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Official PYQ · 1997Q5.4Number Representations (Fixed and Floating Point)MCQ1 mark

GATE CS 1997 Q5.4 — Number Representations (Fixed and Floating Point)

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Given (224)r=(13)r\sqrt{(224)_r} =(13)_r.

The value of the radix rr is:

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Answer: C

To find the value of the radix rr, we convert the given equation into its decimal equivalent.

The given equation is:

(224)r=(13)r\sqrt{(224)_r} = (13)_r

Converting both sides to base 1010:

  • (224)r=2r2+2r+4(224)_r = 2r^2 + 2r + 4
  • (13)r=1⋅r+3=r+3(13)_r = 1 \cdot r + 3 = r + 3

Substituting these back into the equation:

2r2+2r+4=r+3\sqrt{2r^2 + 2r + 4} = r + 3

Squaring both sides to eliminate the square root:

2r2+2r+4=(r+3)22r^2 + 2r + 4 = (r + 3)^22r2+2r+4=r2+6r+92r^2 + 2r + 4 = r^2 + 6r + 9

Rearranging the terms into a standard quadratic equation:

r2−4r−5=0r^2 - 4r - 5 = 0

Factoring the quadratic equation:

(r−5)(r+1)=0(r - 5)(r + 1) = 0

This gives two possible values for rr:

r=5orr=−1r = 5 \quad \text{or} \quad r = -1

Since the radix rr must be a positive integer greater than the largest digit present in the numbers (which is 44), we discard r=−1r = -1.

Thus, the radix is r=5r = 5.

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