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### A radioactive nucleus ${}_Z^AX$ undergoes ....

A radioactive nucleus ${}_Z^AX$ undergoes spontaneous decay in the sequence ${}_Z^AX \to {}_{Z - 1}B \to {}_{Z - 3}C \to {}_{Z - 2}D$ where Z is the atomic number of element X. The possible decay particles in the sequence are:

(1) $\alpha,\beta^-.\beta^+$
(2) $\alpha,\beta^+,\beta^-$
(3) $\beta^+,\alpha,\beta^-$
(4) $\beta^-,\alpha,\beta^+$

Solution

We have the following possible nuclear reactions:

${}_Z^AX \to {}_{Z - 1}^AB + {}_1^0{\beta ^ + }$
${}_{Z - 1}^AB \to {}_{Z - 3}^{A - 4}C + {}_2^4H{e^{2 + }}$
${}_{Z - 3}^{A - 4}C \to {}_{Z - 2}^{A - 4}D + {}_{ - 1}^0{\beta ^ - }$

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