Daily Olympiad: Logical Reasoning - Direction Sense [20260911]

Solution

Correct: B

Let the starting point be the origin (0,0). 1. Walks 10m North: Position becomes (0, 10). 2. Turns right (East) and walks 5m: Position becomes (0+5, 10) = (5, 10). 3. Turns left (North) and walks 10m: Position becomes (5, 10+10) = (5, 20). 4. Turns right (East) and walks 10m: Position becomes (5+10, 20) = (15, 20). The final position is (15, 20) relative to the starting point (0,0). The horizontal displacement is 15m East, and the vertical displacement is 20m North. The direction from the starting point is North-East. The shortest distance is calculated using the Pythagorean theorem: Distance = sqrt((15)^2 + (20)^2) = sqrt(225 + 400) = sqrt(625) = 25m. Therefore, the final position is 25m North-East from the starting point.

Solution

Correct: B

Let Rahul's home be the origin (0,0). 1. Walks 15m West: Position becomes (-15, 0). 2. Turns left (South) and walks 20m: Position becomes (-15, 0-20) = (-15, -20). 3. Turns right (West) and walks 10m: Position becomes (-15-10, -20) = (-25, -20). 4. Turns right (North) and walks 20m: Position becomes (-25, -20+20) = (-25, 0). The final position is (-25, 0) relative to the starting point (0,0). This means he is 25m to the West of his starting point. Therefore, he is in the West direction from his starting point.

Solution

Correct: B

In the morning, after sunrise, the sun is in the East direction. Therefore, shadows always fall towards the West. Rohit's shadow fell exactly to his left. This means Rohit's left side is West. If Rohit's left is West, then he must be facing North (as North is perpendicular to West on the right side when facing North). Rohit and Kartik were standing facing each other. If Rohit is facing North, then Kartik must be facing the opposite direction, which is South.

Solution

Correct: A

In the evening, just before sunset, the sun is in the West direction. Therefore, shadows always fall towards the East. Neha's shadow was exactly to her right. This means Neha's right side is East. If Neha's right is East, then she must be facing South (as South is perpendicular to East on the left side when facing South). Pooja and Neha were talking to each other face to face. If Neha is facing South, then Pooja must be facing the opposite direction, which is North.

Solution

Correct: B

Let the starting point be the origin (0,0). 1. Walks 4 km East: Position becomes (4, 0). 2. Turns left (North) and walks 3 km: Position becomes (4, 0+3) = (4, 3). 3. Turns left (West) and walks 2 km: Position becomes (4-2, 3) = (2, 3). 4. Turns right (North) and walks 3 km: Position becomes (2, 3+3) = (2, 6). The final position is (2, 6) relative to the starting point (0,0). The horizontal displacement is 2 km East, and the vertical displacement is 6 km North. The shortest distance is calculated using the Pythagorean theorem: Distance = sqrt((2)^2 + (6)^2) = sqrt(4 + 36) = sqrt(40) km.

Solution

Correct: C

Let Point B be the origin (0,0). 1. Point A is 10m East of B: A = (10, 0). 2. Point C is 5m North of B: C = (0, 5). 3. Point D is 10m South of C: D = (0, 5-10) = (0, -5). 4. Point E is 5m West of D: E = (0-5, -5) = (-5, -5). We need to find the direction of E with respect to A. Relative coordinates of E from A = (E_x - A_x, E_y - A_y) = (-5 - 10, -5 - 0) = (-15, -5). Since the x-coordinate is negative (-15), E is to the West of A. Since the y-coordinate is negative (-5), E is to the South of A. Therefore, point E is in the South-West direction with respect to point A.

Solution

Correct: B

Let Ravi's home be the origin (0,0). 1. Walks 10 km North: Position becomes (0, 10). 2. Turns left (West) and walks 5 km: Position becomes (0-5, 10) = (-5, 10). 3. Turns left (South) and walks 10 km: Position becomes (-5, 10-10) = (-5, 0). 4. Turns right (West) and walks 10 km: Position becomes (-5-10, 0) = (-15, 0). The final position is (-15, 0) relative to his home (0,0). This means he is 15 km to the West of his home. Therefore, he is 15 km West from his home.

