GATE CS question library

Which one of the following is FALSE?

Consider the directed graph below given. Which one of the following is TRUE?

Let $P$ be quicksort program to sort numbers in ascending order using the first element as the pivot. Let $t_1$ and $t_2$ be the number of comparisons made by P for the inputs $[1 \ 2 \ 3 \ 4 \ 5]$ and $[4 \ 1 \ 5 \ 3 \ 2]$ respectively. Which one of the following holds?

Consider a rooted n node binary tree represented using pointers. The best upper bound on the time required to determine the number of subtrees having exactly $4$ nodes is $O(n^a\log^bn)$. Then the value of $a+10b$ is __________.

Let $G$ be a graph with $n$ vertices and $m$ edges. What is the tightest upper bound on the running time of Depth First Search on $G$, when $G$ is represented as an adjacency matrix?

Consider the following program in C language: #include <stdio.h> main() { int i; int*pi = &i; scanf("%d",pi); printf("%d\n", i+5); } Which one of the following statements is TRUE?

The base (or radix) of the number system such that the following equation holds is____________. $\frac{312}{20} = 13.1$

A machine has a $32\text{-bit}$ architecture, with $1\text{-word}$ long instructions. It has $64$ registers, each of which is $32$ bits long. It needs to support $45$ instructions, which have an immediate operand in addition to two register operands. Assuming that the immediate operand is an unsigned integer, the maximum value of the immediate …

Consider the following Boolean expression for F: $F(P,Q,R,S)= PQ + \bar{P}QR + \bar{P}Q\bar{R}S$ The minimal sum$-$of$-$products form of $F$ is

Let the function $$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3}) \end{vmatrix}$$ where $\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]$ and $f'(\theta)$ denote the …

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4-by-4$ symmetric positive definite matrix is ___________

Consider the following system of equations: $3x + 2y = 1$ $4x + 7z = 1$ $x + y + z = 3$ $x - 2y + 7z = 0$ The number of solutions for this system is ______________

Let $G=(V,E)$ be a directed graph where $V$ is the set of vertices and $E$ the set of edges. Then which one of the following graphs has the same strongly connected components as $G$ ?

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

When a point inside of a tetrahedron (a solid with four triangular surfaces) is connected by straight lines to its corners, how many (new) internal planes are created with these lines?

In a survey, $300$ respondents were asked whether they own a vehicle or not. If yes, they were further asked to mention whether they own a car or scooter or both. Their responses are tabulated below. What percent of respondents do not own a scooter? …

Geneticists say that they are very close to confirming the genetic roots of psychiatric illnesses such as depression and schizophrenia, and consequently, that doctors will be able to eradicate these diseases through early identification and gene therapy. On which of the following assumptions does the statement above rely? Select one:

Round-trip tickets to a tourist destination are eligible for a discount of $10\%$ on the total fare. In addition, groups of $4$ or more get a discount of $5\%$ on the total fare. If the one way single person fare is $Rs$ $100$, a group of $5$ tourists purchasing round-trip tickets will be charged $Rs$ __________

The Palghat Gap (or Palakkad Gap) , a region about $30\;\text{km}$ wide in the southern part of the Western Ghats in India, is lower than the hilly terrain to its north and south. The exact reasons for the formation of this gap are not clear. It results in the neighbouring regions of Tamil Nadu getting more rainfall from the South West monsoon and the …

If $\large\left(z + \dfrac{1}{z}\right)^{2}= 98$, compute $\large \left(z^{2} +\dfrac{1}{z^{2}}\right)$.