The eccentricity of the conic $\frac{5}{r}=2+3 \cos \theta+4 \sin \theta$ is

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

If a circle $C$ passing through the point $(4, 0)$ touches the circle $x^2 + y^2 + 4x - 6y = 12$ externally at the point $(1, -1)$,then the radius of $C$ is

$A$ poster is to be printed on a rectangular sheet of paper of area $18 \ m^2$. The margins at the top and bottom of $75 \ cm$ each and at the sides $50 \ cm$ each are to be left. Then the dimensions i.e. height and breadth of the sheet so that the space available for printing is maximum,are . . . . . . respectively.

An urn $A$ contains $3$ white and $5$ black balls. Another urn $B$ contains $6$ white and $8$ black balls. $A$ ball is picked from $A$ at random and then transferred to $B$. Then,a ball is picked at random from $B$. The probability that it is a white ball is

When a non-volatile solute is added to a solvent,the vapor pressure decreases by $10 \ mm$. The mole fraction of the solute is $0.2$. If the vapor pressure decreases by $20 \ mm$,what will be the mole fraction of the solvent?