Let $f$ be a positive function. Let $I_1 = \int_{1-k}^{k} x f\{x(1-x)\} dx$ and $I_2 = \int_{1-k}^{k} f\{x(1-x)\} dx$,where $2k-1 > 0$. Then $I_1/I_2$ is

To the free end of a spring hanging from a rigid support,a block of mass $m$ is hung and slowly allowed to come to its equilibrium position. Then the stretching in the spring is $d$. If the same block is attached to the same spring and allowed to fall suddenly,the amount of stretching is: (force constant,$k$)

At any instant,two elements $X_1$ and $X_2$ have the same number of radioactive atoms. If the decay constants of $X_1$ and $X_2$ are $10 \lambda$ and $\lambda$ respectively,then the time when the ratio of their atoms becomes $\frac{1}{e}$ will be:

According to molecular orbital theory,the total number of bonding electron pairs in $O_2$ is

$A$ stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any way. $A$ ball of mass $m$ moving with speed $v$ collides elastically with the stick at its end,as shown in the figure. If after the collision the ball comes to rest,then what should be the mass of the ball?

Beryllium is placed above magnesium in Group $2$. What happens when beryllium powder is added to a solution of $MgCl_2$?