In which direction is Karan’s initial position with respect to Rohan’s final position? 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.

To determine the direction of Karan's initial position with respect to Rohan's final position, we can trace the movements of both individuals using a Cartesian coordinate system. Let us define the starting point of Rohan, point W, as the origin $(0,0)$, where the positive x-axis represents East, the negative x-axis represents West, the positive y-axis represents North, and the negative y-axis represents South.

Step 1: Trace Rohan's path to find his final position (point X)
1. Rohan starts at point W $=(0,0)$.
2. He walks 3m East to reach point P:
$P=(3,0)$
3. From point P, he takes a right turn (which is South) and walks 2m to reach point V:
$V=(3,-2)$
4. From point V, he takes a left turn (which is East) and walks 5m to reach point S:
$S=(3+5,-2)=(8,-2)$
5. From point S, he takes a left turn (which is North) and walks 7m to reach point X (Rohan's final position):
$X=(8,-2+7)=(8,5)$

Step 2: Trace Karan's path to find his initial position (point Q)
Let Karan's initial position, point Q, be represented by the coordinates $(x_q,y_q)$.
1. Karan starts walking from point Q in the South direction for 5m to reach point U:
$U=(x_q,y_q-5)$
2. From U, he takes a right turn (which is West) and walks 6m to reach point R:
$R=(x_q-6,y_q-5)$
3. From R, he takes a right turn (which is North) and walks 12m to reach point T:
$T=(x_q-6,y_q-5+12)=(x_q-6,y_q+7)$
4. From T, he takes a left turn (which is West) and walks 4m to reach point X:
$X=(x_q-6-4,y_q+7)=(x_q-10,y_q+7)$

Since we already established that Rohan's final position point X is $(8,5)$, we can equate the coordinates to find Karan's initial position Q:
$x_q-10=8\Rightarrow x_q=18$
$y_q+7=5\Rightarrow y_q=-2$
Thus, Karan's initial position is $Q=(18,-2)$.

Step 3: Find the direction of Q with respect to X
We compare the coordinates of Q $(18,-2)$ with respect to X $(8,5)$:
- The x-coordinate of Q (18) is greater than the x-coordinate of X (8), which means Q is to the East of X.
- The y-coordinate of Q (-2) is less than the y-coordinate of X (5), which means Q is to the South of X.

Combining these directions, Karan’s initial position Q is in the South-east direction with respect to Rohan’s final position X.