If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c = 0$:

1. The equation has imaginary roots.

2. The ratio of the roots of the equation is $1 : \omega$ where $\omega$ is a cube root of unity.

3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$.

Which of the statements given above are correct?

If $x^2 + mx + n$ is an integer for all integral values of $x$, then which of the following is/are correct?

1. $m$ must be an integer

2. $n$ must be an integer

Select the correct answer using the code given below:

Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta \in [0, \pi]$?

Consider the following for the next items that follow:

A quadratic equation is given by (a + b) x2 - (a + b + c) x + k = 0, where a, b, c are real.