Problem 103 - 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: Let \(\alpha_{1}, \alpha_{2}\) and \(\beta_{1}, \beta_{2}\) be the roots of \(a x^{2}\) \(+b x+c=0\) and \(p x^{2}+q x+r=0\) respectively. If the system of equations \(\alpha_{1} y+\alpha_{2} z=0\) and \(\beta_{1} y+\beta_{2} z=0\) has a non-trivial solution, then \(\frac{b^{2}}{q^{2}}=\frac{a c}{p r}\) Reason: The equations \(a_{1} x+b_{1} y=0, a_{2} x+b_{2} y=0\) have a non- trivial solution if \(\left|\begin{array}{ll}a_{1} & b_{1} \\ a_{2} & b_{2}\end{array}\right|=0\).

Step-by-Step Solution

Answer

(B) Assertion is true, Reason is true, but not explaining the Assertion.

Step 1: Verify the Assertion

The assertion states that if the system of equations \(\alpha_{1} y+\alpha_{2} z=0\) and \(\beta_{1} y+\beta_{2} z=0\) has a non-trivial solution, then \(\frac{b^{2}}{q^{2}}=\frac{a c}{p r}\). A non-trivial solution exists if the determinant of the coefficients of \(y\) and \(z\) for both equations is zero, i.e. \(\alpha_{1}\beta_{2} - \alpha_{2}\beta_{1} = 0\). For the quadratic equations, the sum and product of roots relationships give us the relationship \(\frac{b^2}{q^2} = \frac{ac}{pr}\). So, the assertion is true.

Step 2: Evaluate the Reason

The reason given is that equations \(a_{1} x+b_{1} y=0\) and \(a_{2} x+b_{2} y=0\) have a non-trivial solution if the determinant \(\left|\begin{array}{cc}a_{1} & b_{1} \ a_{2} & b_{2}\end{array}\right| = 0\). This is a known property of linear systems: a non-trivial solution exists if the determinant of the coefficient matrix is zero. Hence, the reason is true.

Step 3: Analyze the Explanation

Although both the assertion and reason are true, we must determine if the reason given adequately explains why the assertion is true. The assertion's validity stems from the arrangement of roots and coefficients derived from the quadratic equations and their comparison, not directly from the determinant property used in the reason. Therefore, the reason does not directly explain the assertion although it is itself true.

Step 4: Final Decision

Based on the analysis, both the assertion and reason are true. However, the reason is not a correct explanation of the assertion. Therefore, option B is the correct response.

Key Concepts

Linear Systems

A linear system is a set of equations where each equation is of the first degree, meaning the variables appear only to the first power and there are no products of variables. These systems can be solved using various methods such as substitution, elimination, or using augmented matrices.

When determining the solvability of a linear system, we often evaluate the determinant of the coefficient matrix. For a system of two equations, such as \(a_{1}x + b_{1}y = 0\) and \(a_{2}x + b_{2}y = 0\), the determinant of the coefficient matrix is given by:

• \[\begin{vmatrix}a_{1} & b_{1} \a_{2} & b_{2}\end{vmatrix}\]

If this determinant is zero, it implies the system has a non-trivial solution (a solution other than the trivial zero solution).

This property is used to check whether certain conditions are satisfied for a given set of linear equations. It is a crucial step in understanding when a linear combination of vectors leads to a solution.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally written in the form \(ax^{2} + bx + c = 0\). Solutions to quadratic equations are called 'roots' and can be obtained using various methods including factoring, completing the square, or the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The sum and product of the roots of a quadratic equation can be exploited using Vieta's formulas:
- The sum of the roots \((\alpha_{1} + \alpha_{2})\) is given by \(-\frac{b}{a}\)- The product of the roots \((\alpha_{1}\alpha_{2})\) is \(\frac{c}{a}\)

These relationships are valuable when comparing different quadratic equations, particularly in contexts involving systems of equations or comparing the structure of different polynomials. Knowing the root properties is helpful when examining conditions under which related assertions hold.

Determinants

Determinants are a special number that can be calculated from a square matrix. They provide insight into the properties of a matrix, such as whether it is invertible, and the area or volume scaling factor in geometry. For a \(2 \times 2\) matrix:

• \[\begin{vmatrix}a & b \c & d\end{vmatrix} = ad - bc\]

A determinant offers a way to ascertain whether a system of linear equations has a unique solution. Specifically, a non-zero determinant implies a unique solution exists, while a zero determinant indicates either no solution or infinitely many solutions, suggesting that the equations are dependent.

In the context of quadratic equations and linear systems, determinants serve to check for non-trivial solutions by assessing the condition \(\begin{vmatrix}a & b \ c & d\end{vmatrix} = 0\). Understanding this property allows us to analyze complex mathematical scenarios and establish critical relationships between various elements.