Problem 104 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion (A) is given followed by a Reason. (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(a, b, c\) are in G.P. with common ratio \(r_{1}\) and \(\alpha, \beta, \gamma\) are in G.P. with common ratio \(r_{2}\). If the equations \(a x+\alpha y+z=0, b x+\beta y+z=0, c x+\gamma y+\) \(z=0\) have only trivial solution, then \(r_{1} \neq r_{2}, r_{1}, r_{2} \neq 1\) Reason: The equations \(a_{1} x+b_{1} y+c_{1} z=0, a_{2} x+b_{2} y\) \(+c_{2} z=0, a_{3} x+b_{3} y+c_{3} z=0\) have only trivial solution if \(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\\ a_{3} & b_{3} & c_{3}\end{array}\right| \neq 0\)

Step-by-Step Solution

Answer

(B) Assertion is true; Reason is true, but not a correct explanation for Assertion.

Step 1: Understanding the Assertion

The assertion states that if \(a, b, c\) are in geometric progression (G.P.) with common ratio \(r_1\) and \(\alpha, \beta, \gamma\) are also in G.P. with common ratio \(r_2\), and the equations \(ax+\alpha y+z=0, bx+\beta y+z=0, cx+\gamma y+z=0\) only have trivial solution, then it must be the case that \(r_1 eq r_2\) and \(r_1, r_2 eq 1\). This implies that the given conditions form a certain matrix or determinant that must not equal zero.

Step 2: Evaluating the Reason

The reason given is that for the system of equations \(a_1 x + b_1 y + c_1 z = 0, a_2 x + b_2 y + c_2 z = 0, a_3 x + b_3 y + c_3 z = 0\), only trivial solutions exist if the determinant \(\begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{vmatrix} eq 0\). This condition indicates linear independence of the rows (or columns) of the matrix, meaning the system has only the trivial (zero) solution.

Step 3: Checking the Consistency

Now, we assess the consistency between the assertion and the reason. Given the assertion specifies conditions on ratios \(r_1\) and \(r_2\) for having only the trivial solution, it connects directly to the structure defined by the reason that involves determinant computations for this check. If the determinant is non-zero, it translates to having only the trivial solution in matrix terms.

Step 4: Evaluating Connection Between Assertion and Reason

Observe that while the Reason provides the necessary condition for the equations to have only the trivial solution, it doesn't directly explain why the common ratios \(r_1\) and \(r_2\) must not be equal and why neither should be one. The determinant non-zero condition confirms the presence of only trivial solutions but doesn't rationalize the specifics about the ratios unless derived from other properties of these numbers.

Key Concepts

Geometric Progression

Geometric Progressions (G.P.) are sequences of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18 is a G.P. with a common ratio of 3. In our exercise, the sequences \(a, b, c\) and \(\alpha, \beta, \gamma\) are both examples of geometric progression, with their own distinct common ratios \(r_1\) and \(r_2\).

Geometric progressions are fundamentally significant in solving equations as they imply specific algebraic relationships between the terms. For instances like this exercise, understanding and defining these common ratios helps determine structural conditions of matrices derived from the involved expressions, influencing determinant calculations that play a crucial role in the existence of solutions.

Trivial Solution

A trivial solution in the context of equations and matrices generally refers to the solution where all variables equal zero. In a homogenous system of linear equations, such as \(ax + \alpha y + z = 0\), \(bx + \beta y + z = 0\), and \(cx + \gamma y + z = 0\), the trivial solution implies \(x = 0, y = 0, z = 0\).

This is an important concept because it simplifies understanding whether other potential (non-trivial) solutions exist. When the determinant of the coefficient matrix is non-zero, as further explained in the determinant section, it indicates that only this trivial solution is possible. Hence, determining whether only the trivial solution exists can be crucial in understanding the requirements or restrictions imposed by the problem.

Linear Independence

Linear independence refers to a set of vectors or rows in a matrix where no vector in the set is a linear combination of the others. In the system of equations discussed, the rows \((a, \alpha, 1)\), \((b, \beta, 1)\), and \((c, \gamma, 1)\) can be considered vectors. If these rows are linearly independent, it means the matrix formed by these rows has a non-zero determinant.

Linear independence is significant because it guarantees that only the trivial solution exists in a homogenous system of equations. When solving such systems, verifying the linear independence of the involved vectors confirms the unique solution set, helping determine if the conditions for triviality are met without additional assumptions.

Determinant Non-zero Condition

The determinant non-zero condition is key in matrix algebra, especially when dealing with systems of linear equations. A non-zero determinant indicates that the matrix is invertible, meaning the only solution to the associated homogeneous equation system is the trivial solution \((x = 0, y = 0, z = 0)\).

This exercise uses the determinant to check whether the assertion holds. It starts by forming a matrix with each row being coefficients from each equation. The non-zero determinant ensures that a unique solution is present. However, in our task, it highlights that for trivial solutions, the presence of distinct common ratios \(r_1 eq r_2\) and their inequalities to 1 lead to this non-zero condition being satisfied. Hence, exploring the determinant connects algebraic arrangements in equations with fundamental algebraic properties.