GATE CS 2026 Set 2 Q.44: Machine Instructions and Addressing Modes

Consider a processor that has 16 general purpose registers and it uses 2-byte instruction format for all its instructions. Variable-sized opcodes are permitted. There are three different types of instructions; M-type, R-type, and C-type. Each M-type instruction has 2 register operands and a 6-bit immediate operand. Each R-type instruction has 3 register operands. Each C-type instruction has a register operand and a 6-bit offset value. If there are 2 unique M-type opcodes and 7 unique R-type opcodes, which one of the following options gives the maximum number of unique opcodes possible for C-type instructions?

$8$

$4$

$64$

$16$

Official answer and solution

Official answer: B
From: Official GATE 2026 answer key

Solution

Compute the bit encodings consumed by M-type and R-type instructions out of the $2^{16}$ total 16-bit instruction patterns and divide the remaining space by the operand bit-space of a C-type instruction.

With 16 registers, each register operand requires $\log_2(16) = 4$ bits, and the total instruction length is $2 \text{ bytes} = 16$ bits:

M-type: 2 registers ( $2 \times 4 = 8$ bits) + 6-bit immediate $= 14$ operand bits. Each M-type opcode consumes $2^{14}$ bit patterns.
R-type: 3 registers ( $3 \times 4 = 12$ bits). Each R-type opcode consumes $2^{12}$ bit patterns.
C-type: 1 register ( $4$ bits) + 6-bit offset $= 10$ operand bits. Each C-type opcode consumes $2^{10}$ bit patterns.

Total bit patterns used by 2 M-type and 7 R-type opcodes:

2 \times 2^{14} + 7 \times 2^{12} = 8 \times 2^{12} + 7 \times 2^{12} = 15 \times 2^{12}

Remaining bit patterns out of $2^{16} = 16 \times 2^{12}$ :

16 \times 2^{12} - 15 \times 2^{12} = 1 \times 2^{12} = 4 \times 2^{10}

Therefore, at most $\frac{4 \times 2^{10}}{2^{10}} = 4$ unique C-type opcodes can be supported.

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 2 (CS2)
Question: 44 of 65
Section: Computer Science and Information Technology

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