UGC NET Questions (Paper – 1)

The numeral 1010 in base b expands as 1×b^3 + 0×b^2 + 1×b + 0. Setting this equal to 10 gives b^3 + b = 10. If we try b = 2, we get 2^3 + 2 = 8 + 2 = 10, which satisfies the equation. Therefore, the base must be 2, so 1010₂ equals 10₁₀.

Option A:
Option A, base 8, would make 1010 equal to 1×8^3 + 1×8 = 512 + 8 = 520. This is much larger than 10, so base 8 is not correct. The equality 1010 = 10 does not hold in base 8.

Option B:
Option B, base 2, gives the expansion 1×2^3 + 1×2 = 8 + 2. This sum equals 10, exactly matching the required decimal value. Hence, base 2 is the correct base for which 1010 represents ten.

Option C:
Option C, base 4, yields 1×4^3 + 1×4 = 64 + 4 = 68. This differs greatly from 10, so base 4 does not give the desired equality. Therefore, this option cannot be correct.

Option D:
Option D, base 5, makes 1010 equal to 1×5^3 + 1×5 = 125 + 5 = 130. Since 130 is not equal to 10, base 5 also fails to satisfy the condition.

The decimal system is the standard system used in everyday counting and arithmetic. It employs ten distinct digit symbols from 0 to 9. Because the number of symbols in a positional system equals its base, the base of the decimal system is 10. All place values are powers of 10 such as 1, 10, 100 and 1000.

Option A:
Option A suggests base 2, which is used for binary and involves only digits 0 and 1. This is not consistent with decimal, which uses more symbols. Therefore base 2 cannot be correct.

Option B:
Option B proposes base 4, which would require exactly four distinct digits, often 0 to 3. Since decimal requires ten, base 4 does not match its structure.

Option C:
Option C indicates base 8, which corresponds to the octal system with digits 0–7. It fails to accommodate the digits 8 and 9 that appear in decimal notation.

Option D:
Option D is correct because decimal uses ten symbols and is defined as a base-10 positional system. Every position represents a power of 10, confirming that 10 is its base.

In a positional system with radix r and k digits, the smallest value is all zeros and the largest is all digits equal to r−1. The numeric value of k digits all set to r−1 is (r−1)(1 + r + r² + … + r^(k−1)), which sums to (r−1)(r^k − 1)/(r−1) = r^k − 1. This is the highest value attainable without increasing the number of digits.

Option A:
Option A precisely captures the algebraic result of a geometric series of place values. It recognises that with k digits, the largest magnitude is one less than the next power of the base, r^k. Therefore the formula r^k − 1 is logically consistent with positional representation.

Option B:
Option B, r^(k−1) − 1, only considers up to the (k−1)-th power and would correspond to the maximum value of a (k−1)-digit number instead. It underestimates the representable range for k digits and thus is incorrect.

Option C:
Option C, r^k, corresponds to the value requiring k+1 digits in general, because r^k is written as 1 followed by k zeros in base r. It therefore exceeds the maximum value that fits in exactly k digits.

Option D:
Option D, r^k + 1, extends the overestimation even further and represents a number that is not just larger but two units beyond the next power of r. Such a value cannot be expressed using only k digits in that base.

A positional number system whose base is 3 uses three distinct symbols, typically 0, 1 and 2. Such a base-3 system is known as the ternary system. The prefix “ter” or “tern” is related to the Latin for three, which aligns with the base value. Therefore, the correct term for a base-3 system is ternary.

Option A:
Option A, dual, commonly refers to something involving two parts but is not the standard mathematical term for base 2. Even if loosely related to two, it does not accurately describe a base-3 system.

Option B:
Option B, quinary, refers to a base-5 system that uses five symbols, usually 0 through 4. This is not suitable for a number system that uses only three symbols.

Option C:
Option C is correct because ternary explicitly denotes a system built on the number three. In a ternary number system, each position represents a power of 3. This matches the requirement of a base-3 system.

Option D:
Option D, senary, is used for base 6, where six different digit symbols are employed. Since base 3 uses only three digits, senary is not the correct name.

In 345, the digit 5 is in the units place, 4 in the tens place and 3 in the hundreds place. The tens place corresponds to 10^1, so the value of 4 there is 4×10^1. Therefore, the exponent for the tens place is 1.

Option A:
Option A, exponent 0, refers to the units place where the digit 5 resides. Assigning 10^0 to the tens place would ignore the role of tens in the number.

Option B:
Option B, exponent 2, gives 10^2 = 100, which corresponds to the hundreds place occupied by digit 3. It does not describe the tens place.

