A radioactive element decays by `beta`-emission. A detector records n beta particles in 2 s and in next 2 s, it records 0.75 n beta particles .Find mean life correct to nearest whole number. Give that `log_e2=0.6931 ` and `log_e 3=1.0986`.

A radioactive nucleus X decays to a nucleus Y with a decay constant lambda_X=0.1s^-1, Y further decays to a stable nucleus Z with a decay constant lambda_Y=1//30s^-1 . Initially, there are only X nuclei and their number is N _0=10^20 . Set up the rate equations for the populations of X, Y and Z. The population of Y nucleus as a function of time is given by N_Y(t)={N _0lambda_X//(lambda_X-lambda_Y)}[exp(-lambda_Yt)-exp(-lambda_Xt)]. Find the time at which N_Y is maximum and determine the population X and Z at that instant.

The real numbers x_1, x_2, x_3 satisfying the equation x^3-x^2+beta x+gamma=0 are in A.P. Find the intervals in which beta and gamma lie.

Radioactive element A are being produced at a constant rate alpha . The element has a decay constant lambda . At time t=0, there are N_0 nuclei of the element. If alpha=2N_0lambda , calculate the number of nuclei of A after one half life of A and also limiting value of N as t rarr infty .

The isotope Manganese-56 decays and udergoes beta- particle eission to form the stable isotope Iron-56 .The half life for this decay is 2.6 hours .Initially,at time t=0,a sample of Manganese-56 has a mass of 1.4muG and there is no iron - 56. Determine the time at which the ratio is equal to 9.0.

The isotope Manganese-56 decays and udergoes beta- particle eission to form the stable isotope Iron-56 .The half life for this decay is 2.6 hours .Initially,at time t=0,a sample of Manganese-56 has a mass of 1.4muG and there is no iron - 56. For the smple of Manganese-56 ,determine the initial activity.

Displacement s of a particle at time t is expressed as s=1/2t^3-6t . Find the acceleration at the time when the velocity vanishes (i.e., velocity tends to zero).

Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, aproximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long will it take for half of the original amount to decompose? (given log_e(.989)=-0.01106 and log_e 2=0.6931)

If alpha, beta be the roots of 4x^(2) - 16x + c = 0, c in R such that 1 lt alpha lt 2 and 2 lt beta lt 3 , then the number of integral values of c is