Nucleus A is converted into C through the following reactions, `ArarrB+alpha`
`BrarrC+2 beta`
then,

Options

A. (a) A and B are isotopes
B. (b) A and C are isobars
C. (c) A and B are isobars
D. (d) A and C are isotopes

Explanation

To solve the problem, we need to analyze the nuclear reactions given and determine the relationship between the nuclei involved. Let's break it down step by step.

### Step 1: Analyze the first reaction
The first reaction is:
\[ \text{A} \rightarrow \text{B} + \alpha \]
Here, an alpha particle (\(\alpha\)) is emitted.

- An alpha particle consists of 2 protons and 2 neutrons, which means it has a mass number of 4 and an atomic number of 2.
- If nucleus A has a mass number \(A\) and atomic number \(Z\), after emitting an alpha particle, the mass number and atomic number of nucleus B will be:
- Mass number of B: \(A - 4\)
- Atomic number of B: \(Z - 2\)

### Step 2: Analyze the second reaction
The second reaction is:
\[ \text{B} \rightarrow \text{C} + 2\beta \]
Here, two beta particles (\(\beta\)) are emitted.

- A beta particle is an electron emitted from the nucleus, which means it does not change the mass number but increases the atomic number by 1.
- Since two beta particles are emitted, the atomic number of nucleus C will be:
- Atomic number of C: \((Z - 2) + 2 = Z\)
- The mass number of C remains the same as that of B, which is:
- Mass number of C: \(A - 4\)

### Step 3: Conclusion
From the above analysis, we can summarize the properties of nuclei A, B, and C:
- Nucleus A has mass number \(A\) and atomic number \(Z\).
- Nucleus B has mass number \(A - 4\) and atomic number \(Z - 2\).
- Nucleus C has mass number \(A - 4\) and atomic number \(Z\).

Since nuclei A and C have the same atomic number \(Z\) but different mass numbers, they are isotopes of each other.

### Final Answer
Thus, the correct conclusion is that nuclei A and C are isotopes.

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