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:

The minimum value of $f(x) = x^2 + 4x + 5$ is . . . . . . ,where $x \in R$.

Let $a, b$ and $c$ be three positive real numbers such that the sum of any two of them is greater than the third. All the values of $\lambda$ such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,are given by

If the roots of the equation $ax^2 + x + b = 0$ are real,then the roots of the equation $x^2 - 4\sqrt{ab}x + 1 = 0$ will be

The number of all common roots of the equation $x^4-10x^3+37x^2-60x+36=0$ and the transformed equation obtained by increasing any two distinct roots of it by $1$,keeping the other two roots fixed,is

If the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ has equal roots,where $a + c = 15$ and $b = \frac{36}{5}$,then $a^2 + c^2$ is equal to . . . . . .