Option C:
Option C, exponent 3, would mean a place value of 1000, which is not present in the three-digit number 345. This exponent is not related to the digit 4 in the tens position.

Option D:
Option D is correct because the tens place always carries the weight of 10^1 in decimal positional notation. Thus, 4 in that position contributes 40 to the total value.

The radix of a number system is the base with respect to which positional weights are defined. It determines the number of distinct digit symbols available and the powers used for place values. For example, binary has radix 2 and decimal has radix 10.

Option A:
Option A correctly states that radix and base are synonymous in the context of number systems. Knowing the radix tells us both the set of allowed digits and the multipliers for each position.

Option B:
Option B, maximum digit, is related but not identical because the maximum digit is always radix − 1, not the radix itself. Thus, it conflates two distinct concepts.

Option C:
Option C, number of digits in a number, varies from numeral to numeral and is unrelated to the fixed property of the system’s base.

Option D:
Option D, exponent, refers to the power applied to the base in specific positional weights, but is not the radix itself.

The binary number system is defined as a positional system that uses only two symbols, 0 and 1, so its base must be 2. In any positional number system, the base equals the number of distinct digit symbols available. All place values in binary, such as 1, 2, 4, 8 and so on, are powers of 2. Therefore, the only correct base for the binary system among the options is 2.

Option A:
Option A is correct because binary has exactly two symbols and therefore a base of 2. Every binary position corresponds to 2 raised to some power, which shows that the base is 2. Without base 2, the structure of binary place values would not work correctly.

Option B:
Option B refers to base 8, which is used in the octal system, not in binary. Octal digits range from 0 to 7 and thus require eight different symbols. Therefore base 8 cannot describe a system that has only two digits, 0 and 1.

Option C:
Option C refers to base 10, which belongs to the decimal system, the usual system for everyday counting. Decimal uses ten symbols from 0 to 9, unlike binary which uses only 0 and 1. Hence 10 cannot be the base of the binary system.

Option D:
Option D refers to base 16, which is associated with the hexadecimal system. Hexadecimal uses symbols 0–9 and A–F to represent sixteen distinct values. This makes it unsuitable as a base for binary, which has only two symbols.

With a fixed number of digit positions, a higher base allows each position to hold more distinct values. This expands the total number of combinations, thereby increasing the range of representable values. For example, three digits in base 2 yield 8 values, while three digits in base 10 yield 1000 values. Hence, increasing the base increases the representable range.

Option A:
Option A is correct because the total number of distinct patterns is base^digits, so raising the base while holding digits constant raises this power. This mathematical relationship directly implies an increased range.

Option B:
Option B, decreases, is the opposite of what actually happens. Decreasing the base reduces the number of choices per position and thus shrinks the range.

Option C:
Option C, keeps unchanged, would only be true if either the base or the number of digits did not change. Changing the base without altering digits affects the range.

Option D:
Option D, first increases then decreases, does not apply here because the formula base^digits grows monotonically as the base increases for a fixed positive number of digits.

The base of a positional number system is technically called its radix. Radix denotes how many distinct symbols are used, such as 2 for binary or 10 for decimal. All positional weights are powers of the radix, which is why this term is central in number system discussions. Therefore, radix is the correct technical name for the base.

Option A:
Option A is correct because radix directly refers to the base of a number system and determines the positional weights. When we say base 10 or base 2, we are specifying the radix. In many mathematical texts, the words base and radix are used interchangeably, confirming that radix is the appropriate term.

Option B:
Option B, exponent, refers to the power to which a base is raised in an expression like 2^3. It describes the position in a power expression, not the base itself. Hence, exponent cannot be used as a synonym for base of the number system.

Option C:
Option C, mantissa, is a term used in logarithms and floating-point representation to denote the significant digits. It does not describe how many symbols a number system uses or what its base is. Therefore, mantissa is not an appropriate replacement for the term base.

Option D:
Option D, modulus, generally refers to the divisor in modular arithmetic or the absolute value of a number. While related to number theory, modulus does not mean the same thing as base or radix in positional number systems. So it is not the correct choice here.

Binary numbers are expressed in base 2, so each position corresponds to a power of 2. To convert a binary number to decimal, we multiply each bit by the appropriate power of 2 and add these products. This process accounts for each positional contribution. Therefore, we use powers of 2 in the conversion.

Option A:
Option A, 8, would only be appropriate if we were converting from octal, where the base is 8. Using powers of 8 for binary would give incorrect results.

Option B:
Option B, 10, pertains to decimal positional representation. While the result is expressed in decimal, the positional weights for binary digits must still be powers of 2.

