The value of $a$ for which the quadratic equation $3x^2 + 2(a^2 + 1)x + (a^2 - 3a + 2) = 0$ possesses roots with opposite signs,lies in

$A$ prime number $p$ is called special if there exist primes $p_1, p_2, p_3, p_4$ such that $p = p_1 + p_2 = p_3 - p_4$. The number of special primes is

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$,then $P(x) \cdot Q(x) = 0$ has at least:

The roots of the equation $a(x^2 + 1) - (a^2 + 1)x = 0$ are

Let $p(x) = x^2 - 5x + a$ and $q(x) = x^2 - 3x + b$,where $a$ and $b$ are positive integers. Suppose $\text{HCF}(p(x), q(x)) = x - 1$ and $k(x) = \text{LCM}(p(x), q(x))$. If the coefficient of the highest degree term of $k(x)$ is $1$,then the sum of the roots of $(x - 1) + k(x)$ is:

If roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are discriminants of the given equations respectively).