1. Two planes move along a circle of circumference 1.2 km with constant speeds. When they move in different directions, they meet every 15 seconds and when they move in the same direction, one plane overtakes the other every 60 seconds. Find the speed of the slower plane.

a) 0.04 km/s

b) 0.03 km/s

c) 0.05 km/s

d) 0.02 km/s

Explanation: Let their speeds be x m/sec and y m/sec respectively.

$$\eqalign{ & \frac{{1200}}{{x + y}} = 15 \cr & \Rightarrow x + y = 80.....(i) \cr} $$

$$\eqalign{ & \frac{{1200}}{{x - y}} = 60 \cr & \Rightarrow x - y = 20.....(ii) \cr} $$

Adding (i) and (ii),

2x = 100 or x = 50

Putting x = 50 in (i), we get : y = 30

Speed of slower plane :

= 30 m/sec

= 0.03 km/sec

2. Two joggers left Delhi for Noida simultaneously. The first jogger stopped 42 min later when he was 1 km short of Noida and the other one stopped 52 min later when he was 2 km short of Noida. If the first jogger jogged as many kilometers as the second, and the second as kilometers as first, the first one would need 17 min less than the second. Find the distance between Delhi and Noida?

a) 5 km

b) 15 km

c) 25 km

d) 35 km

Explanation:

$$\eqalign{ & {\text{Speed of first Jogger}} \cr & = {\frac{{ {x - 1} }}{{42}}} \times 60\,{\text{kmph}} \cr & {\text{Speed of }}\,{2^{nd}}\,{\text{jogger}} \cr & = {\frac{{ {x - 2} }}{{52}}} \times 60\,{\text{kmph}} \cr & {\text{Then}}, \cr & {\frac{{x - 2}}{{{s_b}}}} - {\frac{{x - 1}}{{{s_b}}}} \cr & \cr} $$

Now, check option one by one which gives us that option (b) is correct.

3. An ant moved for several seconds and covered 3 mm in the first second and 4 mm more in each successive second than its predecessor. If the ant had covered 1 mm in the first second and 8 mm more in each successive second, then the difference between the path it would cover during the same time and actual path would be more than 6 mm but less than 30 mm. find the time for which the ant moved (in seconds).

a) 5s

b) 4s

c) 6s

d) 2s

Explanation: 3 + 7 + 11 + 15 + . . . . . (1)

1 + 9 + 17 + 25 + . . . . . (2)

The condition is satisfied for the 4 seconds in 2

^{nd}journey, ant covered 17m more than 1

^{st}one.

4. Two trains start from the same point simultaneously and in the same direction. The first train travels at 40 km /h, and the speed of the second train is 25% more than the speed of first train. Thirty minutes later, a third train starts from same point and in the same direction. It over takes the second train 90 minutes later than it overtook the first train. What is the speed of the third train?

a) 20 km/h

b) 40 km/h

c) 60 km/h

d) 50 km/h

Explanation: A _______ B ________ C

Let both the trains start from point A.

At point B third train overtook the first train.

To overtake the first train by third train, Third train needs to cover,

Distance covered by first train in $$1\frac{1}{2}$$ h = distance covered by third train in 1 h.

[as third train has started 30 minutes later]

In this situation distance is constant, then

s × t = d; we get, S α $$\frac{1}{{\text{t}}}$$

Now,

$$\eqalign{ & \frac{{{\text{t}} + \frac{1}{2}}}{{\text{t}}} = \frac{{\text{s}}}{{40}} \cr & \frac{{2{\text{t}} + 1}}{{\text{t}}} = \frac{{\text{s}}}{{40}}\,.\,.\,.\,.\,.\,.\,.\,.\left( 1 \right) \cr} $$

From equation (1),

It is clear that time is in the ratio 3 : 2 then speed will be in 2 : 3 ratios.

Speed of the Third train will be 60 km/h.

5. A racetrack is in the form of a right triangle. The longer of the legs of track is 2 km more than the shorter of the legs (both these legs being on a highway). The start and end points are also connected to each other through a side road. The escort vehicle for the race took the side road and rode with a speed of 30 km/h and then covered the two intervals along the highway during the same time with a speed of 42 km/h. find the length of the race track.

a) 14 km

b) 10 km

c) 24 km

d) 36 km

Explanation: The given conditions are met on a Pythagoras triplet 6, 8, 10.

Since, the racetrack only consists of the legs of the right angle triangle the length must be,

= 6 + 8 = 14 km.

6. An individual is cycling at a speed of 25 km per hour. He catches his predecessor who had started earlier in two hours. What is the speed of his predecessor who had started 3 hours earlier ?

a) 15 kmph

b) 12 kmph

c) 10 kmph

d) 8 kmph

Explanation: The distance covered in two hour,

= 2 × 25 = 50 km

Time taken by first individual = (3h + 2h) = 5h

So, the speed of predecessor

= $$\frac{{50}}{5}$$ = 10 kmph.

7. A 6 cm long cigarette burns up in 15 minutes if no puff is taken.For every puff, it burns three times as fast during the duration of the puff.If the cigarette burns itself in 13 minutes, then how many puffs has the smoker taken if the average puff lasted 3 seconds?

a) 17

b) 18

c) 20

d) 22

Explanation: Let the number of puffs,

$$3x \times \left( {3 \times \frac{1}{{150}}} \right) + \left( {13 \times 60 - 3x} \right) \times \frac{1}{{150}} = 6$$

So, x = 20 puffs.

8. An old man driving bike at 80 km per hour. However being sugar patient, old man could not travel continuously. He takes small breaks each of 2 minutes for every 15 minute of his drive. How much distance the old man will cover in 90 minutes?

a) 112 Km

b) 104 km

c) 89 km

d) 118 km

Explanation: For every 15 minutes he takes a rest of 2 minutes.

Hence, for 90 minutes of drive he would require 12 minutes of rest.

He will be traveling for 90 - 12 = 78 minutes.

In 60 minutes he covers 80 Km.

In 1 minute he would cover $$\frac{{80}}{{60}}$$ Km.

In 78 minutes he would cover $$\frac{{80}}{{60}} \times 78$$ = 104 Km.

9. Two trains start simultaneously from two stations Howrah and Delhi, respectively towards each other on the same track. The distance between the two stations is 560 km and the speeds of trains are 30 kmph and 40 kmph.
Simultaneously with the trains, a sparrow sitting on the top of one of the train starts towards the other and reverses its direction on reaching the other train and so on. If the speed of sparrow is 80 kmph then the distance that the sparrow lies before being crushed between the train is :

a) 70 km

b) 560 km

c) 640 km

d) 650 km

Explanation: Relative speed of the trains = 40 + 30 = 70 kmph.

Time taken by trains to collide,

= $$\frac{{560}}{{70}}$$

= 8 hours

In 8 hours sparrow will cover,

= 8 × 80

= 640 km.

10. Due to the technical snag in the signal system two trains start approaching each other on the same track from two different stations, 240 km away each other. When the train starts a bird also starts moving to and fro between the two trains at 60 kmph touching each train each time. The bird initially sitting on the top of the engine of one of the trains and it moves so till these trains collide. If these trains collide one and half hour after start, then how many kilometers bird travels till the time of collision of trains?

a) 90 km

b) 130 km

c) 120 km

d) 95 km

Explanation: Time taken to collide the trains,

= one and half hour = $$\frac{3}{2}$$ hours

So, in $$\frac{3}{2}$$ hours bird travels,

= $$\frac{{3 \times 60}}{2}$$

= 90 km.