Question

For the clock shown in the figure, if \(O^{*} = O\ Q\ S\ Z\ P\ R\ T\) and \(X^{*} = X\ Z\ P\ W\ Y\ O\ Q\), then which one among the given options is most appropriate for \(P^{*}\)?

• A. P\ U\ W\ R\ T\ V\ X
• B. P\ R\ T\ O\ Q\ S\ U
• C. P\ T\ V\ Q\ S\ U\ W
• D. P\ S\ U\ P\ R\ T\ V

Hint:

When sequences are placed around a circle, assign indices to positions and look for a fixed pattern of index offsets. Once decoded for one example, the same offsets usually generate all others.

Answer 1

The Correct Option is B

Step 1: Fix the circular order.
From the figure, letters are placed around the “clock” in this clockwise order: \[ O,\ P,\ Q,\ R,\ S,\ T,\ U,\ V,\ W,\ X,\ Y,\ Z \ (\text{then back to } O). \] Index these positions as \(1,2,\dots,12\) respectively, so \(O\equiv 1\), \(P\equiv 2\), \(\ldots\), \(Z\equiv 12\).

Step 2: Decode the pattern of the \(*\)-operation using \(O^{*}\).
Given: \[ O^{*}=O,\ Q,\ S,\ Z,\ P,\ R,\ T. \] In index form (with \(O\equiv 1\)): \[ 1,\ 3,\ 5,\ 12,\ 2,\ 4,\ 6. \] Hence, relative to the starting index \(k\), the pattern of offsets is: \[ k+\underbrace{(0,\ +2,\ +4,\ -1,\ +1,\ +3,\ +5)}_{\bmod\ 12}. \] (Here \(-1\) corresponds to \(+11 \bmod 12\).)

Step 3: Validate the pattern with \(X^{*}\).
For \(X\equiv 10\): \[ 10+(0,2,4,-1,1,3,5) \equiv 10,12,2,9,11,1,3, \] which maps to \(X,\ Z,\ P,\ W,\ Y,\ O,\ Q\), exactly the given \(X^{*}\). So the offset pattern is confirmed.

Step 4: Apply the pattern to \(P\).
Since \(P\equiv 2\), compute: \[ 2+(0,2,4,-1,1,3,5) \equiv 2,4,6,1,3,5,7. \] Mapping these indices back to letters: \[ 2\to P,\ 4\to R,\ 6\to T,\ 1\to O,\ 3\to Q,\ 5\to S,\ 7\to U. \] Therefore, \[ P^{*}=P,\ R,\ T,\ O,\ Q,\ S,\ U. \] Final Answer:
\[ \boxed{\text{Option (B)}} \]