Solution

Correct: D

Let's use degrees for directions, with North at 0 degrees (or 360 degrees), East at 90 degrees, South at 180 degrees, and West at 270 degrees. 1. Starts facing South: Initial direction = 180 degrees. 2. Turns 135 degrees clockwise: 180 + 135 = 315 degrees. (315 degrees corresponds to North-West). 3. Turns 90 degrees anti-clockwise: 315 - 90 = 225 degrees. (225 degrees corresponds to South-West, which is exactly between South (180) and West (270)). Therefore, he is now facing South-West.

Solution

Correct: A

Let the starting point be the origin (0,0). 1. Walks 6 km East: Position becomes (6, 0). 2. Turns South and walks 5 km: Position becomes (6, 0-5) = (6, -5). 3. Turns East and walks 6 km: Position becomes (6+6, -5) = (12, -5). 4. Turns North and walks 10 km: Position becomes (12, -5+10) = (12, 5). The final position is (12, 5) relative to the starting point (0,0). The horizontal displacement is 12 km East, and the vertical displacement is 5 km North. The shortest distance is calculated using the Pythagorean theorem: Distance = sqrt((12)^2 + (5)^2) = sqrt(144 + 25) = sqrt(169) = 13 km.

Solution

Correct: A

Let Point Q be the origin (0,0). 1. Point P is 10 km East of Q: P = (10, 0). 2. Point R is 8 km North of P: R = (10, 0+8) = (10, 8). 3. Point S is 10 km West of R: S = (10-10, 8) = (0, 8). 4. Point T is 5 km South of S: T = (0, 8-5) = (0, 3). The final position of T is (0, 3) relative to Q (0,0). Since the x-coordinate is 0 and the y-coordinate is positive (3), point T is directly North of point Q. Therefore, point T is in the North direction with respect to point Q.

Solution

Correct: C

Let North be 0 degrees. 1. Starts facing North: Initial direction = 0 degrees. 2. Turns 45 degrees clockwise: 0 + 45 = 45 degrees (North-East). 3. Turns another 180 degrees clockwise: 45 + 180 = 225 degrees (South-West). 4. Turns 270 degrees anti-clockwise: 225 - 270 = -45 degrees. A negative angle means anti-clockwise rotation. -45 degrees is equivalent to 360 - 45 = 315 degrees. 315 degrees corresponds to North-West. Therefore, he is now facing North-West.

Solution

Correct: A

Let Ankit's house be the origin (0,0). 1. Went 20 km North: Position becomes (0, 20). 2. Turned West and covered 15 km: Position becomes (0-15, 20) = (-15, 20). 3. Turned South and covered 10 km: Position becomes (-15, 20-10) = (-15, 10). 4. Turned East and covered 15 km: Position becomes (-15+15, 10) = (0, 10). The final position is (0, 10) relative to his house (0,0). This means he is 10 km to the North of his house. Therefore, he is in the North direction from his house.

Solution

Correct: C

Let's trace the path backward: 1. Final direction: South. 2. Before this, he turned right. If a right turn led him to face South, then he must have been facing East before the turn. 3. Before that, he turned left. If a left turn led him to face East, then he must have been facing North before the turn. 4. He walked straight from a point and then turned left. So, his initial walking direction was North. Therefore, he started walking towards the North.

Solution

Correct: B

Let Prakash's office be the origin (0,0). 1. Walks 10 km South: Position becomes (0, -10). 2. Turns right (West) and walks 5 km: Position becomes (0-5, -10) = (-5, -10). 3. Turns right (North) and walks 8 km: Position becomes (-5, -10+8) = (-5, -2). 4. Turns left (West) and walks 10 km: Position becomes (-5-10, -2) = (-15, -2). The final position is (-15, -2) relative to his office (0,0). Since the x-coordinate is negative (-15), he is to the West of his office. Since the y-coordinate is negative (-2), he is to the South of his office. Therefore, he is in the South-West direction from his office.

