A : The radius of second orbit of ` he^(+)` is equal to that of first orbit of hydrogen
R : The radius of an orbit in hydrogen like species is directly proportional to n and inversely proportional to Z .

Options

A. IF both assertion & reason are true and the reason is the correct explanation of the assertion then mark
B. IF both assertion & reason are true but the reason is not the correct explanation of the assertion , then mark
C. IF assertion is true statement but reason is false , then mark
D. If both assertion and reason are false statements then mark

Answer

To solve the given question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step 1: Understand the Assertion
The assertion states that the radius of the second orbit of \( \text{He}^+ \) is equal to the radius of the first orbit of hydrogen.

### Step 2: Use the Bohr Model Formula
According to the Bohr model, the radius of the nth orbit for a hydrogen-like atom is given by the formula:
\[ R_n = \frac{0.53 \, n^2}{Z} \, \text{Å} \]
where:
- \( R_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (1 for the first orbit, 2 for the second orbit),
- \( Z \) is the atomic number of the element.

### Step 3: Calculate the Radius for \( \text{He}^+ \)
For \( \text{He}^+ \) (Helium with one electron), the atomic number \( Z = 2 \).
- For the second orbit (\( n = 2 \)):
\[ R_2(\text{He}^+) = \frac{0.53 \times 2^2}{2} = \frac{0.53 \times 4}{2} = \frac{2.12}{2} = 1.06 \, \text{Å} \]

### Step 4: Calculate the Radius for Hydrogen
For hydrogen (\( Z = 1 \)):
- For the first orbit (\( n = 1 \)):
\[ R_1(\text{H}) = \frac{0.53 \times 1^2}{1} = 0.53 \, \text{Å} \]

### Step 5: Compare the Radii
Now we compare the two results:
- \( R_2(\text{He}^+) = 1.06 \, \text{Å} \)
- \( R_1(\text{H}) = 0.53 \, \text{Å} \)

Since \( 1.06 \, \text{Å} \) is not equal to \( 0.53 \, \text{Å} \), the assertion is **false**.

### Step 6: Analyze the Reason
The reason states that the radius of an orbit in hydrogen-like species is directly proportional to \( n \) and inversely proportional to \( Z \).

### Step 7: Correct the Reason
From the formula \( R_n = \frac{0.53 \, n^2}{Z} \):
- The radius is directly proportional to \( n^2 \) (not \( n \)), and inversely proportional to \( Z \). Therefore, this statement is also **false**.

### Conclusion
Both the assertion and the reason are false.

### Final Answer
Both the assertion (A) and the reason (R) are false.