Following statements related to radioactivity are given below:                                 

(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.

(B) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.

(C) Slope of the graph of log_e (no. of undecayed nuclei) Vs, time represents the reciprocal of mean life time   $\left(\tau \right).$

(D) Product of decay constant $\left(\lambda \right)$  and half-life time $\left(T_{1/2}\right)$ is not constant.

<p>We know, N =&nbsp;<math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup></mrow></math></p><p>Where, N Number of un-decayed Nuclei</p><p>No Initial No. Of Nuclei</p><p>N&nbsp;<math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup></mrow></math></p><p><math><mrow><mi>N</mi><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mi>λ</mi><mi>t</mi></mrow></msup></mrow></math></p><p>Taking log both side log N = log N0 + loge&nbsp;<math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup></mrow></math></p><p>log N = log No&nbsp;<math><mrow><mi>λ</mi><mi>t</mi></mrow></math></p><p>Slope =&nbsp;<math><mrow><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>a</mi><mi>v</mi></mrow></msub></mrow></mfrac></mrow></math></p>

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