Twenty six men - 1,2,3,....25 and 26- participate in 10km running race on a circular track of length 100m. All of them start at the same time, from the same point and run in the same direction. Their speeds, taken in the order, are in increasing AP. The time taken by 26 to meet 1, for the first time after they start is 20 sec and the time taken by 13 to complete the race is 52 minutes and 5 seconds. Find the time taken (in seconds), for all the twenty six men to meet for the first time at the starting point.
Answer: B Let 1,2,3,4...are in AP 2 = x+d, 3= x+2d... and so on...let assume 1(variable) = x Given time taken by 26 to meet 1 for the first time is 20 sec i.e 100/(x+25d-x) = 20 => d= 0.5 m/sec Time taken by 13 = x+12d is 52 minutes and 5 seconds i.e 100*100/(x+12d) = 3125 => x= 0.8 m/sec Time taken by all of them to meet for the first time at the starting point is LCM(100/x, 100/x+d, ............100/(x+25d)) => LCM(100,10,100,100,100)/(HCF(0.8, 1, 1.2, 1.4....5.8)) = 100/(0.2) = 500
Q. No. 8:
In a industry, the raw materials and the finished goods are transported on the conveyor belt.There are two conveyor belt, one for carrying parts from P to point Q and another for carrying parts from R to point Q. P,Q and R in that order are in a straight line. Sometimes, the belt serve to transport cart, which can themselves move with respect to the belts. The two belts move at a speed of 0.5 m/s and the cart move at a uniform speed of 2m/s with respect to the belts. A cart goes from point P to R and back to P taking a total of 64s. Find the distance PR in meters. Assume that the time taken by the cart to turn back at R is negligible?
Answer: C Let the speed of the belts be 'a' and that of the trolley is 'b' Let PQ = x and QR = y Time taken for cart to cover PR = x/(a+b) + y/(b-a) Time taken for trolley to cover RP = y/(a+b) + x/(b-a) Total time = (x+y) {1/(a+b) + 1/(b-a)} = 64 s Thus (x+y) = {(22-(0.5)2)/(2*2)} *64 = 60.
Q. No. 9:
The XYZ river flows at 12 km/hr. A boy who can row at 25/18 m/s in still water had to cross it in the least possible time. The distance covered by the boy is how many times the width of the river XYZ?
Answer: C Speed of the river XYZ= 12 kmph Speed of the boy = 5 kmph Let the time taken by the boy to cross the river in still water = T Width of the river = 5T The boy takes the least time when he is travelling directly across the river.But the river current push him in a direction perpendicular to the flow, Distance travelling along the river = 12T Effective distance travelled by boy = [(12T)2+(5T)2]1/2 = 13T => Distance covered by boy/ width of river = 13T/5T = 2.6
Q. No. 10:
A, B and C start simultaneously from X to Y. A reaches Y, turns back and meet B at a distance of 11 km from Y. B reached Y, turns back and meet C at a distance of 9 km from Y. If the ratio of the speeds of A and C is 3:2, what is the distance between X and Y?
Answer: A Let the distance between X and y be d km In the first instance distance travelled by A = d+11 In the first instance distance travelled by B = d-11 The time taken by both is same => (d+11)/A = (d-11)/B => A/B = (d+11)/(d-11)..........(i) In the second instance, distance travelled by B = d+9 While distance travelled by C = d-9 => B/C = (d+9)/(d-9)......(ii) From (i) and (ii) A/C = A/B * B/C = 3/2(Given) Thus ,By solving we get d= 1 or 99
Q. No. 11:
Mukesh, Suresh and Dinesh travel from Delhi to Mathura to attend Janmashtmi Utsav. They have a bike which can carry only two riders at a time as per traffic rules. Bike can be driven only by Mukesh. Mathura is 300km from Delhi. All of them walk at 15km/h. All of them start their journey from Delhi simultaneously and are required to reach Mathura at the same time. If the speed of bike is 60km/h, then what is the shortest possible time in which all three can reach Mathura at the same time ?
Answer: B Mukesh starts from Delhi (say A). He has to take of the other two (say Dinesh) on his bike, take him upto a certain point(say C) drop him there and return for Suresh. Mean while Suresh starts walking. Suresh and Mukesh meet at (say B) Mukesh picks up Suresh at B and turn towards mathura. All of them arrive together at Mathura (say D). A..................B......................C.........................D
As, M drives at 60km/h and S (as well as D) walk at 15 km/h. AC+CB = 4(AB) BC+CB = 3(AB) => CB = 1.5(AB) Let AB = 2, BC =3, (Also CD =2) Actually, AB = 600/7, BC = 900/7 and CD = 600/7 Time taken = 600/(7*15) + (900+600)/(7*60) => 40/7 + 25/7 = 65/7 hr.
Q. No. 12:
Two motorists Anil and Sunil are practicing with two different sports car; Ferrari and Maclarun, on the circular racing track, for the car racing tournament to be held next month. Both Anil and Sunil start from the same point on the circular track. Anil completes one round of the track in 1 min and Sunil takes 2 min to complete a round. While Anil maintains speed for all the rounds, Sunil halves his speed after the completion of each round. How many times Anil and Sunil will meet between 6th round and 9th round of Sunil (6th and 9th round is excluded)? Assume that the speed of Sunil remains steady throughout each round and changes only after the completion of that round.
Answer: C Time taken by Sunil for 1 st round = 2 min 2nd round = 4min 3rd round = 8 min 4th round = 16 min 5th round = 32 min 6th round = 64 min 7th round = 128 min 8th round = 256 min => Anil tales one minute for every round. He meets 127 times in 7th and 255 times in 8th round Total meet = 127 + 255 = 382