Rahul leaves his home and first walks 5 km towards the East.\nHe then turns in the South-East direction and walks 10 km, after which he turns in the North-East direction and walks another 10 km.\nFinally, he moves straight towards the North for 10 km.\nIn which direction is he now located from his starting point (home)?

Introduction / Context:
This is a more advanced direction sense question because it involves movements not only along the main North–South–East–West directions but also along diagonal directions, namely South-East and North-East. We must determine the final direction of Rahul from his starting point after several such moves. To solve it accurately, we need to interpret diagonal movements as equal components along two perpendicular axes and then combine all horizontal and vertical displacements to identify the final quadrant in which Rahul lies relative to his home.

Given Data / Assumptions:

• Rahul starts from his home, which we treat as the origin.
• He walks 5 km towards the East initially.
• He then turns towards the South-East direction and walks 10 km.
• Next, he turns towards the North-East direction and walks another 10 km.
• Finally, he moves 10 km straight towards the North.
• South-East is taken as a diagonal direction equally inclined between South and East, and North-East is equally inclined between North and East.
• We assume ideal 45 degree diagonal directions so that each diagonal movement has equal North–South and East–West components.

Concept / Approach:
To handle diagonal movements, each 10 km step in a 45 degree direction is broken into two perpendicular components using basic trigonometry. For a 45 degree diagonal, the horizontal and vertical components are equal and can be taken as 10 / √2 km each. We then treat all movements as vectors, sum the East–West components and the North–South components separately, and examine the sign and relative size of the net components. If both net components are positive (East and North), the final direction will be North-East, and similarly for other quadrants.

Step-by-Step Solution:
Step 1: Place home at (0, 0). After walking 5 km East, Rahul is at (5, 0). Step 2: Walking 10 km South-East means moving 10 / √2 km East and 10 / √2 km South. Numerically, 10 / √2 is approximately 7.07 km. Step 3: From (5, 0), add East component +7.07 and South component −7.07 to get approximately (12.07, −7.07). Step 4: Walking 10 km North-East from this point adds +7.07 km East and +7.07 km North. So the new position is about (12.07 + 7.07, −7.07 + 7.07) = (19.14, 0.00). Step 5: Finally, walking 10 km North adds 10 km to the y-coordinate, giving a final position of roughly (19.14, 10). Step 6: At the end, Rahul's x-coordinate is positive (East) and his y-coordinate is positive (North), which clearly places him in the North-East direction relative to the starting point.

Verification / Alternative check:
Even without heavy calculation, we can reason qualitatively. Rahul first moves East, then diagonally South-East (adding East and South components), then diagonally North-East (adding East and North components that cancel the earlier South component), and finally moves North. The Southward and Northward diagonal components cancel approximately, leaving an overall strong Eastward displacement and a final Northward component from the last move. Hence, his net displacement is to the East and North of the origin, which must correspond to North-East. A rough sketch using equal diagonals at 45 degrees will visually confirm this conclusion.

Why Other Options Are Wrong:
West is impossible because Rahul moves predominantly East or diagonal with an East component in every stage; there is no move with a net West component. East alone ignores the clear Northward movement at the end that lifts him above the starting level. North-West would require a West component, which does not exist here. South-East would place him below the starting point, but the final 10 km North move makes his net vertical displacement positive (North), not negative (South). Thus only North-East matches both the qualitative and quantitative analysis.

Common Pitfalls:
A common error is to treat diagonal movements as if they occur entirely in one direction, for example, counting 10 km South-East as 10 km South and 10 km East simultaneously, which overestimates the displacement. Another mistake is to focus on the last movement alone and ignore the cumulative effect of earlier diagonals on the final position. When dealing with such questions, it is helpful to break diagonals into components or at least to sketch approximate 45 degree lines to see in which quadrant the endpoint lies.

Final Answer:
Rahul is finally located in the North-East direction from his starting point (home).