# GATE CS practice set · 28 October 2026

10 original GATE CS & IT questions from GATE Daily, 14 marks, about 18 minutes. Solve them on the page: https://gatedaily.in/daily/2026-10-28/

Scoring follows GATE: a wrong 1-mark MCQ costs 1/3 mark and a wrong 2-mark MCQ costs 2/3. MSQ and NAT have no negative marking and no partial credit (an MSQ is right only with exactly the right options). Skipped scores zero.

Answers and solutions: submit on the page, or through GATE Daily's MCP server (https://gatedaily.in/mcp, tool submit_answers), which scores the set and returns every solution.

### Q.1 · General Aptitude · Algebra and functions · MCQ, 1 mark (choose one option)

Let $r$ be a root of the equation $x^2 - 4x + 1 = 0$.

The value of $r^3 + \dfrac{1}{r^3}$ is

(A) $14$
(B) $52$
(C) $64$
(D) $76$

### Q.2 · Engineering Mathematics · Systems of equations, rank and vector spaces · MSQ, 1 mark (choose every correct option)

Let $M$ be a $4 \times 5$ matrix with real entries. The set of all solutions of $Mx = 0$ is exactly the set of linear combinations $\alpha u + \beta v$ ($\alpha, \beta \in \mathbb{R}$), where

$$u = [1\;\,0\;\,2\;\,0\;\,1]^T, \qquad v = [0\;\,1\;\,1\;\,1\;\,0]^T.$$

Which of the following statements is/are TRUE?

(A) The rank of $M$ is $3$.
(B) The four rows of $M$ are linearly dependent.
(C) For every $b \in \mathbb{R}^4$, the system $Mx = b$ has at least one solution.
(D) The system $M^T y = 0$ has only the solution $y = 0$.

### Q.3 · Discrete Mathematics · Sets and cardinality · MSQ, 1 mark (choose every correct option)

Let $a$ and $b$ be two distinct objects, neither of which is a set, and let

$$A = \{\, a,\; \{b\},\; \{a, b\} \,\}.$$

$\mathcal{P}(A)$ denotes the power set of $A$. Which of the following statements is/are TRUE?

(A) $\{a, \{b\}\} \in \mathcal{P}(A)$
(B) $\{b\} \in \mathcal{P}(A)$
(C) $\{\emptyset, \{a\}\} \subseteq \mathcal{P}(A)$
(D) $\{a, b\} \subseteq A$

### Q.4 · Digital Logic · Flip-flops and sequential circuits · MCQ, 2 marks (choose one option)

A synchronous circuit has three positive-edge-triggered flip-flops $A$, $B$ and $C$ driven by a common clock, wired as follows ($\overline{X}$ is the complement output of flip-flop $X$):

| Flip-flop | Type | Input(s) connected to |
|---|---|---|
| $A$ | JK | $J_A = \overline{C}$, $K_A = B$ |
| $B$ | D | $D_B = A$ |
| $C$ | T | $T_C = B$ |

The JK flip-flop holds for $J=K=0$, resets for $J=0, K=1$, sets for $J=1, K=0$ and toggles for $J=K=1$. Initially $(A, B, C) = (0, 0, 0)$, and the circuit is then clocked indefinitely.

Which ONE of the following values of $(A, B, C)$ can NEVER occur?

(A) $(1, 1, 0)$
(B) $(0, 1, 1)$
(C) $(1, 1, 1)$
(D) $(1, 0, 0)$

### Q.5 · Computer Organization & Architecture · Data-path and control · MSQ, 2 marks (choose every correct option)

A processor runs at $2$ GHz. On a program of $10^9$ instructions its CPI without any memory stalls is $1.5$. Every instruction is fetched once through the instruction cache, whose miss rate is $2\%$. $30\%$ of the instructions are loads or stores, each making exactly one access through the data cache, whose miss rate is $5\%$. Every miss in either cache stalls the processor for the full memory access time of $40$ ns; there are no other stalls.

Which of the following statements is/are TRUE?

(A) The average number of memory-stall cycles per instruction is $2.8$.
(B) The program takes $2.15$ seconds to execute.
(C) Making only the data cache perfect (no data-cache misses, instruction cache unchanged) speeds the program up by a factor greater than $1.5$.
(D) Raising the clock to $4$ GHz, with the CPI without stalls, the miss rates and the $40$ ns memory access time unchanged, makes the program run more than $1.5$ times faster.

### Q.6 · Programming & Data Structures · Arrays · MCQ, 1 mark (choose one option)

Consider the following C program. Assume a 2-D array is stored contiguously in row-major order, so pointer arithmetic that runs past the end of one row continues into the next row.

```c
#include <stdio.h>
int main(void) {
    int a[3][4] = { {2, 5, 8, 1},
                    {7, 3, 6, 4},
                    {9, 0, 11, 10} };
    printf("%d %d", *(*(a + **a) + a[0][3]), *(a[0] + 6));
    return 0;
}
```

What does the program print?

(A) `0 6`
(B) `1 6`
(C) `0 8`
(D) `1 8`

### Q.7 · Algorithms · Searching and selection · MCQ, 1 mark (choose one option)

An unordered array holds $64$ distinct integers. What is the minimum number of comparisons between array elements that suffices, in the worst case, to find both the largest and the second-largest element?

(A) $68$
(B) $69$
(C) $94$
(D) $125$

### Q.8 · Theory of Computation · CFLs, DCFLs and PDAs · MCQ, 1 mark (choose one option)

Let $w^R$ denote the reverse of a string $w$. Which ONE of the following languages is context-free but NOT deterministic context-free?

(A) $\{\, w\,c\,w^R \mid w \in \{a,b\}^* \,\}$
(B) $\{\, w\,w^R \mid w \in \{a,b\}^* \,\}$
(C) $\{\, a^n b^m c^n \mid n, m \ge 0 \,\}$
(D) $\{\, a^n b^n c^n \mid n \ge 0 \,\}$

### Q.9 · Databases · Transactions and serializability · NAT, 2 marks (answer with a number)

Two transactions are given as

$$T_1:\; r_1(A)\; w_1(A)\; r_1(B)\; w_1(B)$$
$$T_2:\; r_2(C)\; w_2(B)\; r_2(A)\; w_2(C)$$

where $r_i(V)$ and $w_i(V)$ denote a read and a write of data item $V$ by $T_i$. A schedule interleaves all eight operations, keeping each transaction's operations in the order shown. The number of such schedules (serial ones included) that are conflict serializable is ________.

### Q.10 · Computer Networks · Congestion control · NAT, 2 marks (answer with a number)

A TCP sender uses a maximum segment size (MSS) of $2$ KB and an initial slow-start threshold of $64$ KB; the receiver's advertised window is large enough never to limit it. Transmission proceeds in rounds of one RTT, and round 1 is sent with a congestion window (cwnd) of $1$ MSS. While cwnd is below the threshold it doubles every round (slow start); once cwnd reaches the threshold it grows by $1$ MSS per round.

All segments of rounds 1 to 4 are acknowledged. During round 5 a segment is lost and the sender detects it by a timeout: it sets the threshold to half of round 5's cwnd and cwnd to $1$ MSS, and round 6 is sent with this cwnd. Every segment from round 6 onwards is acknowledged.

The cwnd (in KB) used in round 12 is ________.

Note: $1\text{K} = 2^{10}$