Question

Given below are two statements:
Statement I : If the roots of a quadratic equation are 2 and 3, then the equation is \(x^2-5x-6=0.\)
Statement II: If the roots of \(4x^2+3kx+9=0\) are real and distinct, then \(k≤-4\) or \(k≥4.\)
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Answer 1

The Correct Option is B

Let's critically evaluate both statements to determine their correctness: 

1. Statement I: If the roots of a quadratic equation are 2 and 3, then the equation is \(x^2-5x-6=0.\)
   • The general form of a quadratic equation with roots \(r_1\) and \(r_2\) is \(x^2 - (r_1 + r_2)x + r_1r_2 = 0\).
   • Using the given roots 2 and 3:
     • Sum of roots, \(r_1 + r_2 = 2 + 3 = 5\).
     • Product of roots, \(r_1r_2 = 2 \times 3 = 6\).
   • Thus, the equation is \(x^2 - 5x + 6 = 0\), not \(x^2-5x-6=0\) as stated.
   • Therefore, Statement I is incorrect.
2. Statement II: If the roots of \(4x^2+3kx+9=0\) are real and distinct, then \(k≤-4\) or \(k≥4.\)
   • For a quadratic equation \(ax^2 + bx + c = 0\) to have real and distinct roots, the discriminant must be greater than zero.
   • The discriminant, \(D = b^2 - 4ac\).
   • For \(4x^2+3kx+9=0\), \(a = 4\), \(b = 3k\), and \(c = 9\).
   • Calculating the discriminant:
     • \(D = (3k)^2 - 4 \times 4 \times 9 = 9k^2 - 144\).
     • To have real and distinct roots: \(9k^2 - 144 > 0\).
     • Solving this inequality:
       • \(9(k^2 - 16) > 0\) ⇒ \((k - 4)(k + 4) > 0\).
       • The solution to this inequality is \(k < -4\) or \(k > 4\), not \(k ≤ -4\) or \(k ≥ 4\).
   • Thus, Statement II is also incorrect.

In conclusion, both Statement I and Statement II are incorrect.