NIMCET 2010 — Computer PYQ

The correct answer is: (d) None

Calculation

1. Apply the Base Property Rule:

In any number system with base $b$, every digit $d$ in the number must satisfy 0 \le d < b.

• For $(12x{)}_3$, the digit $x$ must be less than the base $3$. So, x < 3.

• For $(123{)}_x$, the digits $1,2,$ and $3$ must be less than the base $x$. The largest digit is $3$, so x > 3.

2. Check for Consistency:

• From the first term: $x\in \{0,1,2\}$

• From the second term: x > 3 (i.e., $x\in \{4,5,6,...\}$)

There is no value of $x$ that satisfies both x < 3 and x > 3 simultaneously.

3. Verification via Equations:

Even if we ignore the base rules and solve the equation by converting both sides to decimal (Base 10):

Solving the quadratic equation:

$x=-4$ or $x=3$.

• $x=-4$ is impossible as a base cannot be negative.

• $x=3$ is impossible because in base $3$, we cannot have a digit "3" in the number $(123{)}_x$.

Therefore, no valid value for $x$ exists.