If $\sin (\alpha)$ and $\cos (\alpha)$ are the roots of the equation $a x^{2}+b x+c=0$, then $b^{2}$ is

Given below are two statements:

Statement (I): (x² + 3x + 1) = (x - 2)² is not a quadratic equation.

Statement (II): The nature of roots of quadratic equations x² + 2x√3 + 3 = 0 are real and equal.

In light of the above statements, choose the most appropriate answer from the options given below.

If $\alpha$ and $\beta$ are the roots of the equation $a x^{2}+b x+c=0$, then value of

$\frac{1}{a \alpha+b}+\frac{1}{a \beta+b}$ is :

If the roots of the equation $x^{2}-p x+54=0$ are in the ratio $2: 3$, then value of $p$ is:

The graph of a polynomial $y = f(x)$ is shown in figure below, then the number of its zeros is:

The factors of polynomial $x^{3}-6 x^{2}+11 x-6$ are :

Amit and Alok attempted to solve a quadratic equation. Amit made a mistake in writing down the constant term and ended up with roots $(4, 3)$. Alok made a mistake in writing down coefficient of $x$ to get roots $(3, 2)$. The correct roots of the equation are: