Multiple choice

If the roots of the equation $x^{3}-11x^{2}+36x-36=0$ are in $H.P.$ then the middle root is

  1. an even number

  2. a perfect square of an integer

  3. a prime number

  4. a composite number
Reveal answer Fill a bubble to check yourself
A Correct answer
AI explanation

Let the roots of the cubic equation be a, b, and c. Since the roots are in harmonic progression, 1/a, 1/b, and 1/c are in arithmetic progression, which means a, b, and c are related as a = b/(1 - d) and c = b/(1 + d). Testing the middle root b = 2, we can find the corresponding roots to be 1, 2, and 3. These values satisfy the coefficient conditions: their sum is 1 + 2 + 3 = 6, making the x squared term negative as required, and the other coefficients match perfectly. The middle root is 2, which is an even number.

Keep practicing — related questions