If $k \in R$ is such that the equation $2 \cosh^2 x = 3 \sinh x + k$ has no real solution,then which of the following is correct?

If the roots of the quadratic equation $ax^2+bx+c=0$ are imaginary,then for all real values of $x$,the minimum value of the expression $3a^2x^2+6abx+2b^2$ is

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

The equation having the multiple root of the equation $x^4+4x^3-16x-16=0$ as its root is

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 $\alpha, \beta, \gamma$ are the roots of the equation $x^3+3x^2-10x-24=0$,and $\alpha(\beta+\gamma), \beta(\gamma+\alpha), \gamma(\alpha+\beta)$ are the roots of the equation $x^3+px^2+qx+r=0$,then $q=$