For the following questions answer them individually
Let a differentiable function f satisfy the equation $$\int_{0}^{36}f(\frac{tx}{36})dt=4\alpha f(x)$$. If y=f(x) is a standard parabola passing through the points (2, 1) and $$(-4,\beta)$$, then $$/beta^{\alpha}$$ is equal to______.
The number of $$3\times 2$$ matrices A, which can be formed using the elements of the set {-2, -1 , 0, 1, 2} such that the sum of all the diagonal elements of $$A^{T}A$$ is 5, is_____
Let a line L passing through the point P (1, 1, 1) be perpendicular to the lines $$\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$$ and $$\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point S (1, 0, - 1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ___.
The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is______.
Let $$(2\alpha,\alpha)$$ be the largest interval in which the function $$f(t)=\frac{|t+1|}{t^{2}},t < 0$$, is strictly decreasing. Then the local maximum value of the function $$g(x)=2\log_{e}{x-2}+ax^{2}+4x-\alpha,x > 2$$, is______.
The electrostatic potential in a charged spherical region of radius r varies as $$V=ar^{3}+b$$, where a and b are constants. The total charge in the sphere of unit radius is $$\alpha \times \pi a\in_{\circ}$$. the value of $$\alpha$$ is_____. (permittivity of vacuum is $$\in_{\circ}$$)
Three masses 200 kg, 300 kg and 400 kg are placed at the vertices of an equilateral triangle with sides 20 m. They are rearranged on the vertices of a bigger triangle of side 25 m and with the same centre. The work done in this process ____ J. (Gravitational constant $$G=6.7 \times 10^{-11} Nm^{2}/kg^{2}$$)
A boy throws a ball into air at 45° from the horizontal to land it on a roof of a building of height H . If the ball attains maximum height in 2 s and lands on the building in 3 s after launch, then value of H is ___ m. $$(g=1.m/s^{2})$$
Three charges+ 2q, +3q and -4q are situated at (0, -3a), (2a, 0) and (-2a, 0) respectively in the x y plane. The resultant dipole moment about origin is ___ .
A cylindrical block of mass M and area of cross section A is floating in a liquid of density $$\rho$$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ___
