4
Hemant, departing from his house in the east, arrived at a crossing on his way to the university. At this crossing, the road on his left leads to a theatre, and the road directly ahead leads to a hospital. In which direction is the university located?
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Solution: Step 1: Hemant starts from his house in the east and arrives at a crossing. This means he approached the crossing from the East, or if he is at the crossing, his house is to the East relative to the crossing. Thus, he would be facing West if his house is behind him.
Step 2: If Hemant is facing West at the crossing:
* The road to his left would be pointing South (West -> Left is South).
* The road straight ahead would be pointing West.
Step 3: The problem states that the road to his left (South) leads to a theatre, and the road straight ahead (West) leads to a hospital.
Step 4: The only remaining main direction is to his right, which would be North (West -> Right is North).
Step 5: Therefore, the university is in the North direction.
5
Village Q is located North of village P. Village R is situated East of village Q. Village S is positioned to the left of village P. In what direction is village S with respect to village R?
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Solution: Step 1: Let's set village P as the origin (0,0).
Step 2: 'Village Q is to the North of P.' So, Q is at (0, y_Q) where y_Q > 0. For simplicity, let Q = (0,1).
Step 3: 'Village R is in the East of Q.' So, R is at (x_R, 1) where x_R > 0. For simplicity, let R = (1,1).
Step 4: 'Village S is to the left of P.' In geographical context, 'left of' typically means West of. So, S is at (x_S, 0) where x_S < 0. For simplicity, let S = (-1,0).
Step 5: Now, we need to find the direction of S (-1,0) with respect to R (1,1).
Step 6: To go from R (1,1) to S (-1,0):
* The x-coordinate changes from 1 to -1 (decrease), indicating Westward movement.
* The y-coordinate changes from 1 to 0 (decrease), indicating Southward movement.
Step 7: Therefore, village S is in the South-West direction with respect to village R.
8
One evening, before sunset, Rekha and Hema were conversing face to face. If Hema's shadow appeared exactly to Hema's right, what direction was Rekha facing?
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Solution: Step 1: Determine the sun's position and the general direction of shadows in the evening. In the evening, the sun sets in the West.
Step 2: Consequently, any shadow will fall towards the East.
Step 3: The problem states that Hema's shadow was exactly to Hema's right.
Step 4: This implies that Hema's right side is pointing towards the East.
Step 5: If a person's right side is East, that person must be facing North. So, Hema is facing North.
Step 6: Rekha and Hema are talking to each other face to face. If Hema is facing North, then Rekha, who is facing Hema, must be facing the opposite direction.
Step 7: Therefore, Rekha was facing South.
9
After traversing 6 km, I turned right and then walked 2 km. Subsequently, I turned left and walked 10 km. If, at the journey's end, I was moving North, what was my initial starting direction?
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Solution: Step 1: Let's analyze the sequence of movements in reverse, knowing the final facing direction.
Step 2: In the end, I was moving towards the North.
Step 3: Before moving North, I turned to the left. To face North after turning left, I must have been facing East prior to that turn.
Step 4: Before facing East, I turned to the right. To face East after turning right, I must have been facing North prior to that turn.
Step 5: This 'North' was the direction I was facing after completing the initial 6 km walk. Thus, the direction in which I started my journey and walked the first 6 km was North.
Step 6: Based on standard directional logic, if the final direction is North, and the net effect of a Right turn followed by a Left turn is no change in orientation, then the initial direction must also be North.
Note: The provided correct answer is 'South'. However, based on the logical derivation of the problem statement, the starting direction would be North. If the correct answer is indeed 'South', there might be an alternative interpretation of the turns or a specific context not provided in the problem statement that leads to this conclusion. For this solution, we adhere to the standard logical interpretation.
10
Shyam begins walking 5 km East, then turns left and walks 6 km. He turns right again and walks 9 km. Finally, he turns right once more and walks 6 km. What is Shyam's current straight-line distance from his starting point?
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Solution: Step 1: Assume Shyam's starting point is the origin (0,0).
Step 2: Walks 5 km East: Position (5,0). Facing East.
Step 3: Turns left (North) and walks 6 km: Position (5,6). Facing North.
Step 4: Turns right (East) and walks 9 km: Position (5+9, 6) = (14,6). Facing East.
Step 5: Turns right (South) and walks 6 km: Position (14, 6-6) = (14,0). Facing South.
Step 6: Shyam's final position is (14,0). His starting point is (0,0).
Step 7: The distance from the starting point is the straight-line distance between (14,0) and (0,0).
Step 8: Distance = sqrt((14-0)^2 + (0-0)^2) = sqrt(14^2) = 14 km.
13
Rohit walked 25 m South. He then turned left and walked 20 m. Next, he turned left again and walked 25 m. Finally, he turned right and walked 15 m. What is Rohit's distance and direction from his starting point?
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Solution: Step 1: Assume Rohit's starting point is the origin (0,0).
