Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$ have integral roots such that $P(6) = 3$. Then $a$ cannot be equal to:

If $a$ and $b$ are odd integers, then the roots of the equation $2ax^2 + (2a + b)x + b = 0$, where $a \neq 0$, are

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0$ where $0 < \theta < 45^\circ$,and $\alpha < \beta$. Then $\sum_{n=0}^\infty (\alpha^n + \frac{(-1)^n}{\beta^n})$ is equal to

The graph of $y = ax^2 + bx + c$ is shown. Which of the following does $NOT$ hold good?

The equation $x^4-x^3-6x^2+4x+8=0$ has two equal roots. If $\alpha$ and $\beta$ are the other two roots of this equation,then $\alpha^2+\beta^2=$

For the equation $x^2-5|x|-14=0$,which of the following is true?