Which of the following is/are true ?

Options

A. (a) `1000^(1000) gt 1002^(999)`
B. (b) `1000^(1000) lt 1002^(999)`
C. (c) `1000^(1002) lt 1002^(1000)`
D. (d) `1000^(1002) gt 1002^(1000)`

Explanation

To solve the problem, we need to determine the validity of the given options based on the relationship between powers of numbers. We will analyze two comparisons: \(1002^{999}\) vs \(1000^{1000}\) and \(1002^{1000}\) vs \(1000^{1002}\).

### Step 1: Compare \(1002^{999}\) and \(1000^{1000}\)

We start by calculating the ratio:

\[ \frac{1002^{999}}{1000^{1000}} = \frac{1002^{999}}{1000^{999} \cdot 1000} = \frac{1002^{999}}{1000^{999}} \cdot \frac{1}{1000} \]

This can be rewritten as:

\[ \frac{1002}{1000}^{999} \cdot \frac{1}{1000} \]

### Step 2: Simplify the Expression

Next, we simplify \(\frac{1002}{1000}\):

\[ \frac{1002}{1000} = 1 + \frac{2}{1000} \]

Thus, we have:

\[ \frac{1002^{999}}{1000^{1000}} = \left(1 + \frac{2}{1000}\right)^{999} \cdot \frac{1}{1000} \]

### Step 3: Apply Binomial Theorem

Using the Binomial Theorem, we can approximate:

\[ \left(1 + \frac{2}{1000}\right)^{999} \approx 1 + 999 \cdot \frac{2}{1000} = 1 + \frac{1998}{1000} \]

This gives us:

\[ \left(1 + \frac{2}{1000}\right)^{999} \approx 1.998 \]

### Step 4: Final Calculation

Now substituting back:

\[ \frac{1002^{999}}{1000^{1000}} \approx \frac{1.998}{1000} \]

Since \(1.998 < 1\), we conclude:

\[ 1002^{999} < 1000^{1000} \]

### Step 5: Conclusion for Option A

Thus, Option A is **true**.

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### Step 6: Compare \(1002^{1000}\) and \(1000^{1002}\)

Now we will compare \(1002^{1000}\) and \(1000^{1002}\):

\[ \frac{1002^{1000}}{1000^{1002}} = \frac{1002^{1000}}{1000^{1000} \cdot 1000^{2}} = \frac{1002^{1000}}{1000^{1000}} \cdot \frac{1}{1000^{2}} \]

### Step 7: Simplify Again

This can be rewritten as:

\[ \left(\frac{1002}{1000}\right)^{1000} \cdot \frac{1}{1000^{2}} = \left(1 + \frac{2}{1000}\right)^{1000} \cdot \frac{1}{1000^{2}} \]

### Step 8: Apply Binomial Theorem Again

Using the Binomial Theorem again:

\[ \left(1 + \frac{2}{1000}\right)^{1000} \approx 1 + 1000 \cdot \frac{2}{1000} = 3 \]

### Step 9: Final Calculation for Option D

Substituting back:

\[ \frac{1002^{1000}}{1000^{1002}} \approx \frac{3}{1000^{2}} \]

Since \(3 < 1000^{2}\), we conclude:

\[ 1002^{1000} < 1000^{1002} \]

### Step 10: Conclusion for Option D

Thus, Option D is also **true**.

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### Summary of Results

- Option A: **True**
- Option B: **False**
- Option C: **False**
- Option D: **True**