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Two cities C1 and C2 are connected on the opposite end of a long straight parallel track. The cities are connected by a train service as well as bus service. The trains leave with constant speed v for either city at regular frequency of one train every x minute. The buses ply on a parallel road at a constant speed of 30 km h-1. A bus passenger going from City C1 to City C2 observes a train going past him every 20 minutes, while a train goes in the opposite direction every 10 minutes what are the value of x and v?​​​​​​

(A) x = 15 min, v = 90 km h-1

(B) x = 13 min 20 s, v = 90 km h-1

(C) x = 15 min, v = 75 km h-1

(D) x = 13 min 20 s, v = 70 km h-1

The correct option of this question is (B).

Solution : If v (in km h-1) is the constant speed of the trains then the distance between the successive trains

$v × \frac {x}{60} = \frac {vx}{60}km$

When a train moves in the same direction as that of the bus passenger,

$=\frac {vx/60}{v-30} = \frac {20}{60}$

Or, vx = 20(v-30)              .......(i)

When a train moves in a direction opposite to the bus passenger,

$\frac {vx/60}{(v+30)} = \frac {10}{60}$

Or, vx = 10(v+30).         .........(ii)

From eqns. (i) and (ii),

20(v-30) = 10(v+30)

Or, v =90 km h-1

From eqn. (i),

90x = 20(90-30) = 1200

Or, x = 13 min 20 s

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