UGC NET Questions (Paper – 1)

Many decimal fractions cannot be represented exactly in binary using a finite number of bits. The fraction 0.1₁₀ is one such example; its binary expansion repeats endlessly, similar to how 1/3 is repeating in decimal. This leads to an infinite repeating pattern like 0.0001100110011…₂. Hence, 0.1 in decimal is a non-terminating repeating binary fraction in general.

Option A:
Option A captures the key property that 0.1₁₀ does not have a finite binary representation. Instead, its binary expansion is repeating, which causes rounding errors in digital computations. This conceptual understanding is crucial for numerical analysis and floating-point representation.

Option B:
Option B, terminating finite, is incorrect because only fractions whose denominators factor into powers of 2 can terminate in binary; 0.1 has denominator 10, which introduces a factor of 5. Therefore, it cannot terminate in base 2.

Option C:
Option C, claiming an exact pattern of 0.0001₂, would correspond to 1/16 = 0.0625 in decimal, which is significantly different from 0.1. This indicates a misunderstanding of positional values.

Option D:
Option D, stating 0.1₂ with two bits, misinterprets notation, as 0.1₂ equals 1/2 = 0.5 in decimal. It therefore confuses the base notation and equates two different quantities.

To convert 345 to binary, we repeatedly divide by 2 and record remainders. This process yields the bit pattern 101011001 when read from last remainder to first. We can verify by expanding 101011001 as 2^8 + 2^6 + 2^4 + 2^3 + 2^0. This equals 256 + 64 + 16 + 8 + 1 = 345, confirming that 101011001 is correct.

Option A:
Option A, 100101001, corresponds to a different decimal value. If expanded, it sums to 256 + 32 + 8 + 1 = 297. Since 297 is not equal to 345, this binary representation is not correct for the given decimal number.

Option B:
Option B, 101011001, represents 256 + 64 + 16 + 8 + 1 = 345. This matches the target decimal value exactly. Both the division method and place value expansion confirm this match. Therefore, this is the correct 9-bit binary representation of 345.

Option C:
Option C, 101001011, gives yet another decimal value. Its expansion is 256 + 64 + 8 + 2 + 1 = 331. Because 331 differs from 345, this option cannot be the correct binary encoding for 345.

Option D:
Option D, 100011001, expands to 256 + 16 + 8 + 1 = 281. This again fails to equal 345. As a result, this binary pattern does not represent the required decimal number and must be rejected.