રેડિયોએક્ટિવ તત્વ બે પ્રક્રિયાથી ક્ષય પામે છે,તેમના અર્ધઆયુ \(T _{1 / 2}^{(1)}\) અને \(T _{1 / 2}^{(2)}\) છે, તો પરિણામી અર્ધઆયુ કેટલો થાય?

A આપેલ પૈકી એક પણ નહીં
B \(T _{1 / 2}= T _{1 / 2}^{(1)}+ T _{1 / 2}^{(2)}\)
C \(T _{1 / 2}=\frac{ T _{1 / 2}^{(1)} T _{1 / 2}^{(2)}}{ T _{1 / 2}^{(1)}+ T _{1 / 2}^{(2)}}\)
D \(T _{1 / 2}=\frac{ T _{1 / 2}^{(1)}+ T _{1 / 2}^{(2)}}{ T _{1 / 2}^{(1)}- T _{1 / 2}^{(2)}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(T _{1 / 2}=\frac{ T _{1 / 2}^{(1)} T _{1 / 2}^{(2)}}{ T _{1 / 2}^{(1)}+ T _{1 / 2}^{(2)}}\)

Step-by-step Solution

Detailed explanation

\(\lambda_{ eq }=\lambda_{1}+\lambda_{2}\) \(\frac{1}{ T _{1 / 2}}=\frac{1}{ T _{1 / 2}^{(1)}}+\frac{1}{ T _{1 / 2}^{(2)}}\) \(T _{1 / 2}=\frac{ T _{1 / 2}^{(1)} T _{1 / 2}^{(2)}}{ T _{1 / 2}^{(1)}+ T _{1 / 2}^{(2)}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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સૂચિ \(-I\) અને સૂચિ \(-II\) મેળવો.

સૂચિ \(-I\)	સૂચિ \(-II\)
\((A)\) કોણીય વેગમાન	\((I)\) \(\left[ ML ^2 T ^{-2}\right]\)
\((B)\) ટોર્ક	\((II)\) \(\left[ ML ^{-2} T ^{-2}\right]\)
\((C)\) તણાવ	\((III)\) \(\left[ ML ^2 T ^{-1}\right]\)
\((D)\) દબાણ પ્રચલન	\((IV)\) \(\left[ ML ^{-1} T ^{-2}\right]\)

નીચે આપેલા વિકલ્પોમાંથી સાચો વિકલ્પ પસંદ કરો.

JEE Mains 2023 Medium

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