`A overset(lambda_(1)) rarr B overset (lambda_(2))rarr C`
t=0 `N_(0)` 0 0
` t N_(1) N_(2) N_(3)` In the above radioactive decay C is stable nucleus. Then:

Options

A. rate of decay of `A` will first increase and then decreases
B. number of nuclei of `B` first increase and then decreases
C. if `lambda_(2) gt lambda_(1)`, then activity of `B` will always be higher than activity of `A`
D. if `lambda gt gt lambda_(2)`, then number of nucleus of `C` will always be less than number of nucleus of `B`.

Explanation

To solve the problem of radioactive decay from nucleus A to B and finally to stable nucleus C, we will analyze the decay processes and the changes in the number of nuclei over time.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions:**
At time \( t = 0 \):
- The number of nuclei of A, \( N_1(0) = N_0 \)
- The number of nuclei of B, \( N_2(0) = 0 \)
- The number of nuclei of C, \( N_3(0) = 0 \)

2. **Decay Process:**
The decay process can be represented as:
- \( A \overset{\lambda_1}{\rightarrow} B \)
- \( B \overset{\lambda_2}{\rightarrow} C \)

Here, \( \lambda_1 \) is the decay constant for A, and \( \lambda_2 \) is the decay constant for B.

3. **Rate of Change of Nuclei:**
The rate of change of the number of nuclei can be described by the following equations:
- For A:
\[ \frac{dN_1}{dt} = -\lambda_1 N_1 \]
- For B:
\[ \frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2 \]
- For C (since C is stable, it only accumulates):
\[ \frac{dN_3}{dt} = \lambda_2 N_2 \]

4. **Behavior of Nuclei Over Time:**
- As time progresses, \( N_1 \) will decrease due to its decay into B.
- Initially, \( N_2 \) will start at zero and will increase as A decays into B.
- Eventually, \( N_2 \) will reach a maximum point when the production of B (from A) equals the decay of B (into C).

5. **Finding the Maximum of B:**
- The maximum number of nuclei of B occurs when:
\[ \frac{dN_2}{dt} = 0 \Rightarrow \lambda_1 N_1 = \lambda_2 N_2 \]
- This indicates that the activity of B will first increase (as A decays into B) and then decrease (as B decays into C).

6. **Conclusion:**
- The correct statement is that the number of nuclei of B will first increase and then decrease. This is due to the fact that B is produced from A and then consumed to form C.

### Final Answer:
The correct statement is that the number of nuclei of B will first increase and then decrease.

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