Detailed Solution:
Step 1: Interest from Bank A
Principal deposited in Bank A = ₹10,000
Simple interest rate at Bank A = 5% per annum
Let the number of years of investment in Bank A = \( t_A \)
Interest from Bank A (\( SI_A \)) is calculated as:
\[ SI_A = \frac{10,000 \cdot 5 \cdot t_A}{100} = 500 \cdot t_A \]
Step 2: Total amount from Bank A deposited in Bank B
Total amount after maturity from Bank A = Principal + Interest = \( 10,000 + 500 \cdot t_A \)
This becomes the principal deposited in Bank B (\( P_B \)).

Step 3: Interest from Bank B
Bank B invests this total amount for 5 years at a simple interest rate of 6%.
Interest from Bank B (\( SI_B \)) is:
\[ SI_B = \frac{(10,000 + 500 \cdot t_A) \cdot 30}{100} \] Simplifying:
\[ SI_B = 3000 + 150 \cdot t_A \]
Step 4: Use the ratio of interests
The interests are given in the ratio 10:13:
\[ \frac{SI_A}{SI_B} = \frac{10}{13} \] Substituting values of \( SI_A \) and \( SI_B \):
\[ \frac{500 \cdot t_A}{3000 + 150 \cdot t_A} = \frac{10}{13} \]
Step 5: Solve for \( t_A \)
Cross-multiply to eliminate the fraction:
\[ 13 \cdot 500 \cdot t_A = 10 \cdot (3000 + 150 \cdot t_A) \] Simplify:
\[ 6500 \cdot t_A = 30,000 + 1500 \cdot t_A \] \[ 5000 \cdot t_A = 30,000 \] \[ t_A = \frac{30,000}{5000} = 6 \]
Final Answer: The investment period in Bank A is \( t_A = 6 \) years.