Which one of the following is correct ?
(i) `13^31gt31^13`
(ii) `10^100lt100^10`
(iii) `23^2lt32^2`

Options

A. a)(i) and (ii)
B. b)(i) and (iii)
C. c)(ii) and (iii)
D. d)(i) only

Answer

To solve the question, we need to analyze each of the three statements given and determine which one is correct.

### Step 1: Analyze the first statement
**Statement (i):** \( 13^{31} > 31^{13} \)

To compare these two expressions, we can take the logarithm of both sides:

\[ \log(13^{31}) > \log(31^{13}) \]

Using the property of logarithms, we can simplify this:

\[ 31 \cdot \log(13) > 13 \cdot \log(31) \]

Now, we can calculate the values of \( \log(13) \) and \( \log(31) \):

- \( \log(13) \approx 1.113943 \)
- \( \log(31) \approx 1.491654 \)

Now substituting these values:

\[ 31 \cdot 1.113943 > 13 \cdot 1.491654 \]

Calculating both sides:

- Left side: \( 31 \cdot 1.113943 \approx 34.525 \)
- Right side: \( 13 \cdot 1.491654 \approx 19.391 \)

Since \( 34.525 > 19.391 \), the first statement is **true**.

### Step 2: Analyze the second statement
**Statement (ii):** \( 10^{100} < 100^{10} \)

We can rewrite \( 100 \) as \( 10^2 \):

\[ 100^{10} = (10^2)^{10} = 10^{20} \]

Now we compare:

\[ 10^{100} < 10^{20} \]

This is incorrect because \( 10^{100} \) is much larger than \( 10^{20} \). Therefore, the second statement is **false**.

### Step 3: Analyze the third statement
**Statement (iii):** \( 23^2 < 32^2 \)

Calculating both sides:

- \( 23^2 = 529 \)
- \( 32^2 = 1024 \)

Now we compare:

\[ 529 < 1024 \]

This statement is **true**.

### Conclusion
From our analysis:
- Statement (i) is true.
- Statement (ii) is false.
- Statement (iii) is true.

Thus, the correct statements are (i) and (iii).

### Final Answer
The correct options are (i) \( 13^{31} > 31^{13} \) and (iii) \( 23^2 < 32^2 \).

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