Daily Olympiad: Logical Reasoning - Direction Sense [20260614]

Solution

Correct: B

Let the starting point be (0,0). 1. Walks 10 meters North: Position becomes (0, 10). 2. Turns right (East) and walks 15 meters: Position becomes (0+15, 10) = (15, 10). 3. Turns right (South) and walks 10 meters: Position becomes (15, 10-10) = (15, 0). 4. Turns left (East) and walks 5 meters: Position becomes (15+5, 0) = (20, 0). Final position is (20, 0). So, he is 20 meters to the East from his starting point.

Solution

Correct: B

Let the starting point for both be (0,0). For Person A: 1. Walks 10 meters North: (0, 10). 2. Turns right (East) and walks 5 meters: (0+5, 10) = (5, 10). A's final position is (5, 10). For Person B: 1. Walks 5 meters West: (-5, 0). 2. Turns right (North) and walks 10 meters: (-5, 0+10) = (-5, 10). B's final position is (-5, 10). To find the direction of B with respect to A, we calculate B_coordinates - A_coordinates: (-5 - 5, 10 - 10) = (-10, 0). Since the x-coordinate is negative and y-coordinate is zero, B is 10 meters to the West of A.

Solution

Correct: A

In the evening, after sunset, the sun is in the West. Therefore, shadows fall towards the East. Anil's shadow fell exactly to his left side. This means Anil's left side is East. If Anil's left side is East, then Anil must be facing South (because if you face South, East is to your left and West is to your right). Anil and Gopal were standing with their backs towards each other. If Anil is facing South, then Gopal must be facing the opposite direction, i.e., North.

Solution

Correct: B

Let the starting point be (0,0). 1. Walks 10 meters North: (0, 10). 2. Turns right (East) and walks 20 meters: (20, 10). 3. Turns right (South) and walks 10 meters: (20, 0). 4. Turns right (West) and walks 5 meters: (15, 0). His final position is (15, 0). This means he is 15 meters to the East of his starting point.

Solution

Correct: A

Let's consider directions in degrees clockwise from North (0 degrees): North = 0/360, East = 90, South = 180, West = 270 North-East = 45, South-East = 135, South-West = 225, North-West = 315 1. South-East (135 degrees) becomes North (0 degrees). This is a 135-degree anti-clockwise rotation (135 - 135 = 0). Or a 225-degree clockwise rotation (135 + 225 = 360 = 0). 2. North-East (45 degrees) becomes West (270 degrees). This is a 45-degree anti-clockwise rotation (45-135 = -90, -90+360 = 270 degrees). This doesn't match a fixed rotation. Let's recheck the angles. North-East (45) to West (270) is 225 degrees anti-clockwise (270 - 45 = 225). Or 135 degrees clockwise (360 - 225 = 135 degrees clockwise, or 45 + 135 = 180 which is South. So 45 + X = 270, X = 225 clockwise. No, that's not right. 45 - X = 270. X = 45-270 = -225, so 225 degrees anti-clockwise). Let's assume a consistent anti-clockwise rotation: SE (135 deg) -> N (360 deg or 0 deg). Difference = 360 - 135 = 225 degrees anti-clockwise. NE (45 deg) -> W (270 deg). Difference = 270 - 45 = 225 degrees anti-clockwise. The rotation is 225 degrees anti-clockwise. Now apply this to East (90 degrees): 90 - 225 = -135 degrees. Adding 360 to get a positive angle: -135 + 360 = 225 degrees. 225 degrees corresponds to South-West. Alternatively, considering clockwise rotation: SE (135) to N (0/360) is a 225-degree clockwise rotation (360-135=225). NE (45) to W (270) is 225-degree clockwise rotation (45+225 = 270). So, it's a 225-degree clockwise rotation. East (90 degrees) + 225 degrees = 315 degrees. 315 degrees is North-West.

