Let the speed of the stream be s km/h.
Speed of the motor boat 24 km / h
Speed of the motor -boat upstream 24 s
Speed of the motor boat downstream 24 s
According to the given condition,



Since, speed of the stream cannot be negative, the speed of the stream is 8 km/h.

Let the average speed of train for the first 54 km be x km/h

⇒ Average speed for the next 63 km = ( x + 6) km/h

We know




⇒ 117x + 324 = 3 ( x² + 6x)

⇒ 117x + 324 = 3x² + 18x

⇒ 3x² - 99x -324 = 0 ..(i)

Taking common from the above equation (i)

we have

x² - 33x - 108 = 0

⇒ x² - 36x + 3x -108 = 0

⇒ x (x - 36) + 3 (x - 36) = 0

⇒ (x - 36) (x +3 )=0

⇒ x -36 = 0 Or x + 3 =0

x = 36 or x = -3

 The speed of the train cannot be negative. Thus, x = 36

Hence, the speed of the train to cover 54 km or its first speed is 36 km/h.

Given points are A( k+1, 2k), B(3k, 2k+3), and C(5k-1, 5k)

These points will be collinear, if area of the triangle formed by them is zero.

We have,

i.e.