1. P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is.
a) 9 b) 5 c) 4 d) 6 e) 7
2. R walks 20 m North. Then he turns right and walks 30 m. Then he turns right and walks 35 m. Then he turns left and walks 15 m. Then he again turns left and walks 15 m.
Determine finally in which direction and how many metres away is he from his original position?
A) 30 metres West B) 15 metres West C) 45 metres East D) 45 metres West E) 30 metres East
3. Ajoy stood facing East. He walks 10m, turns left, walks 5m, turns right, walks 2m. What is the shortest distance between start and end point, and what direction is Ajoy facing?
Options:
A) 13m, North B) 12m, North C) 11m, North D) 13m, West E) 12m, West
Solutions
1. Answer 7
One can travel from P to S either via PQS or PRS or PQRS= 12 + 4n+ 3mn= 62 .
4n+ 3mn= 50——-(1)
One can travel from Q to R either via QR or QPR or QSR= m+ 12+ 4n= 27.
m+4n= 15———–(2)
Subtracting equation(2) from equation(1), we get 3mn-m= 35
m(3n-1)= 35 = 7*5
The number of direct paths between Q and R is m= 7
2. Answer: Option C.. R starts by walking 20 metres north, reaching (0, 20). He then turns right (east) and walks 30 metres to reach (30, 20). Turning right again (south), he walks 35 metres to reach (30, -15). He then turns left (east) and walks 15 metres to (45, -15), followed by another left turn (north) and walks 15 metres to reach his final position at (45, 0). His original position was (0, 0). So, he is now 45 metres east of his starting point. Thus, the final direction and distance from the starting point is 45 metres East.
3. Correct answer -13 meters, North
Ajoy starts facing East.
He turns right (now facing North again) and walks 2 meters.
He turns left (now facing North) and walks 10 meters.
He turns left again (now facing West) and walks 5 meters.
