NIMCET 2008 — Computer PYQ

Step 1: Convert base $r$ numbers to decimal (base 10)

The positional value of a number in base $r$ is calculated by multiplying each digit by $r^n$, where $n$ is the position starting from 0.

• $(224{)}_r=2\cdot r^2+2\cdot r^1+4\cdot r^0=2r^2+2r+4$

• $(13{)}_r=1\cdot r^1+3\cdot r^0=r+3$

Step 2: Set up the equation

Substitute these decimal equivalents back into the original equation:

Step 3: Solve for $r$

Square both sides of the equation to remove the square root:

Expand the right side using the identity $(a+b{)}^2=a^2+2ab+b^2$:

Bring all terms to one side to form a quadratic equation:

Step 4: Factor the quadratic equation

Find two numbers that multiply to $-5$ and add to $-4$. These are $-5$ and $+1$:

This gives two possible values for $r$:

• $r=5$

• $r=-1$

Step 5: Verify the valid radix

A radix must be a positive integer. Furthermore, the digits used in the number $(224{)}_r$ (specifically the digit 4) require that r > 4.

• $r=5$ satisfies these conditions.

• $r=-1$ is mathematically impossible for a base.

Final Answer:

The correct option is (d) 5.