Statement-1(Assertion and Statement -2 (reason) Each of these examples also has four alternative choices, only one of which is the correct answer. You have select the correct choice as given below.
Statement -1 If `N=(1/0.4)^20`, then N contains 7 digit before decimal.
Statement -2 Characteristic of the logarithm of N to the base 10 is 7.

Options

A. Statement -1 if true, Statement-2 is true, Statement-2 is a correct explanation for Statement -1
B. Statement -1 if true, Statement-2 is true, Statement-2 is not a correct explanation for Statement -1
C. Statement -1 if true, Statement-2 is false
D. Statement -1 if false , Statement-2 is true

Answer

To solve the given problem, we need to analyze both statements and determine their validity step by step.

### Step 1: Calculate N
Given:
\[ N = \left(\frac{1}{0.4}\right)^{20} \]

To simplify \( N \):
\[ N = \left(\frac{1}{0.4}\right)^{20} = (2.5)^{20} \]
since \( \frac{1}{0.4} = 2.5 \).

### Step 2: Find the logarithm of N
We will take the logarithm (base 10) of \( N \) to find its characteristic:
\[ \log_{10}(N) = \log_{10}((2.5)^{20}) \]
Using the power rule of logarithms:
\[ \log_{10}(N) = 20 \cdot \log_{10}(2.5) \]

### Step 3: Calculate \( \log_{10}(2.5) \)
We can express \( 2.5 \) as:
\[ 2.5 = \frac{5}{2} \]
Thus,
\[ \log_{10}(2.5) = \log_{10}(5) - \log_{10}(2) \]

Using approximate values:
- \( \log_{10}(5) \approx 0.699 \)
- \( \log_{10}(2) \approx 0.301 \)

So,
\[ \log_{10}(2.5) \approx 0.699 - 0.301 = 0.398 \]

### Step 4: Calculate \( \log_{10}(N) \)
Now substituting back:
\[ \log_{10}(N) = 20 \cdot 0.398 = 7.96 \]

### Step 5: Determine the characteristic of \( N \)
The characteristic of a logarithm is the integer part. Therefore, for \( \log_{10}(N) = 7.96 \):
- The characteristic is \( 7 \).

### Step 6: Determine the number of digits before the decimal
The number of digits before the decimal in \( N \) can be determined from \( N = 10^{\log_{10}(N)} \):
- Since \( \log_{10}(N) \approx 7.96 \), we can express \( N \) as:
\[ N \approx 10^{7.96} \]
This means \( N \) is approximately \( 10^7 \times 10^{0.96} \).

Since \( 10^{0.96} \) is approximately \( 9.12 \) (as \( 10^{0.96} \) is slightly less than \( 10 \)), we can conclude:
- The number of digits before the decimal in \( N \) is \( 8 \) (from \( 10^7 \) and the leading digit from \( 10^{0.96} \)).

### Conclusion
- **Statement 1**: "N contains 7 digits before the decimal" is **False** (it actually contains 8 digits).
- **Statement 2**: "Characteristic of the logarithm of N to the base 10 is 7" is **True**.

### Final Answer
The correct choice is that Statement 1 is false and Statement 2 is true.

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