Multiple choice

If $\displaystyle a,\alpha _{1},\alpha _{2},...,\alpha _{2n-1},b$ are in AP $\displaystyle a,\beta _{1},\beta _{2},...,\beta _{2n-1},b $ in GP and a, $\displaystyle a,\gamma _{1},\gamma _{2},...,\gamma _{2n-1},b$ are in HP where a, b are positive then the equation $\displaystyle \alpha _{n}x^{2}-\beta _{n}x+\gamma _{n}=0$ has

  1. real and equal roots

  2. real and unequal roots

  3. imaginary roots

  4. roots which are in AP
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In AP, GP, HP sequences with same endpoints, the middle terms satisfy alpha_n * gamma_n = beta_n^2. The discriminant of alpha_n x^2 - beta_n x + gamma_n = 0 is D = beta_n^2 - 4 * alpha_n * gamma_n. Since beta_n^2 = alpha_n * gamma_n, D = -3 * alpha_n * gamma_n. Since a, b are positive, all terms are positive, so D < 0, implying imaginary roots.

AI explanation

For the given arithmetic progression, geometric progression, and harmonic progression, the nth terms are alpha_n = (a + b)/2, beta_n = sqrt(ab), and gamma_n = 2ab/(a + b). The given quadratic equation is alpha_n*x^2 - beta_n*x + gamma_n = 0. To determine the nature of the roots, we calculate the discriminant, D = (-beta_n)^2 - 4(alpha_n)(gamma_n). Substituting the nth terms gives D = (sqrt(ab))^2 - 4((a + b)/2)(2ab/(a + b)), which simplifies to ab - 8ab = -7ab. Since a and b are positive, the product ab is positive, making the discriminant negative, so the equation has imaginary roots.

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