Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ 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 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 _________.

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 __________.

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 ______________.

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

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

If the maximum value of the term independent of $t$ in the expansion of $\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0$, is , 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 $f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M 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 ___________.