Number Systems, Binary Arithmetic and Data Representation PYQs for GATE CS

Solve 5+ Number Systems, Binary Arithmetic and Data Representation previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Number Systems and Binary Arithmetic

Chapter Journey

Step 1: Radix Conversion

The foundation of how computers count. Mastering integer conversions across any base.

Step 2: Binary Fractions

Extending the logic to the fractional world. Handling precision and repeating fractions.

Step 3: Two's Complement

The hardware standard for negative numbers. Detecting when arithmetic exceeds limits.

The Core Idea: Radix and Positional Weight

The Anatomy of a Number System

Every system is defined by its radix ( $r$ ), which dictates the available digits ${0, 1, \dots, r - 1}$ .

d_{2}

d_{1}

d_{0}

r^{2}

r^{1}

r^{0}

(N)_{r} = i = 0 \sum n d_{i} r^{i}

Positions are counted from right to left, starting at $i = 0$ .

Number Systems, Binary Arithmetic and Data Representation: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Digital Logic · 2024_Set2 MSQ

Which of the following is/are EQUAL to 224 in radix-5 (i.e., base-5) notation?

A.
64 in radix-10
B.
100 in radix-8
C.
50 in radix-16
D.
121 in radix-7

Correct Answer:

["A","B","D"]

Step-by-Step Solution

Insight: Convert the given Base 5 number to Base 10, then convert each option to Base 10 to check for equality.

Exam route:

$(224)_{5} = 2 (25) + 2 (5) + 4 (1) = 50 + 10 + 4 = 6 4_{10}$ .
Check options in Base 10:

A) $6 4_{10} = 64$ . (Match)

B) $(100)_{8} = 1 (64) = 64$ . (Match)

C) $(50)_{16} = 5 (16) = 80$ . (No match)

D) $(121)_{7} = 1 (49) + 2 (7) + 1 (1) = 49 + 14 + 1 = 64$ . (Match)

Learning route:

This is a multiple-select question testing base equivalence. The most efficient strategy is to anchor everything to Base 10.

Step 1: Convert the target number $(224)_{5}$ to Base 10.

$2 \times 5^{2} + 2 \times 5^{1} + 4 \times 5^{0} = 2 (25) + 10 + 4 = 6 4_{10}$ .

Step 2: Evaluate each option in Base 10.

Option A: $6 4_{10}$ is already in Base 10. Value is 64. (Equal)

Option B: $(100)_{8} = 1 \times 8^{2} = 6 4_{10}$ . (Equal)

Option C: $(50)_{16} = 5 \times 1 6^{1} + 0 = 8 0_{10}$ . (Not equal)

Option D: $(121)_{7} = 1 \times 7^{2} + 2 \times 7^{1} + 1 \times 7^{0} = 49 + 14 + 1 = 6 4_{10}$ . (Equal)

The correct options are A, B, and D.

Question 2 · Digital Logic · 2023 NAT

A particular number is written as 132 in radix-4 representation. The same number in radix-5 representation is __________.

Correct Answer:

110.00

Step-by-Step Solution

Insight: Route the conversion through Base 10 to avoid direct base-4 to base-5 arithmetic errors.

Exam route:

Convert $(132)_{4}$ to Base 10: $1 (16) + 3 (4) + 2 (1) = 30$ .
Convert $3 0_{10}$ to Base 5: $30 \div 5 = 6$ R $0$ ; $6 \div 5 = 1$ R $1$ ; $1 \div 5 = 0$ R $1$ . Read bottom-up: $110$ .

Learning route:

The question asks for a cross-base conversion. The golden rule for arbitrary bases is to always route through Base 10.

Step 1: Evaluate $(132)_{4}$ in Base 10 using positional weights. The weights are $4^{2} = 16$ , $4^{1} = 4$ , $4^{0} = 1$ .

$1 \times 16 + 3 \times 4 + 2 \times 1 = 16 + 12 + 2 = 3 0_{10}$ .

Step 2: Convert $3 0_{10}$ to Base 5 using repeated division.

Divide 30 by 5: quotient 6, remainder 0.

Divide 6 by 5: quotient 1, remainder 1.

Divide 1 by 5: quotient 0, remainder 1.

Reading the remainders from bottom to top gives $(110)_{5}$ .

Verification: $(110)_{5} = 1 (25) + 1 (5) + 0 (1) = 3 0_{10}$ . Matches.

Question 3 · Digital Logic · 2022 MCQ

Let R1 and R2 be two 4-bit registers that store numbers in 2’s complement form. For the operation R1+R2, which one of the following values of R1 and R2 gives an arithmetic overflow?

A.
R1 = 1011 and R2 = 1110
B.
R1 = 1100 and R2 = 1010
C.
R1 = 0011 and R2 = 0100
D.
R1 = 1001 and R2 = 1111

Correct Answer:

Step-by-Step Solution

Insight: Overflow in 2's complement only occurs when adding two numbers of the same sign yields a result with a different sign.

Exam route:

Check the sign bits (MSB) of R1 and R2.

A) 1011 (-), 1110 (-). Same sign. Sum = 11001 $\to$ 1001 (-). No overflow.

B) 1100 (-), 1010 (-). Same sign. Sum = 10110 $\to$ 0110 (+). OVERFLOW.

