant of the coefficient matrix must be zero. The matrix is:\[\begin{pmatrix}a & \alpha & 1 \b & \beta & 1 \c & \gamma & 1 \end{pmatrix}\]and its determinant must equal zero.

Step 3: Evaluate Determinant

Calculate the determinant of the matrix:\[\left| \begin{array}{ccc}a & \alpha & 1 \b & \beta & 1 \c & \gamma & 1 \end{array} \right| = a(\beta - \gamma) - \alpha(b - c) + 1(b\gamma - c\beta).\]Since \(b = ar_1\), \(c = ar_1^2\), \(\beta = \alpha r_2\), and \(\gamma = \alpha r_2^2\), substitute these into the expression and simplify. The determinant, expressed in terms of \(a\), \(\alpha\), \(r_1\), and \(r_2\), should equal zero given the initial conditions.

Step 4: Analyze Simplified Determinant

After substituting and simplifying, you get:\[\alpha a^2 r_1^2 (1 - r_2) + \alpha r_2^2 a (1 - r_1) = 0.\]For this to hold true for the zero solution condition (determinant = 0), each term must zero out. This implies either \(r_1 = 1\), \(r_2 = 1\), or \(r_1 = r_2\). Each corresponding term's coefficient or difference must equal zero.

Step 5: Determine Which Statements are Possible

The options given must be considered in light of whether they contradict the zero determinant requirement. Since \(r_1\), \(r_2\), and their difference can equal various values subject to conditions, consider options:1. If \(r_1 = 1\), then the first term zeroes.2. If \(r_2 = 1\), then terms are not necessarily zero.3. If \(r_1 = r_2\), then both terms potentially zero. Therefore, statements A, B, and C contradict the zero solution condition.

Key Concepts

System of Equations

Understanding a system of equations is crucial when solving problems involving multiple variables. Such a system consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this context, we have the equations:

• \( ax + \alpha y + z = 0 \)
• \( bx + \beta y + z = 0 \)
• \( cx + \gamma y + z = 0 \)

These equations have been set to zero, meaning we are dealing with a homogeneous system. A homogeneous system will always have at least the trivial solution, where all variables equal zero. However, in this exercise, we need to ensure that this is the only solution, pointing to certain dependencies among the coefficients.

Determinant of Matrix

A matrix is a rectangular array of numbers, and its determinant is a special value that can tell us important information about the system of equations it represents. For the given problem, the coefficient matrix is represented as:\[\begin{pmatrix}a & \alpha & 1 \b & \beta & 1 \c & \gamma & 1 \\end{pmatrix}\]The determinant of this matrix should be zero to ensure that the system of equations only has a trivial solution (zero solution). The determinant is calculated using the formula:\[a(\beta - \gamma) - \alpha(b - c) + 1(b\gamma - c\beta)\]This formula allows you to check if the coefficients would yield a non-trivial solution or not. By ensuring the determinant is zero, we confirm the only solution possible is the trivial solution.

Trivial Solution

In linear algebra, a trivial solution refers to the simplest type of solution to a system of equations, where all variables are set to zero. For the given set of equations, the trivial solution \((x = 0, y = 0, z = 0)\) is desired in this problem. This occurs specifically when the determinant of the coefficients' matrix is zero, indicating that the system is neither independent nor inconsistent.

• In independent systems, multiple solutions may exist; the zero determinant prevents this.
• In inconsistent systems, no solutions exist at all, which isn't possible with zero determinant.
• Thus, a zero determinant guides the system only to have the trivial solution available.

This understanding is crucial when interpreting the conditions needed for the coefficients in our problems, ensuring no other solutions satisfy the equation other than all-zero values.

Common Ratio in Sequences

In a geometric progression, the common ratio is what links each term to the one following it. For a sequence \( a, b, c \) in such a progression:

• \( b = a \cdot r_1 \)
• \( c = b \cdot r_1 = a \cdot r_1^2 \)

This relationship demonstrates the multiplying effect of the common ratio \( r_1 \) through the sequence.Similarly, for the second sequence \( \alpha, \beta, \gamma \), the common ratio \( r_2 \) is key:

• \( \beta = \alpha \cdot r_2 \)
• \( \gamma = \beta \cdot r_2 = \alpha \cdot r_2^2 \)

Understanding the common ratio helps make sense of the dependencies among terms, crucial for simplifying the determinant equation from the matrix. Recognizing when these ratios are equal or equal to one can radically change the types of solutions one expects in the system of equations.