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प्रश्न
| A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/hr more than the original speed. If it takes 3 hours to complete the total journey, then based on this information, answer the following questions: |
1. If the original speed of the bus be $$x$$ km/hr, then time taken by the bus to travel the next given distance is :
(a) $$\frac{75}{x}$$ hours
(b) $$\frac{90}{x}$$ hours
(c) $$\left(\frac{90}{x + 10}\right)$$ hours
(d) $$\left(\frac{90}{x - 10}\right)$$ hours
2. The quadratic equation for the given information, if the original speed of the bus be $$x$$ km/hr, is :
(a) $$x^2 + 45x - 250 = 0$$
(b) $$x^2 - 45x - 250 = 0$$
(c) $$x^2 - 75x - 450 = 0$$
(d) $$x^2 - 45x + 250 = 0$$
3. The original speed of the bus is :
(a) 50 km/hr
(b) 40 km/hr
(c) 75 km/hr
(d) 60 km/hr
4. The speed of the bus during which it travels the distance of 90 km is :
(a) 70 km/hr
(b) 50 km/hr
(c) 60 km/hr
(d) 85 km/hr
5. The time taken by the bus to travel a distance of 510 km with the new speed is :
(a) 8 hours
(b) $$8\frac{1}{2}$$ hours
(c) $$10\frac{1}{5}$$ hours
(d) $$12\frac{3}{4}$$ hours
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उत्तर
1. The correct option is (c).
Reasoning:
Next given distance = 90 km
Speed for this distance is 10 km/hr more than the original speed x km/hr, so the new speed = (x + 10) km/hr
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
Therefore, time taken = \[\left(\frac{90}{x + 10}\right) \text{ hours}\]
Option (a) is the time for the first part of the journey.
Option (b) is the time for 90 km at the original speed.
2. The correct option is (b).
Reasoning: Total time taken = 3 hours: \[\frac{75}{x} + \frac{90}{x + 10} = 3\]
Dividing both sides by 3: \[\frac{25}{x} + \frac{30}{x + 10} = 1\]
\[\frac{25(x + 10) + 30x}{x(x + 10)} = 1\]
\[\frac{25x + 250 + 30x}{x^2 + 10x} = 1\]
\[55x + 250 = x^2 + 10x\]
\[x^2 + 10x - 55x - 250 = 0\]
\[x^2 - 45x - 250 = 0\]
3. The correct option is (a).
Reasoning: Solving the quadratic equation:
\[x^2 - 45x - 250 = 0\]
\[x^2 - 50x + 5x - 250 = 0\]
\[x(x - 50) + 5(x - 50) = 0\]
\[(x - 50)(x + 5) = 0\]
\[x = 50 \quad \text{or} \quad x = -5\]
Since speed cannot be negative, \[x = 50\text{ km/hr}\].
Hence, the original speed of the bus is 50 km/hr.
4. The correct option is (c).
Reasoning:
Original speed = \[x = 50\text{ km/hr}\]
Speed during the 90 km journey = \[(x + 10)\text{ km/hr}\]
\[\text{Speed} = 50 + 10 = 60\text{ km/hr}\]
5. The correct option is (b).
Reasoning:
Distance = 510 km
New speed = 60 km/hr
\[\text{Time taken} = \frac{\text{Distance}}{\text{Speed}}\]
\[= \frac{510}{60}\]
\[= \frac{17}{2}\]
\[= 8\frac{1}{2}\text{ hours}\]
