Four of the following five are alike in a certain way and thus form a group. Find the one which doesn’t belong to the group?

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 answer is P,X.

To solve this problem, we need to map the movements of Rohan and Karan to determine the positions of all points on a Cartesian coordinate plane. Let us set an arbitrary coordinate for the starting point of one of the paths, say W, and trace the relative coordinates of all other points.

1. Rohan's Path:
Let point W be at the origin: $W=(0,0)$.
- Rohan starts walking from W in the East direction for 3m to reach P.
East is along the positive x-axis:
$P=(0+3,0)=(3,0)$.
- From P, he takes a right turn (which is South, negative y-axis) and walks 2m to reach V:
$V=(3,0-2)=(3,-2)$.
- From V, he takes a left turn (which is East, positive x-axis) and walks 5m to reach S:
$S=(3+5,-2)=(8,-2)$.
- From S, he takes a left turn (which is North, positive y-axis) and walks 7m to reach X:
$X=(8,-2+7)=(8,5)$.

2. Karan's Path:
We know Karan's path ends at point X, which has coordinates $(8,5)$. Let's work backwards from X to find the coordinates of T, R, U, and Q.
- From T, Karan takes a left turn and walks 4m to reach X. Since we are tracing backwards, we need to reverse the directions.
Let's trace Karan's path forwards from Q first using variable coordinates $Q=(x_q,y_q)$:
- Q to U: South direction, walks 5m.
$U=(x_q,y_q-5)$.
- U to R: From U (facing South), he takes a right turn (which is West, negative x-axis) and walks 6m.
$R=(x_q-6,y_q-5)$.
- R to T: From R (facing West), he takes another right turn (which is North, positive y-axis) and walks 12m.
$T=(x_q-6,y_q-5+12)=(x_q-6,y_q+7)$.
- T to X: From T (facing North), he takes a left turn (which is West, negative x-axis) and walks 4m to reach X.
$X=(x_q-6-4,y_q+7)=(x_q-10,y_q+7)$.

Since we know $X=(8,5)$:
$x_q-10=8\Rightarrow x_q=18$
$y_q+7=5\Rightarrow y_q=-2$
So, the coordinates of the points in Karan's path are:
- $Q=(18,-2)$
- $U=(18,-7)$
- $R=(12,-7)$
- $T=(12,5)$
- $X=(8,5)$

3. Analyze the Direction Patterns for Each Option:
We need to find the relative direction of the second point with respect to the first point for each option:
- W,V: W is $(0,0)$ and V is $(3,-2)$. V is to the East and South of W. Thus, V is in the South-East direction of W.
- X,R: X is $(8,5)$ and R is $(12,-7)$. R is to the East (12 > 8) and South (-7 < 5) of X. Thus, R is in the South-East direction of X.
- P,X: P is $(3,0)$ and X is $(8,5)$. X is to the East (8 > 3) and North (5 > 0) of P. Thus, X is in the North-East direction of P.
- X,U: X is $(8,5)$ and U is $(18,-7)$. U is to the East (18 > 8) and South (-7 < 5) of X. Thus, U is in the South-East direction of X.

In all options except P,X, the second point lies in the South-East direction of the first point. In P,X, the second point lies in the North-East direction of the first point. Therefore, P,X does not belong to the group.