If $(x-1)$ is a factor of the polynomial $x^{5}-4 x^{3}+2 x^{2}-3 x+k=0$,then the value of $k$ is:

Let $f(x) = ax^2 + bx + c$,where $a, b, c$ are integers. Suppose $f(1) = 0$,$40 < f(6) < 50$,$60 < f(7) < 70$,and $1000t < f(50) < 1000(t+1)$ for some integer $t$. Then,the value of $t$ is

If the roots of the equation $x^2 - 2cx + ab = 0$ are real and unequal,then the roots of the equation $x^2 - 2(a + b)x + a^2 + b^2 + 2c^2 = 0$ are

If $f(x) \in \mathbb{Q}[x]$ is a non-zero polynomial such that all its roots are irrational,then the degree of $f(x)$ is

If the roots of the equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ are equal,then $a, b, c$ are in

The sum of squares of the roots of the equation $x^{2/3} + x^{1/3} - 2 = 0$ is