Solution

Correct: A

In a standard clock: - At 3 o'clock, the hour hand points East and the minute hand points North. - At 9 o'clock, the hour hand points West and the minute hand points North. According to the problem statement: At 3 o'clock, the minute hand points towards the East. In a standard clock, the minute hand points North at 3 o'clock. If North is now East, this implies a 90-degree clockwise rotation of the entire compass. Applying this rotation: - Standard North becomes East. - Standard East becomes South. - Standard South becomes West. - Standard West becomes North. Now, consider 9 o'clock. In a standard clock, the hour hand points towards the West at 9 o'clock. Since, due to the rotation, Standard West has become North, the hour hand will point towards North at 9 o'clock in this rotated setup.

Solution

Correct: A

Let point A be the origin (0,0). 1. Walks 20 meters North: Position becomes (0, 20). 2. Turns left (West) and walks 30 meters: Position becomes (0-30, 20) = (-30, 20). 3. Turns right (North) and walks 10 meters: Position becomes (-30, 20+10) = (-30, 30). 4. Turns right (East) and walks 30 meters: Position becomes (-30+30, 30) = (0, 30). The final position is (0, 30) relative to point A (0,0). This means he is 30 meters directly North of point A. Therefore, he is 30 meters North from point A.

Solution

Correct: B

Let's determine the rotation. We can use cardinal directions in 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°) becomes North (0°/360°). To go from 135° to 0°/360°, you move 135° anti-clockwise or 225° clockwise. (360 - 135 = 225° clockwise; 135 - 0 = 135° anti-clockwise) 2. North-East (45°) becomes West (270°). To go from 45° to 270°, you move 225° clockwise. (270 - 45 = 225° clockwise) Both examples indicate a consistent rotation of 225 degrees clockwise for all directions. Now, we need to find what West (270°) will become after a 225-degree clockwise rotation: 270° + 225° = 495°. Since a circle is 360°, we subtract 360° from 495° to find the equivalent angle: 495° - 360° = 135°. 135° corresponds to South-East. Therefore, West will become South-East.

Solution

Correct: C

Let Point B be the origin (0,0). 1. A is to the East of B: A = (x, 0) where x > 0 (e.g., A = (5,0)). 2. C is to the South of B: C = (0, -y) where y > 0 (e.g., C = (0,-3)). 3. D is to the West of C: D = (-z, -y) where z > 0 (e.g., D = (-2,-3)). So, using example values, A = (5,0) and D = (-2,-3). To find the direction of D with respect to A, we treat A as the new origin. Relative coordinates of D from A = (D_x - A_x, D_y - A_y) = (-z - x, -y - 0) = (-(z+x), -y). Since (z+x) is positive (both z and x are positive distances), the x-coordinate of D relative to A is negative, indicating West. Since y is positive, the y-coordinate of D relative to A is negative, indicating South. Therefore, point D is in the South-West direction with respect to point A.

Solution

Correct: B

Let Ramesh's initial position be the origin (0,0). 1. Walks 20 meters North: Position becomes (0, 20). 2. Turns right (East) and walks 30 meters: Position becomes (0+30, 20) = (30, 20). 3. Turns right (South) and walks 35 meters: Position becomes (30, 20-35) = (30, -15). 4. Turns left (East) and walks 15 meters: Position becomes (30+15, -15) = (45, -15). 5. Turns left (North) and walks 15 meters: Position becomes (45, -15+15) = (45, 0). The final position is (45, 0) relative to his initial position (0,0). The horizontal displacement is 45 meters East, and the vertical displacement is 0 meters. Therefore, the shortest distance between his initial and final positions is 45 meters.

Solution

Correct: B

Let the starting point be the origin (0,0). 1. Walks 10m towards East: Position becomes (10, 0). 2. Takes a right turn (South) and walks 10m: Position becomes (10, 0-10) = (10, -10). 3. Takes a left turn (East) and walks 10m: Position becomes (10+10, -10) = (20, -10). 4. Takes a left turn (North) and walks 10m: Position becomes (20, -10+10) = (20, 0). The final position is (20, 0) relative to the starting point (0,0). The horizontal displacement is 20m East, and the vertical displacement is 0m. Therefore, the minimum distance between his starting and final point is 20 meters.