Multiple choice

Statement 1: If $z_1, z_2$ are the roots of the quadratic equation $az^2+bz+c=0$ such that $Im(z_1z_2)\neq 0$, then at least one of a, b, c is imaginary. Statement 2: If quadratic equation having real coefficients has complex roots, then roots are always conjugate to each other.

  1. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

  2. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

  3. Statement 1 is true and Statement 2 is false.

  4. Statement 1 is false and Statement 2 is true.
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A Correct answer
Explanation

Statement 1 is true because if the coefficients are real, the roots must be real or conjugate pairs. If the roots are conjugate pairs (z, z_bar), their product is |z|^2, which is real, so Im(z1z2) would be 0. Thus, for the imaginary part to be non-zero, the coefficients cannot all be real. Statement 2 is a standard property of quadratic equations with real coefficients.

AI explanation

Statement 2 correctly states that complex roots of a real-coefficient quadratic equation always occur in conjugate pairs. For Statement 1, if z1 and z2 are roots of az^2 + bz + c = 0, their product is z1 * z2 = c/a. If Im(z1 * z2) is not zero, the product c/a must have a non-zero imaginary part, meaning the coefficient c must be complex or imaginary. Thus, at least one of a, b, or c must be non-real, proving Statement 1 is true and that Statement 2 is the correct explanation for it.

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