Raj travelled from a point X straight to Y at a distance of 80 metres. He turned right and walked 50 metres, then again turned right and walked 70 metres. Finally, he turned right and walked 50 metres

Question

Raj travelled from a point X straight to Y at a distance of 80 metres. He turned right and walked 50 metres, then again turned right and walked 70 metres. Finally, he turned right and walked 50 metres. How far is he from the starting point ?

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Correct — B, 10 metres. Put X at the origin of a grid and take the first leg as due north; the direction chosen does not matter, because every turn in the problem is a right turn and the shape is therefore fixed whichever way you start. Raj walks 80 metres north from X to Y, so Y is at (0, 80). He turns right, which now faces him east, and walks 50 metres to (50, 80). He turns right again, now facing south, and walks 70 metres to (50, 10). He turns right a third time, now facing west, and walks 50 metres to (0, 10). His final position is (0, 10) and his starting point is (0, 0), so he is 10 metres away — and the two points are on the same north-south line, so no square root is needed at all. The structure behind the arithmetic is worth seeing, because it makes the answer obvious without any coordinates. Four legs and three right turns trace a rectangle that almost closes. The two side legs are both 50 metres, one east and one west, so they cancel exactly and leave no east-west displacement. The two vertical legs are 80 metres north and 70 metres south, and they do not cancel: the gap between them, 80 minus 70, is 10 metres, and that is the entire answer. Whenever a direction-sense problem gives four legs with three same-hand turns, the distance from the start is simply the difference between the two opposite pairs — here the sideways pair cancels and the vertical pair differs by 10.

Why the other options are wrong — (a) The answer you get if the third leg were 60 metres instead of 70 — that is, the option produced by misreading one figure. It also tempts a candidate who has grasped that the answer is a difference of lengths but subtracts the wrong pair. The two 50-metre legs cancel; the difference that survives is 80 minus 70.  (c) Corresponds to no consistent reading of the moves at all. It is the decoy for a candidate who abandons the trace and estimates a plausible-looking middle value, and its presence is a reminder that on this item type a guess is worse than a blank on a negatively marked paper.  (d) Another arithmetically plausible but unsupported figure — it would arise from treating the sideways legs as unequal, or from subtracting 50 from 80. Since the two 50-metre legs are equal and opposite, they contribute nothing to the final displacement.

Concept: Direction-sense problems are coordinate geometry disguised as a story, and the reliable method is to make the disguise explicit. Fix the starting point at the origin, decide once which way is north, and track the position after each leg as a pair of coordinates. Right turns cycle the facing in the order north, east, south, west and back to north; left turns cycle it the other way. Displacement east and west are added and subtracted along one axis, north and south along the other, and the straight-line distance from the start is the hypotenuse of the two net displacements — the Pythagorean step, which is only needed when both net displacements are non-zero. The commonest configurations are worth recognising on sight: four legs with three same-hand turns give a near-rectangle, so the answer is a difference of two lengths; an L-shaped path with one turn gives a genuine right triangle, where the answer is a square root and the numbers are almost always a 3-4-5 or 5-12-13 triple.

State PCS 2025 Q32 — Science & Technology PYQ with Answer

State PCS 2025 Q32 — Science & Technology PYQ with Answer