PQRSTU is a regular hexagon drawn on the ground. Prashant stands at P and he starts jumpimg from vertex to vertex beginning from P. From any vertex of the hexagon except S, which is opposite to A, he may jump to any adjacent vertices. When he reaches S, he stops. Let Sn be the number of distinct paths of exactly n jumps ending at S. What is the value of S2k, where k is an integer?
Answer: A The least number of jumps required to reach S without any backtracking is 3. Every time there is back tracking, there is an increase of two jumps. hence the number of jumps is always odd. Whenever the number of jumps is even, there is no paths to reach S. Since '2k' is even S2k =0
Q. No. 8:
A total of 15 teams participated in a tournament. Each team plays with every other team exactly once. A team gets 3 points for a win, 2 points for a draw and 1 point for a loss. The team which scored the least got 21 points. The scores of all the teams were distinct and at ;east one match played by winning team was drawn. Which of the following is always true for the winning team?
Answer: B Number of points scored by any team when it plays with another =4. Total number of match played = 105 Total number of points of all the matches played = 420. As the losing team scored 21 points and the number of points scored by the teams are distinct integers, only if their scores are consecutive. The total number of points of all matches played would be 420. Hence winning team have scored 35 points. Let the number of matches won, drawn and lost by the winning team be W,D and L respectively. => 3W+2D+L= 35. and, W+D+L= 14 Subtracting the second eq from the first eq we get, 2W+D=21 Also, W+D <= 14. (W,D) can be (10,1) or (9,3) or (8,5) or (7,7) The respective value of L is 3 or 2 or 1 or 0.
Q. No. 9:
Firoz has a certain amount of money in only Re 1 and Rs 10 notes. the number of Re 1 notes multiplied by the number of Rs 10 notes is equal to the total money (in Rs) that he has. The number of Rs 10 notes is less than ten. Which of the following is a possible figure for the total number of notes he can have?
Answer: A Let Firoz have n Rs10 notes and m Re 1 notes. Given 10n+m = nm => m = 10n/(n-1) Since, n<10 the possible values of n for which m is an integer are 2,3 and 6 Corresponding m= 20, 15 and 12 n+m is either 22 or 18.
Q. No. 10:
A lock has a code consisting of five distinct letters, from among the first eleven words of English alphabet. The lock opens, if any three letters of the code match occurring not in the same position.What is the least number of five-letter codes, one should try to be sure that the lock is opened?
Answer: D If the following five five-letters codes are tried then the lock will open, no matter what will be the original code ABCDE FGHIJ FGHIK FGHJK FGIJK
Q. No. 11:
In a group, each person has at most two A. No person has less than three C. Considering all the persons in the group there are more A than G, more G than B and more B than persons. find the minimum number of persons in the groups?
Answer: C Let the total number of person be 'p'. hence, a>g>b>p Going by the choices, p must be at least 2, If p=2, a<=4. but to satisfy the above inequality , a must be atleast 5. hence p is not equal to 2. If p =3, to satisfy the inequality above, a must be at least 6. As, a<=6. It can be satisfied minimum value of f = 3.
Q. No. 12:
Fortuner, the latest SUV by Toyata Motors, consumes diesel at the rate of 1/400*{(1000/x) + x} L/km, when driven at the speed of x km/hr. If the diesel is Rs35/L and the driver is paid at the rate of Rs 125 per hour, then find the approximate optimal speed (in km/hour) of Fortuner that will minimize the total cost of the round trip of 800 km.
Answer: A Expenditure of diesel for 1 km = 1/400*{(1000/x) + x} Expenditure of diesel for 800 km = 2*{(1000/x) + x} Expenditure on diesel = 35 * 2*{(1000/x) + x} Time of journey = (800/x) hour Driver's pay = (125*800)/x rupees Total cost = 70 [{1000/x} + x] + (125*800)/x Now, if we go through the options, Then total cost at speed 49 km/hr => Rs 6899.38 55 km/hr => Rs 6940.90 50 km.hr => Rs 6900.00 53 km/hr => Rs 6917.55