The sum of all the real values of $x$ satisfying the equation $2^{(x - 1)(x^2 + 5x - 50)} = 1$ is

Let $t$ be a real number such that $t^2 = at + b$ for some positive integers $a$ and $b$. Then,for any choice of positive integers $a$ and $b$,$t^3$ is never equal to:

The equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4x - 1}$ has

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3+bx+c=0$. If $\beta \gamma=1=-\alpha$,then $b^3+2c^3-3\alpha^3-6\beta^3-8\gamma^3$ is equal to $......$.

If a polynomial $f(x)$ is divided by $(x + 1)$,$(x - 2)$,and $(x + 2)$,the remainders are $6$,$3$,and $15$ respectively. Find the remainder when $f(x)$ is divided by $(x + 1)(x - 2)(x + 2)$.

The integral values of $a$ for which the quadratic equation $(x - a)(x - 10) + 1 = 0$ has integral roots are