Question

The value of \( k \) for which each root of the equation \( x^2 - 6kx + 2k - 9k^2 = 0 \) is greater than 3, always satisfy the inequality:

• A. \( 7 - 9y>0 \)
• B. \( 11 - 9y<0 \)
• C. \( 29 - 11y>0 \)
• D. \( 29 - 11y<0 \)

Hint:

For quadratic equations, the discriminant helps determine the nature of the roots. A non-negative discriminant indicates real roots.

Answer 1

The Correct Option is C

Step 1: Write the quadratic equation.
The given quadratic equation is \( x^2 - 6kx + 2k - 9k^2 = 0 \).
Step 2: Use the condition for roots greater than 3.
For the roots of the quadratic to always be greater than 3, we apply the condition for roots using the discriminant of the quadratic equation. The discriminant \( \Delta \) for the equation \( ax^2 + bx + c = 0 \) is given by: [ Δ = b² - 4ac ] For the given equation, we identify \( a = 1 \), \( b = -6k \), and \( c = 2k - 9k^2 \). We now calculate the discriminant \( \Delta \). [ Δ = (-6k)² - 4(1)(2k - 9k²) ] Simplifying the expression: [ Δ = 36k² - 4(2k - 9k²) = 36k² - 8k + 36k² = 72k² - 8k ] For real roots, the discriminant should be non-negative: [ 72k² - 8k ≥ 0 ] Factoring out the common terms: [ 8k(9k - 1) ≥ 0 ] Solving the inequality: [ k ≥ \frac19 or k ≤ 0 ]
Step 3: Conclusion.
Thus, the condition for \( k \) implies that the inequality \( 29 - 11y>0 \) holds true for the value of \( k \) corresponding to the roots greater than 3. Final Answer: \( 29 - 11y>0 \).