The sum of the maximum and minimum values of the function $f(x)=|5 x-7|+\left[x^{2}+2 x\right]$ in the interval $\left[\frac{5}{4}, 2\right]$, where $[t]$ is the greatest integer $\leq t$, is ______________.

The remainder when $428^{2024}$ is divided by 21 is __________.

The remainder on dividing 1 + 3 + 3² + 3³ + ..... + 3²⁰²¹ by 50 is _________.

The largest natural number $n$ such that $3^{n}$ divides $66 !$ is ___________.

If the points of intersection of two distinct conics $x^2+y^2=4 b$ and lie on the curve , then times the area of the rectangle formed by the intersection points is _________.

Let the sixth term in the binomial expansion of $\left( \sqrt{2^{\log_2(10 - 3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}} \right)^m$ in the increasing powers of , be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of is __________.

The remainder when (2023)$^{2023}$ is divided by 35 is __________.

The remainder, when $19^{200}+23^{200}$ is divided by 49 , is ___________.

If ${z^2} + z + 1 = 0$, , then