If f(t) = _ , - t ^ ,t dx 1 + x^2 , then f'(1) is

with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS _1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_1 P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is. . . . . . .

→

If $\lim _{t \rightarrow 0}\left(\int_0^1(3 x+5)^t d x\right)^{\frac{1}{t}}=\frac{\alpha}{5 e }\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to _________ .

→

If the system of equations $x + y+ z = 5$ ; $x + 2y + 3z = 9$ ; $x + 3y + \alpha z = \beta $ has infinitely many solutions, then $\beta - \alpha $ equals

→

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left(x^{n}+\frac{2}{x^{5}}\right)^{7}$ is $939 ,$ then the sum of all the possible integral values of $n$ is

→

The number of integers $k$ for which the equation $x^3-27 x+k=0$ has at least two distinct integer roots is

→

$i\times (j\times k)+j\times (k\times i)\,+k\times (i\times i)$ equals

→

Let $5$ digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is $...............$.