If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of ${\left( {2{x^3} + {3 \over x}} \right)^{10}}$ is , 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 __________.

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 __________.

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

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is , then is equal to ________.

The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to ___________.

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

Let the sum of the coefficients of the first three terms in the expansion of ${\left( {x - {3 \over {{x^2}}}} \right)^n},x \ne 0.~n \in \mathbb{N}$, be 376. Then the coefficient of is __________.