Prove that all the common elementary gases ate diatomic.

A variable circle which always touches the line x+y-2=0 at (1, 1) cuts the circle x^2+y^2+4x+5y-6=0 . Prove that all the common chords of intersection pass through a fixed point. Find that points.

Under same conditions of temperature and pressure the vapour density of an elementary gas is 5 times than that of oxygen. Calculate the relative atomic mass of the gas if it is diatomic in nature.

Prove that 0 is the identity element of the binary operation ** on ZZ^(+) defined by x**y=x+y for all x,yinZZ^(+) .

Prove that the identity element of the binary opeartion ** on RR defined by a**b = min. (a,b) for all a,binRR , does not exist.

Prove that all the point on the curve y=sqrt(x+sinx) at which the tangent is parallel to x-axis lie on parabola.

Prove that all odd-ordered central moments are zero for a symmetrical distribution.

Give examples to prove that microbes release gases during metabolism.

Prove that , of all rectangles of given area , the square has the least perimeter.

If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

Prove that the difference between the first term and the fourth term of a G.P. Is at least times as great as the difference between the second and the third terms. Assume that both the first term and common ratio are positive and the common ratio != 1 .