If the points of intersection of two distinct conics $x^2+y^2=4 b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3 x^2$, then $3 \sqrt{3}$ times the area of the rectangle formed by the intersection points is _________.

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 ______________.

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 __________.

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 __________.

Let the function $f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$, be decreasing in and increasing in . A tangent to the parabola at a point on it passes through the point but does not pass through the point . If the equation of the normal at is : , then is equal to ________________.

Let $[t]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7}$ is , then is equal to ___________.

The coefficient of $x^7$ in ${(1 - x + 2{x^3})^{10}}$ is ___________.

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4^th day is 30, then number of infected students on 8^th day will be __________.

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

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

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

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}\left(1+x+x^2\right)^{2007}$ is equal to _________.

Let $\alpha>0$, be the smallest number such that the expansion of $\left(x^{\frac{2}{3}}+\frac{2}{x^{3}}\right)^{30}$ has a term . Then is equal to ___________.

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135, 30 and $\frac{10}{3}$, respectively, then is equal to __________.