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### A heavy nucleus Q of half-life 20 minutes ...

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is

(A) 50          (B) 75          (C) 350          (D) 525

Solution

If $N_0$ be the initial number of nuclei and N the number after n half-lives, then

$\frac {N}{N_0}=(\frac {1}{2})^n$

Here, $\frac {N}{1000}=(\frac {1}{2})^3$

$\therefore N = 125$

Total number of decays = 1000 - 125 = 875

Total number of alpha-decays = 60% of 875 = 525

Hence, Option (D).

### 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 = ${}^{12}{C_6} = 924$