Solution

Correct: C

Let the starting point for both be (0,0). For Person P: 1. Walks 10 meters West: (-10, 0). 2. Turns left (South) and walks 5 meters: (-10, -5). P's final position is (-10, -5). For Person Q: 1. Walks 5 meters East: (5, 0). 2. Turns right (South) and walks 10 meters: (5, -10). Q's final position is (5, -10). To find the direction of P with respect to Q, we calculate P_coordinates - Q_coordinates: (-10 - 5, -5 - (-10)) = (-15, -5 + 10) = (-15, 5). Since the x-coordinate is negative and the y-coordinate is positive, the direction is North-West.

Solution

Correct: A

Let the starting point be (0,0). 1. Walks 20 meters North: (0, 20). 2. Turns right (East) and walks 30 meters: (30, 20). 3. Turns right (South) and walks 35 meters: (30, 20-35) = (30, -15). 4. Turns left (East) and walks 15 meters: (30+15, -15) = (45, -15). 5. Turns left (North) and walks 15 meters: (45, -15+15) = (45, 0). His final position is (45, 0). This means he is 45 meters to the East of his starting point.

Solution

Correct: A

In the evening, the sun is in the West. Therefore, all shadows fall towards the East. P's shadow fell exactly to Q's right. This means Q's right side is East. If Q's right side is East, then Q must be facing South (because if you face South, East is to your left and West is to your right). P and Q are talking face to face. If Q is facing South, then P, who is facing Q, must be facing North.

Solution

Correct: A

Let the starting point be (0,0). 1. Goes 5 meters South: (0, -5). 2. Turns left (East) and walks 8 meters: (8, -5). 3. Turns right (South) and walks 3 meters: (8, -5-3) = (8, -8). 4. Turns right (West) and walks 4 meters: (8-4, -8) = (4, -8). His final position is (4, -8). To find the shortest distance from the starting point (0,0) to (4,-8), we use the Pythagorean theorem: Distance = sqrt((x2-x1)^2 + (y2-y1)^2) Distance = sqrt((4-0)^2 + (-8-0)^2) Distance = sqrt(4^2 + (-8)^2) Distance = sqrt(16 + 64) Distance = sqrt(80) Distance = sqrt(16 * 5) Distance = 4*sqrt(5) meters.

Solution

Correct: B

Let's assume North as 0 degrees. 1. Starts facing North: 0 degrees. 2. Turns 45 degrees clockwise: 0 + 45 = 45 degrees (North-East). 3. Turns 180 degrees clockwise: 45 + 180 = 225 degrees (South-West). 4. Turns 90 degrees anti-clockwise: 225 - 90 = 135 degrees. 135 degrees corresponds to South-East.

Solution

Correct: C

Let the starting point be (0,0). 1. Walks 10m North: (0, 10). 2. Turns right (East) and walks 20m: (20, 10). 3. Turns right (South) and walks 5m: (20, 5). 4. Turns left (East) and walks 10m: (30, 5). 5. Turns left (North) and walks 5m: (30, 10). 6. Turns right (East) and walks 10m: (40, 10). His final position is (40, 10). This position is to the North-East of his starting point (0,0).

Solution

Correct: A

Let Village B be the origin (0,0). 1. Village A is 20km North of B: A = (0, 20). 2. Village C is 15km East of B: C = (15, 0). 3. Village D is 10km South of C: D = (15, 0-10) = (15, -10). 4. Village E is 15km West of D: E = (15-15, -10) = (0, -10). Now, we need to find the direction and distance of Village A (0, 20) from Village E (0, -10). Coordinates of A relative to E: (0-0, 20 - (-10)) = (0, 30). This means A is 30km North of E.

Solution

Correct: A

Let the starting point be (0,0). 1. Walks 12 meters North: (0, 12). 2. Turns right (East) and walks 15 meters: (15, 12). 3. Turns left (North) and walks 8 meters: (15, 12+8) = (15, 20). 4. Turns left (West) and walks 3 meters: (15-3, 20) = (12, 20). 5. Turns left (South) and walks 20 meters: (12, 20-20) = (12, 0). His final position is (12, 0). The shortest distance from the starting point (0,0) to (12,0) is 12 meters.