Option C:
Option C, 16, corresponds to hexadecimal, a base-16 system. Hex digits are weighted by powers of 16, not appropriate for direct binary conversion.

Option D:
Option D is correct because binary uses base 2 and each position is 2^n for some integer n. Converting by multiplying bits by powers of 2 is the standard method used in computing.

In a positional number system, the numeric value of each digit is determined by its position relative to other digits and the base of the system. The same digit symbol can represent different values depending on where it is placed in the number. For example, in decimal, the digit 2 in 20 and 200 has different values because of their positions. Therefore, position is the key factor governing the digit's value.

Option A:
Option A refers to font style, which is only a presentation attribute and does not change the numeric value. Whether a number is written in bold or italics, its quantitative meaning is unchanged. Hence font style has no role in determining numerical value.

Option B:
Option B mentions shape, but the geometric style of a digit symbol does not affect its numeric interpretation. Even if the typography changes slightly, the underlying digit remains the same. Thus shape is not the basis for place value in positional systems.

Option C:
Option C is correct because positional systems assign place values based on the position of digits relative to the base. Each position corresponds to a power of the base, and the digit multiplies that positional weight. Without positional dependence, we would not distinguish between units, tens and hundreds.

Option D:
Option D refers to colour, which is not a mathematical property of numbers. Changing the colour of a digit has no effect on its value or its role in calculations. Therefore colour cannot determine the value of a digit in any number system.

In a positional decimal system, the rightmost digit is in the units place. Its positional weight is 10^0, which equals 1. Multiplying the digit by 10^0 leaves its value unchanged, reflecting that it counts units. Hence the exponent for the rightmost digit is 0.

Option A:
Option A is correct because any number raised to the power 0 equals 1, and the units place must have a weight of 1. This matches how we interpret the rightmost digit in any decimal number.

Option B:
Option B, exponent 2, would give a weight of 10^2 = 100, which belongs to the hundreds place, not the units place. Assigning 10^2 to the rightmost digit would distort the positional structure.

Option C:
Option C, exponent 3, yields 10^3 = 1000, corresponding to the thousands place. This is far from the unit's role and clearly inappropriate.

Option D:
Option D, exponent 1, represents the tens place, where each digit counts groups of ten. The rightmost digit is not in this position, so exponent 1 is not correct for it.

By definition, a positional number system with base r uses exactly r distinct digit symbols. These symbols typically range from 0 to r - 1. The base describes how many unique values a single digit can take. Therefore, r symbols are necessary and sufficient to represent numbers in that base.

Option A:
Option A is correct because the base and the number of symbols are conceptually linked. In base 10 we use 10 symbols, and in base 2 we use 2 symbols. This pattern generalises to base r requiring r symbols.

Option B:
Option B suggests r - 1 symbols, which would only cover values from 0 to r - 2. This would leave one value unrepresented and break the completeness of the system.

Option C:
Option C proposes r + 1 symbols, which would introduce redundancy beyond what is required by the base. Having more symbols than the base allows would complicate representations unnecessarily.

Option D:
Option D indicates 2r symbols, again providing many more symbols than required by the base. This would not align with standard definitions of positional number systems.

In binary, each digit position corresponds to a power of 2, reflecting the base of the system. The rightmost bit is 2^0, the next is 2^1, then 2^2 and so on. This structure allows binary to represent any integer using only the digits 0 and 1. Therefore, each position represents a power of 2.

Option A:
Option A suggests powers of 8, which would be correct in an octal system, not in binary. Octal uses base 8, whereas binary is strictly base 2.

Option B:
Option B indicates powers of 10, which define the decimal system. Decimal numbers are based on 10 distinct digits, unlike binary.

Option C:
Option C is correct because binary's base is 2 and all positional weights are powers of 2. This is fundamental to how digital systems interpret bit patterns.

Option D:
Option D, 16, is the base of the hexadecimal system where each position is a power of 16. That system uses digits and letters to represent values.

In a positional system, the same digit can represent different values depending on where it appears. Its weight is determined by its position relative to the radix point. Each position corresponds to a specific power of the base. Therefore, the weight of a digit depends primarily on its position in the numeral.

Option A:
Option A, value, is the digit itself, but weight is not determined by digit alone; position controls the power of the base used.

Option B:
Option B is correct because each position has a place value (base^0, base^1, base^2, etc.), which determines the weight.

Option C:
Option C, symbol, is another way to refer to the digit; it does not determine the positional weight by itself.

Option D:
Option D, base, affects the system globally, but the specific weight of a digit in a numeral is determined directly by its position (as a power of the base).