Explanation:

Here is the analysis of the problem:

- The total simple interest (SI) is Rs 8,160 for 10 years.

- Interest rates and periods:

- 5% p.a. for the first 2 years.

- 9% p.a. for the next 3 years.

- 13% p.a. for the final 5 years (after 5 years).

Calculation:

- Calculate interest for each period:

1. 2 years at 5%: $\frac{2\times 5}{100}=0.10$ times the principal.

2. 3 years at 9%: $\frac{3\times 9}{100}=0.27$ times the principal.

3. 5 years at 13%: $\frac{5\times 13}{100}=0.65$ times the principal.

- Total interest rate over 10 years: $0.10+0.27+0.65=1.02$.

- Total SI = 1.02 times the principal = Rs 8,160.

- Solve for principal $P$:

$$

1.02 \times P = 8160 \quad \Rightarrow \quad P = \frac{8160}{1.02} = 8,000

$$

Answer:

- Option 1: 8,000 is correct.