GATE CS practice set · 26 October 2026

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Q.1Aptitude · Probability and counting

Five different books $P$ , $Q$ , $R$ , $S$ and $T$ are to be placed in a single row on a shelf. In how many of the arrangements is book $P$ not in the leftmost position and book $Q$ not in the rightmost position?

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Q.2Maths · Limits, continuity and differentiability

The value of

x \to 1 lim \frac{x ^{3} - 3 x + 2}{x ^{4} - 2 x ^{2} + 1}

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Q.3Discrete · Matching, colouring and planarity

A matching in an undirected graph is a set of edges in which no two edges share an endpoint; the empty set of edges is also a matching.

Let $C_{9}$ be the cycle on the 9 vertices $v_{1}, v_{2}, \dots, v_{9}$ , with the 9 edges $v_{1} v_{2}, v_{2} v_{3}, \dots, v_{8} v_{9}$ and $v_{9} v_{1}$ .

The number of matchings in $C_{9}$ is ______.

(Answer in integer)

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Q.4Digital · Minimization and K-maps

Consider the 4-variable Boolean function

F (w, x, y, z) = Σ m (0, 2, 5, 7, 8, 10, 13, 15)

where $w$ is the most significant bit and $z$ the least significant bit, and there are no don't-care terms. Which one of the following is the minimal sum-of-products form of $F$ ?

(Here $^{'}$ denotes complement.)

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Q.5COA · Pipelining: timing and speedup

A processor's instruction execution is divided into five stages with these delays:

Stage	IF	ID	EX	MEM	WB
Delay (ps)	200	150	380	290	120

In the pipelined implementation each stage is followed by a pipeline register whose delay of 20 ps adds to the clock period. A designer splits EX into two consecutive stages EX1 and EX2 with delays 200 ps and 180 ps, giving a six-stage pipeline with the same 20 ps register delay per stage.

One instruction is issued per clock cycle and there are no stalls or hazards. The speedup of the six-stage pipeline over the five-stage pipeline when executing 500 independent instructions is ______ (rounded to two decimal places).

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Q.6DS · Binary heaps

The array $A [1..10]$ below is turned into a min-heap in place by the standard bottom-up build-heap procedure: for $i = ⌊ n /2 ⌋$ down to $1$ , sift $A [i]$ down, at each step swapping it with its smaller child as long as that child is smaller than it. The children of $A [i]$ are $A [2 i]$ and $A [2 i + 1]$ .

Index	1	2	3	4	5	6	7	8	9	10
Value	45	12	78	33	9	60	21	5	40	18

Which one of the following is the array $A [1..10]$ after build-heap finishes?

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Q.7Algorithms · Divide and conquer

Match each algorithm in List-I with the design paradigm in List-II that it follows.

List-I	List-II
(P) Bellman-Ford algorithm for single-source shortest paths	(i) Divide and conquer
(Q) Kruskal's algorithm for a minimum spanning tree	(ii) Dynamic programming
(R) Strassen's matrix multiplication	(iii) Greedy method
(S) Solving the N-Queens problem by placing queens one column at a time and undoing a placement when no safe square remains	(iv) Backtracking

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Q.8TOC · Regular expressions and regular languages

Let $L$ be the set of all strings over the alphabet ${a, b}$ , including the empty string $ε$ , that do not contain $bb$ as a substring. Which one of the following regular expressions denotes exactly $L$ ?

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Q.9OS · Paging and virtual memory

A system uses 39-bit virtual addresses, a page size of 4 KB and a three-level page table ( $1 KB = 2^{10}$ bytes, $1 MB = 2^{20}$ bytes). A page-table entry is 8 bytes at every level, and every page table at every level occupies exactly one page.

The outermost (first-level) table is allocated when a process is created. A second-level or third-level table is allocated only if it contains at least one valid entry.

A process has exactly two regions of its virtual address space mapped, and every page in both regions is valid:

a code-and-data region from virtual address 1 MB up to, but not including, 7 MB;
a stack occupying the highest 1 MB of the virtual address space.

The total memory occupied by this process's page tables, over all three levels, is ______ KB.

(Answer in integer)

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Q.10DBMS · Transactions and serializability

Consider the following schedule $S$ of two transactions $T_{1}$ and $T_{2}$ . Here $r_{i} (Z)$ and $w_{i} (Z)$ denote a read and a write of data item $Z$ by $T_{i}$ , and $c_{i}$ denotes the commit of $T_{i}$ . Assume a single-version database in which each read returns the value of the latest preceding write to that item, or its initial value if no write precedes it.

S : r_{1} (X); w_{1} (X); r_{2} (X); r_{1} (Y); w_{1} (Y); c_{1}; r_{2} (Y); w_{2} (X); c_{2}

Which of the following statements about $S$ is/are TRUE?

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Monday’s GATE CS set

Solution MCQ · 1 mark · −⅓ if wrong

Solution MCQ · 1 mark · −⅓ if wrong

Solution NAT · 2 marks · no negative marking

Solution MCQ · 2 marks · −⅔ if wrong

Solution NAT · 2 marks · no negative marking

Solution MCQ · 2 marks · −⅔ if wrong

Solution MCQ · 1 mark · −⅓ if wrong

Solution MCQ · 1 mark · −⅓ if wrong

Solution NAT · 2 marks · no negative marking

Solution MSQ · 1 mark · no negative marking

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