Let a curve $y=f(x), x \in(0, \infty)$ pass through the points and . If the tangent at any point to the given curve cuts the -axis at the point such that , then is equal to __________.

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

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

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

Remainder when $64^{32^{32}}$ is divided by 9 is equal to ________.

Let the coefficients of the middle terms in the expansion of $\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}$ and , respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then is equal to __________.

If the tangent to the curve $y=x^{3}-x^{2}+x$ at the point is also tangent to the curve at the point (2, 1), then is equal to __________.

If the coefficient of x¹⁰ in the binomial expansion of ${\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}$ is , where l, k N and l is co-prime to 5, then k is equal to _____________.

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 , where is the greatest integer , is ______________.