A person climbs a hill in a straight path from point ‘O’ on the ground in the direction of north-east and reaches a point ‘A’ after travelling a distance of 5 km. Then, from the point ‘A’ he moves to point ‘B’ in the direction of north-west. Let the distance AB be 12 km. Now, how far is the person away from the starting point ‘O’ ?

(a) 7 km
(b) 13 km ✓ UPSC's answer
(c) 17 km
(d) 11 km

Why the answer is (b)

['- The person travels 5 km from O to A in the North-East direction, which is at a 45-degree angle to both the North and East axes.', '- From A, the person travels 12 km to B in the North-West direction, which is at a 45-degree angle to the North and West axes.', '- The angle between the North-East vector (OA) and the North-West vector (AB) is 90 degrees, forming a right-angled triangle OAB with the right angle at A.', '- Using the Pythagorean theorem, the distance OB is calculated as the square root of (5^2 + 12^2).', '- This results in the square root of (25 + 144), which is the square root of 169, equal to 13 km.', '- Therefore, the person is 13 km away from the starting point O, corresponding to option (b).']

Why the other options are wrong

(a) 7 km: 7 km is the result of incorrectly subtracting the two distances (12 - 5) instead of using the Pythagorean theorem for perpendicular vectors.
(c) 17 km: 17 km is the result of incorrectly adding the two distances (12 + 5) instead of recognizing the right-angled triangle formed by the perpendicular directions.
(d) 11 km: 11 km is an arbitrary value that does not correspond to any valid geometric calculation for the given distances and angles.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.