Correct Answer: Option C: 42
To solve this problem, we need to determine the original speed of the train. Let's denote the original speed of the train as $x$ km/hr.
     Step 1: Calculate the time taken to travel the first part of the journey.
         ○ The train travels 63 km at the original speed $x$.
         ○ Time taken for this part = $\frac{63}{x}$ hours.
     Step 2: Calculate the time taken to travel the second part of the journey.
         ○ The train travels 72 km at a speed of $x + 6$ km/hr.
         ○ Time taken for this part = $\frac{72}{x + 6}$ hours.
     Step 3: Set up the equation for the total time.
         ○ According to the problem, the total time for the journey is 3 hours.
         ○ Therefore, the equation is:
       $$\frac{63}{x} + \frac{72}{x + 6} = 3$$
     Step 4: Solve the equation.
         ○ To solve this equation, find a common denominator and simplify:
       $$\frac{63(x + 6) + 72x}{x(x + 6)} = 3$$
       $$63x + 378 + 72x = 3x(x + 6)$$
       $$135x + 378 = 3x^2 + 18x$$
       $$3x^2 + 18x - 135x - 378 = 0$$
       $$3x^2 - 117x - 378 = 0$$
         ○ Divide the entire equation by 3 to simplify:
       $$x^2 - 39x - 126 = 0$$
     Step 5: Solve the quadratic equation.
         ○ Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -39$, and $c = -126$.
         ○ Calculate the discriminant:
       $$b^2 - 4ac = (-39)^2 - 4 \times 1 \times (-126) = 1521 + 504 = 2025$$
         ○ Since the discriminant is a perfect square, calculate the roots:
       $$x = \frac{39 \pm \sqrt{2025}}{2}$$
       $$x = \frac{39 \pm 45}{2}$$
         ○ The possible solutions are:
       $$x = \frac{39 + 45}{2} = 42 \quad \text{and} \quad x = \frac{39 - 45}{2} = -3$$
         ○ Since speed cannot be negative, the original speed of the train is 42 km/hr.
     Therefore, the correct answer is Option C: 42.