In how many ways can $6$ persons be selected from $4$ officers and $8$ constables,if at least one officer is to be included?

The curve between charge density and distance near a $P-N$ junction will be:

When a certain weight is suspended from a long uniform wire,its length increases by $1 \ cm$. If the same weight is suspended from another wire of the same material and length but having a diameter half of the first one,the increase in length will be ......... $cm$.

If $P$ is a point such that the ratio of the square of the lengths of the tangents from $P$ to the circles $x^2+y^2+2x-4y-20=0$ and $x^2+y^2-4x+2y-44=0$ is $2:3$,then the locus of $P$ is a circle with centre :

If $\int \frac{\sin x}{\cos x(1+\cos x)} d x=f(x)+c$,then $f(x)$ is equal to

$A$ mass $M$ attached to a horizontal spring executes $S.H.M.$ of amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio $\left(\frac{A_1}{A_2}\right)$ is