প্রশ্ন: a6 - b6 এর একটি উৎপাদক কোনটি?

সমাধান:
a6 - b6
= (a3)2 - (b3)2
= (a3 + b3)(a3 - b3)
= (a + b)(a2 - ab + b2)(a - b)(a2 + ab + b2)
= (a + b)(a - b)(a2 - ab + b2)(a2 + ab + b2)

1. Let the train length = x meters.
   Distances covered while crossing the bridges:

   • First bridge: $x + 420$ meters

   • Second bridge: $x + 200$ meters

2. Time taken:

   • First bridge = 60 sec → speed = $(x + 420)/60$

   • Second bridge = 40 sec → speed = $(x + 200)/40$

3. Since speed is constant:

$$\frac{x + 420}{60} = \frac{x + 200}{40}$$

4. Solve for x:

$$40(x + 420) = 60(x + 200)$$ $$40x + 16800 = 60x + 12000$$ $$20x = 4800 \implies x = 240$$

Answer: The train is 240 meters long

প্রশ্ন: যদি m + n = x এবং mn = 6x2 হয়, তবে m3 + n3 = কত?

সমাধান:
দেওয়া আছে,
m + n = x এবং mn = 6x2

আমরা জানি,
m3 + n3 = (m + n)3 - 3mn(m + n)
= x3 - 3 × 6x2 × x
= x3 - 18x3
= - 17x3

∴ m3 + n3 = - 17x3