Step 2: Walks 25 m South: Position (0, -25). Facing South.
Step 3: Turns left (East) and walks 20 m: Position (20, -25). Facing East.
Step 4: Turns left (North) and walks 25 m: Position (20, -25 + 25) = (20, 0). Facing North.
Step 5: Turns right (East) and walks 15 m: Position (20 + 15, 0) = (35, 0). Facing East.
Step 6: Rohit's final position is (35,0). His starting point is (0,0).
Step 7: The distance from the starting point is the magnitude of the final position's coordinates, which is 35 m.
Step 8: The direction from the starting point is determined by the coordinates (35,0), which is purely along the positive X-axis.
Step 9: Therefore, Rohit is 35 m East from his starting point.
14
A person starts from their house, walks 10 km south, then 5 km west, takes a left turn, and walks 7 km to reach school. In which direction is the house from the school?
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Solution: Step 1: The person starts at their house.
Step 2: They walk 10 km south.
Step 3: Then, they walk 5 km west.
Step 4: After that, they walk 7 km south.
Step 5: To find the direction from the school to the house, reverse the steps.
Step 6: First, go 7 km north.
Step 7: Then, go 5 km east.
Step 8: Finally, go 10 km north.
Step 9: This results in being 5 km east and 17 km north of the starting point.
Step 10: Therefore, the house is North-East of the school.
16
Y is situated East of X, and X is North of Z. If P is South of Z, what is the direction of P relative to Y?
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Solution: Step 1: Visualize the positions relative to each other.
Step 2: 'X is in the North of Z': Imagine Z at the bottom, and X directly above it.
Step 3: 'Y is in the East of X': Imagine Y to the right of X. So, from a top-down view, you have Z, then X directly above Z, and Y to the right of X.
Step 4: 'P is in the South of Z': Imagine P directly below Z.
Step 5: Now, determine the direction of P from Y. From Y, you would need to move left (West) and then down (South) to reach P.
Step 6: Therefore, P is in the South-West direction of Y.
17
Sundar runs 20 m East, then turns right and runs 10 m. He then turns right again and runs 9 m. After another right turn, he runs 5 m. Subsequently, he turns left and runs 12 m, and finally, he turns right and runs 6 m. In which direction is Sundar currently facing?
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Solution: Step 1: Sundar starts by running towards East. Current facing direction: East.
Step 2: Turns right. When facing East, a right turn means facing South. Current facing direction: South.
Step 3: Turns right again. When facing South, a right turn means facing West. Current facing direction: West.
Step 4: Turns right again. When facing West, a right turn means facing North. Current facing direction: North.
Step 5: Turns left. When facing North, a left turn means facing West. Current facing direction: West.
Step 6: Turns right. When facing West, a right turn means facing North. Current facing direction: North.
Step 7: Therefore, Sundar is finally facing North.
19
Amit started walking with his back facing the sun. After some time, he turned left, then turned right, and then turned left again. In which direction is Amit currently walking?
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Solution: Step 1: Analyze the initial condition 'positioning his back towards the sun'.
* If it is morning (sun in the East), facing with back to the sun means Amit is facing West.
* If it is evening (sun in the West), facing with back to the sun means Amit is facing East.
Step 2: Analyze the sequence of turns: 'turned left, then turned right, and towards the left again'.
* A left turn (L) followed by a right turn (R) cancels out (L + R = no net change in orientation).
* So, the sequence 'L, then R, then L' is equivalent to a single Left turn from the initial facing direction.
Step 3: Apply the net turn to both possible initial directions.
* Case 1: If Amit started facing West (morning scenario), after a single Left turn, he would be facing South (West -> Left is South).
* Case 2: If Amit started facing East (evening scenario), after a single Left turn, he would be facing North (East -> Left is North).
Step 4: Since the time of day (morning or evening) is not specified, there are two possible final directions.
Step 5: Therefore, Amit is now going either North or South.
20
Four friends, A, B, C, and D, reside in the same area. B's house is situated to the East of A's house, but also to the North of C's house. C's house is located to the West of D's house. In which direction is D's house from A's house?
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Solution: Step 1: Let's set A's house as the reference point, (0,0).
Step 2: 'B is in the East of A's house.' This means B is to the right of A. Let A = (0,0) and B = (x,0) where x > 0. For simplicity, let B = (1,0).
Step 3: 'B is in the North of C's house.' This means C is to the South of B. If B = (1,0), then C = (1,y) where y < 0. For simplicity, let C = (1,-1).
Step 4: 'C is in the West of D's house.' This means D is to the East of C. If C = (1,-1), then D = (z,-1) where z > 1. For simplicity, let D = (2,-1).
Step 5: Now, compare D's position (2,-1) with A's position (0,0).
Step 6: The positive x-coordinate (2) indicates East, and the negative y-coordinate (-1) indicates South.
Step 7: Therefore, D's house is in the South-East direction of A's house.