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 _________.

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

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 __________.

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 $[t]$ 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 __________.

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

The constant term in the expansion of ${\left( {2x + {1 \over {{x^7}}} + 3{x^2}} \right)^5}$ is ___________.

If the area enclosed by the parabolas $\mathrm{P_1:2y=5x^2}$ and is equal to the area enclosed by and , then is equal to ____________.

The mean of the coefficients of $x, x^{2}, \ldots, x^{7}$ in the binomial expansion of is ___________.

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 __________.