Which among the following statement(s) is/are true?

I. Total distance covered by Karan is more than the total distance covered by Rohan.

II. Point W is in the west of point Q

III. The shortest distance between point T and point S is $\sqrt{65}$m

Study the following information carefully and answer the questions given below:

Rohan starts walking from point W in the east direction after walking 3m he reaches at point P. From there he takes a right turn and walks 2m to reach at point V. From point V he takes a left turn and walks 5m to reach at point S. From point S he takes left turn and walks point 7m to reach at point X. Karan starts walking from point Q in south direction after walking 5m he reaches at point U. From there he takes a right turn and walks 6m to reach at point R. From there he takes again right turn and walks 12m to reach at point T. Now, from point T he takes a left turn and walks 4m to reach at point X.

The correct option is Both I and III.

Let us analyze the paths taken by Rohan and Karan step-by-step to verify the statements.

1. Rohan's Path:
- Starts at point W.
- Walks 3m East to reach point P. (W → P = 3m)
- Turns right (South) and walks 2m to reach point V. (P → V = 2m)
- Turns left (East) and walks 5m to reach point S. (V → S = 5m)
- Turns left (North) and walks 7m to reach point X. (S → X = 7m)
Total distance covered by Rohan = 3m + 2m + 5m + 7m = 17m.

2. Karan's Path:
- Starts at point Q.
- Walks 5m South to reach point U. (Q → U = 5m)
- Turns right (West) and walks 6m to reach point R. (U → R = 6m)
- Turns right (North) and walks 12m to reach point T. (R → T = 12m)
- Turns left (West) and walks 4m to reach point X. (T → X = 4m)
Total distance covered by Karan = 5m + 6m + 12m + 4m = 27m.

3. Evaluating the Statements:

Statement I: Total distance covered by Karan is more than the total distance covered by Rohan.
Karan's distance (27m) is greater than Rohan's distance (17m). Therefore, Statement I is true.

Statement II: Point W is in the west of point Q.
Let us find the coordinates of the points by setting the final common point X as the origin (0, 0).
Working backward for Rohan from X(0, 0):
- X is 7m North of S, so S is at (0, -7).
- S is 5m East of V, so V is at (-5, -7).
- V is 2m South of P, so P is at (-5, -5).
- P is 3m East of W, so W is at (-8, -5).
Working backward for Karan from X(0, 0):
- X is 4m West of T, so T is at (4, 0).
- T is 12m North of R, so R is at (4, -12).
- R is 6m West of U, so U is at (10, -12).
- U is 5m South of Q, so Q is at (10, -7).
Comparing W(-8, -5) and Q(10, -7):
W is to the North-West of Q, not strictly in the West. Therefore, Statement II is false.

Statement III: The shortest distance between point T and point S is $\sqrt{65}$m.
Coordinates of T are (4, 0) and coordinates of S are (0, -7).
Using the distance formula:
Distance = $\sqrt{{\left(4-0\right)}^2+{\left(0-\left(-7\right)\right)}^2}$
Distance = $\sqrt{4^2+7^2}$
Distance = $\sqrt{16+49}$ = $\sqrt{65}$m.
Therefore, Statement III is true.

Since Statements I and III are true, the correct option is "Both I and III".