The sum of all the real values of $x$ satisfying the equation $(x^2-7x+11)^{x^2-6x-7}=1$ is

The number of real solutions of the equation $|x|^2 - 3|x| + 2 = 0$ is

If the set of all $a \in R$,for which the equation $2x^2 + (a-5)x + (15-3a) = 0$ has no real root,is the interval $(\alpha, \beta)$,and $X = \{x \in Z : \alpha < x < \beta\}$,then $\sum_{x \in X} x^2$ is equal to

If $2 \sin^3 x + \sin 2x \cos x + 4 \sin x - 4 = 0$ has exactly $3$ solutions in the interval $[0, \frac{n \pi}{2}]$,$n \in N$,then the roots of the equation $x^2 + nx + (n-3) = 0$ belong to :

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 . . . . . .

Let $f(x) = Ax^2 + Bx$ and $g(x) = Lx^2 + Mx + N$. Given that $f(2) - g(2) = 1$,$f(3) - g(3) = 4$,and $f(4) - g(4) = 9$. Then a root of $f(x) - g(x) = 0$ is