Explanation:

Let's solve the equations and find the correct option:

- Scheme A: The rate is given by the equation \(x^2 - 3x - 10 = 0\). Solving gives us \(x = 5\) or \(x = -2\). We take \(x = 5\) since it's positive.

- Scheme B: Solve \(y^2 - y - 6 = 0\), which gives \(y = 3\) or \(y = -2\). Choose \(y = 3\).

- Interest from Scheme A: Riya's profit from Scheme A is 160% of Diya's investment of Rs. 15000. So, Profit = \(1.6 \times 15000 = 24000\).

- Scheme C Rate: Solve \(z^2 - 10z + 24 = 0\) to get \(z = 6\) or \(z = 4\). Choose \(z = 6\) as it's greater than the rate from Scheme A (5).

- Scheme C Interest Calculation with simple interest formula as time (40) > rate (6):

$$

\text{Interest} = \frac{\text{Principal} \times 6 \times 40}{100} = 24000

$$

Solving, Principal = \( \frac{24000 \times 100}{6 \times 40} = 10000 \).

- Total Amount from Scheme C:

$$

\text{Total Amount} = 10000 + 24000 = 34000

$$

- Correct Option: \(\textbf{Option 3: 34000} \)