Problem 137 - A Complete Resource Book in Mathematics for JEE Main

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: If the equation \(x^{2}+2(k+1) x+9 k-5=0\) has only negative roots, then \(k \leq 6\) Reason: The equation \(f(x)=0\) will have both roots negative if and only if (i) Discriminant \(\geq 0\), (ii) Sum of roots \(<0\), (iii) Product of roots \(>0\)

Step-by-Step Solution

Answer

(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A)

Step 1: Analyze the Assertion

The assertion claims that for the quadratic equation \(x^{2} + 2(k+1)x + 9k - 5 = 0\) to have only negative roots, \(k\) must be less than or equal to 6. Our task will involve verifying if this condition is necessary for the roots to be negative.

Step 2: Examine the Given Condition for Roots

According to the condition given for the equation \(f(x) = 0\) to have both roots negative, the following must be true: 1. Discriminant (\(D\)) \(\geq 0\)2. Sum of the roots \(< 0\)3. Product of the roots \(> 0\). We will need to evaluate these conditions for the given quadratic equation.

Step 3: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Here, \(a = 1\), \(b = 2(k+1)\), and \(c = 9k - 5\). Therefore, \(D = [2(k+1)]^2 - 4(1)(9k-5)\). Simplifying gives \(D = 4(k+1)^2 - 36k + 20\). Evaluate to check \(D \geq 0\).

Step 4: Evaluate the Sum of Roots Condition

The sum of the roots of the quadratic equation, given by \(-b/a\), is \(-2(k+1)\). For both roots to be negative, \(-2(k+1) < 0\) must hold true. This simplifies to \(k+1 > 0\), which implies \(k > -1\).

Step 5: Evaluate the Product of Roots Condition

For the product of the roots, given by \(c/a\), we have \(9k - 5\). This must be greater than 0 for both roots to be negative. Hence, \(9k - 5 > 0\) simplifies to \(k > 5/9\).

Step 6: Synthesize the Conditions

Both \(-2(k+1)<0\) and \(9k-5>0\) need to be true for both roots to be negative, i.e., \(k > -1\) and \(k > 5/9\). Since \(k > 5/9\) is the stricter bound, \(k > 5/9\) must hold. Also, for \(k > 5/9\), it naturally follows that \(k \leq 6\) meets the required condition from the Assertion.

Step 7: Verify the Reason

According to the Reason, for the equation \(f(x)=0\) to have both roots negative, the discriminant should be \(\geq 0\), the sum of the roots should be less than zero, and the product of the roots should be positive. These criteria have been used above in Steps 3, 4, and 5 to verify the Assertion. Hence, the Reason is a correct explanation for the Assertion.

Step 8: Conclusion

The Assertion is true, the Reason is also true, and the Reason is the correct explanation for the Assertion.

Key Concepts

Discriminant of Quadratic Equation

To determine the nature of the roots of a quadratic equation, the discriminant plays a crucial role. It is represented by the symbol \(D\) and is calculated from the quadratic equation \(ax^2 + bx + c = 0\) using the formula: \(D = b^2 - 4ac\).
The value of \(D\) can help us understand the type of solutions a quadratic equation has:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there are two equal real roots.
• If \(D < 0\), the roots are complex and not real.

For the equation \(x^2 + 2(k+1)x + 9k - 5 = 0\), the discriminant is calculated as \(D = [2(k+1)]^2 - 4(9k - 5)\). Simplifying this gives \(D = 4(k+1)^2 - 36k + 20\).
Checking if \(D \geq 0\) helps us ensure that the roots are real, which is a prerequisite for them to be both negative.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be found using different methods such as factoring, completing the square, or applying the quadratic formula. The quadratic formula is the most general method: \[x = \frac{-b \pm \sqrt{D}}{2a}\] where \(D\) is the discriminant of the quadratic equation.
In the context of our problem, knowing whether these roots are negative requires evaluating the sum and product conditions:

• The sum of the roots is \(-b/a\).
• The product of the roots is \(c/a\).

To ensure both roots \(x_1\) and \(x_2\) are negative, we need:

• A sum of roots that is less than zero i.e., \(-b/a < 0\).
• A product of roots greater than zero i.e., \(c/a > 0\).

By confirming these conditions along with \(D \geq 0\), we confirm the nature of the roots.

Sum and Product of Roots

In any quadratic equation \(ax^2 + bx + c = 0\), fundamental relationships exist between the coefficients and the roots, \(x_1\) and \(x_2\):

• The sum of the roots \(x_1 + x_2\) is given by \(-b/a\).
• The product of the roots \(x_1 \cdot x_2\) is given by \(c/a\).

Understanding these relationships helps us swiftly determine the main characteristics of the roots without even finding them explicitly. If we need both roots to be negative, specific conditions about their sum and product are required:

• The sum of the roots being negative implies \(-b/a < 0\), ensuring they both tilt to the left on a number line.
• The product of the roots being positive, \(c/a > 0\), assures both roots are in the same negative domain.

Applying these insights to \(x^2 + 2(k+1)x + 9k - 5 = 0\) ensures that for the roots to be negative, conditions such as \(k > -1\) and \(k > 5/9\) must be met, confirming \(k \leq 6\).
This also aligns with the assertion that these criteria ensure both roots are negative.