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### The half life period of radioactive element x is same ....

The half life period of radioactive element x is same as the mean life time of another radioactive element y. Initially they have the same number of atoms. Then:

(A) x and y have same decay rate initially and later on different decay rate.
(B) x and y decay at the same rate always.
(C) x will decay faster than y.
(D) y will decay faster than x.

Solution

Given, ${({t_{1/2}})_x} = {({t_{mean}})_y}$

$\therefore \frac{{0.693}}{{{\lambda _x}}} = \frac{1}{{{\lambda _y}}}$

$\Rightarrow {\lambda _x} = 0.693{\lambda _y}$

$\Rightarrow {\lambda _x} < {\lambda _y}$

If N is same, $A \propto \lambda$

$\therefore A_x < A_y$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$