C) 0011 (+), 0100 (+). Same sign. Sum = 0111 (+). No overflow.

D) 1001 (-), 1111 (-). Same sign. Sum = 11000 $\to$ 1000 (-). No overflow.

Learning route:

In a 4-bit 2's complement system, the range is -8 to +7. Overflow happens if the true mathematical sum exceeds this range.

The golden rule for detection: Overflow can ONLY happen when adding two numbers of the same sign. If the operands have the same sign, but the result's sign bit is different, overflow has occurred.

Let's evaluate the options:

A) R1 = 1011 (-5), R2 = 1110 (-2). Both negative. Sum = -7. Binary: 1011 + 1110 = 11001. Discard carry $\to$ 1001 (-7). Sign matches. Valid.

B) R1 = 1100 (-4), R2 = 1010 (-6). Both negative. True sum = -10 (out of range). Binary: 1100 + 1010 = 10110. Discard carry $\to$ 0110 (+6). Two negatives gave a positive. OVERFLOW.

C) R1 = 0011 (+3), R2 = 0100 (+4). Both positive. Sum = +7. Binary: 0011 + 0100 = 0111 (+7). Sign matches. Valid.

D) R1 = 1001 (-7), R2 = 1111 (-1). Both negative. Sum = -8. Binary: 1001 + 1111 = 11000. Discard carry $\to$ 1000 (-8). Sign matches. Valid.

Question 4 · Digital Logic · 2021_Set2 NAT

If $x$ and $y$ are two decimal digits and $(0.1101)_{2} = (0.8 x y 5)_{10}$ , the decimal value of $x + y$ is __________.

Correct Answer:

3.00

Step-by-Step Solution

Insight: Convert the fully known binary fraction to Base 10, then expand the decimal fraction with unknowns and match coefficients.

Exam route:

$(0.1101)_{2} = 1/2 + 1/4 + 1/16 = 0.5 + 0.25 + 0.0625 = 0.812 5_{10}$ .
$(0.8 x y 5)_{10} = 0.8 + x /100 + y /1000 + 5/10000 = 0.8005 + 0.01 x + 0.001 y$ .
Equate: $0.8125 = 0.8005 + 0.01 x + 0.001 y ⟹ 0.012 = 0.01 x + 0.001 y$ .
Multiply by 1000: $12 = 10 x + y$ . Since $x, y$ are single digits, $x = 1, y = 2$ .
$x + y = 1 + 2 = 3$ .

Learning route:

This is a mixed-base fractional equation. The most efficient strategy is to convert the fully known side to Base 10 first.

Step 1: Convert $(0.1101)_{2}$ to Base 10.

The weights are $2^{- 1} = 0.5$ , $2^{- 2} = 0.25$ , $2^{- 3} = 0.125$ , $2^{- 4} = 0.0625$ .

$1 (0.5) + 1 (0.25) + 0 (0.125) + 1 (0.0625) = 0.812 5_{10}$ .

Step 2: Expand the right side $(0.8 x y 5)_{10}$ using decimal positional weights.

$0.8 + x (1 0^{- 2}) + y (1 0^{- 3}) + 5 (1 0^{- 4}) = 0.8 + 0.01 x + 0.001 y + 0.0005 = 0.8005 + 0.01 x + 0.001 y$ .

Step 3: Equate the two Base 10 values.

$0.8125 = 0.8005 + 0.01 x + 0.001 y$

$0.0120 = 0.01 x + 0.001 y$

Multiply the entire equation by 1000 to clear decimals:

$12 = 10 x + y$

Since $x$ and $y$ are single decimal digits (0-9), the only solution is $x = 1$ and $y = 2$ .

Step 4: Calculate $x + y = 1 + 2 = 3$ .

Question 5 · Digital Logic · 2021_Set1 MCQ

Let the representation of a number in base 3 be 210. What is the hexadecimal representation of the number?

A.
15
B.
21
C.
D2
D.
528

Correct Answer:

Step-by-Step Solution

Insight: Route through Base 10 to convert from Base 3 to Hexadecimal (Base 16).

Exam route:

$(210)_{3} = 2 (9) + 1 (3) + 0 (1) = 2 1_{10}$ .
$21 \div 16 = 1$ R $5$ . Result is $1 5_{16}$ .

Learning route:

The question requires converting a Base 3 number to Hexadecimal. Since 3 and 16 are not powers of the same base, we must use Base 10 as an intermediate.

Step 1: Convert $(210)_{3}$ to Base 10. The positional weights for Base 3 are $3^{2} = 9$ , $3^{1} = 3$ , $3^{0} = 1$ .

$2 \times 9 + 1 \times 3 + 0 \times 1 = 18 + 3 + 0 = 2 1_{10}$ .

Step 2: Convert $2 1_{10}$ to Base 16. Divide 21 by 16. The quotient is 1 and the remainder is 5.

Reading the remainders gives $1 5_{16}$ .

Verification: $(15)_{16} = 1 (16) + 5 (1) = 2 1_{10}$ . Matches.

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Number Systems, Binary Arithmetic and Data Representation PYQs for GATE CS

Chapter Roadmap: Number Systems and Binary Arithmetic

Chapter Journey

The Core Idea: Radix and Positional Weight

The Anatomy of a Number System

Number Systems, Binary Arithmetic and Data Representation: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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