Solution

Correct: B

In the morning, after sunrise, the sun is in the East. Therefore, all shadows fall towards the West. Q's shadow fell exactly to P's right. This means P's right side is West (where the shadow is). If P's right side is West, then P must be facing South (because if you face South, West is to your right and East is to your left). If P is facing South, then Q, who is face to face with P, must be facing North.

Solution

Correct: A

Let Point A be the origin (0,0). 1. Point B is 7m North of A: B = (0, 7). 2. Point C is 5m East of B: C = (0+5, 7) = (5, 7). 3. Point D is 7m South of C: D = (5, 7-7) = (5, 0). 4. Point E is 3m West of D: E = (5-3, 0) = (2, 0). Now, we need to find the shortest distance between Point A (0,0) and Point E (2,0). Distance = sqrt((2-0)^2 + (0-0)^2) = sqrt(2^2 + 0^2) = sqrt(4) = 2 meters.

Solution

Correct: B

Let's trace the path backwards from the final direction. 1. Final direction: North. 2. Before walking North for 5m, he took a Left turn. If he turned left to face North, he must have been walking East before the turn. 3. Before walking East for 8m, he took a Right turn. If he turned right to face East, he must have been walking North before the turn. 4. Before walking North for 5m, he took a Left turn. If he turned left to face North, he must have been walking East before the turn. 5. Before walking East for 10m (straight), he started from a point. So, he started walking East. Let's verify by moving forward: - Start: East - 10m straight: walking East. - Turns left: now walking North. - Walks 5m: still walking North. - Turns right: now walking East. - Walks 8m: still walking East. - Turns left: now walking North. - Walks 5m: still walking North. (Final direction matches the given condition). Therefore, he started walking East.

Solution

Correct: C

Let's use degrees clockwise from North (0 degrees). North-West = 315 degrees. 1. Turns 90 degrees clockwise: 315 + 90 = 405 degrees. Since a full circle is 360 degrees, 405 - 360 = 45 degrees. (This is North-East). 2. Turns 180 degrees anti-clockwise: 45 - 180 = -135 degrees. Adding 360 to get a positive angle: -135 + 360 = 225 degrees. (This is South-West). 3. Turns another 90 degrees in the same (anti-clockwise) direction: 225 - 90 = 135 degrees. 135 degrees corresponds to South-East.

Solution

Correct: B

Let Point B be the origin (0,0). 1. Point A is 10m East of B: A = (10, 0). 2. Point C is 15m South of B: C = (0, -15). 3. Point D is 10m West of C: D = (0-10, -15) = (-10, -15). 4. Point E is 5m North of D: E = (-10, -15+5) = (-10, -10). To find the direction of E with respect to A, we calculate E_coordinates - A_coordinates: (-10 - 10, -10 - 0) = (-20, -10). Since both the x and y coordinates are negative, E is to the South-West of A.

Solution

Correct: A

In the morning, the sun is in the East. Rahul started walking towards the sun, so he started walking East. Let's trace his path: 1. Started walking East. 2. Turned right: Now walking South. 3. Again turned right: Now walking West. 4. Turned left: Now walking South. So, at the final point, he is walking South. In the morning, shadows fall towards the West. If he is walking South, his right side is West, and his left side is East. Since his shadow (which falls West) was to his right, this confirms that he was indeed walking South at that point. Therefore, his initial direction of walking was East.

Solution

Correct: D

Let the starting point be (0,0). 1. Walks 2km South: (0, -2). 2. Turns West and walks 3km: (-3, -2). 3. Turns North and walks 6km: (-3, -2+6) = (-3, 4). 4. Turns East and walks 7km: (-3+7, 4) = (4, 4). 5. Turns South and walks 2km: (4, 4-2) = (4, 2). His final position is (4, 2). To find the shortest distance from the starting point (0,0) to (4,2), we use the Pythagorean theorem: Distance = sqrt((x2-x1)^2 + (y2-y1)^2) Distance = sqrt((4-0)^2 + (2-0)^2) Distance = sqrt(4^2 + 2^2) Distance = sqrt(16 + 4) Distance = sqrt(20) km. Also, sqrt(20) can be simplified as sqrt(4 * 5) = 2*sqrt(5) km. Both A and C represent the same value.