GATE CS question library

Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z | x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z | w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ denotes that $’a \text{ divides } b’$ , where $a$ and $b$ are integers. Consider the following sets: …

There are $n$ unsorted arrays: $A_1, A_2, \dots, A_n$. Assume that $n$ is odd.Each of $A_1, A_2, \dots, A_n$ contains $n$ distinct elements. There are no common elements between any two arrays. The worst-case time complexity of computing the median of the medians of $A_1, A_2, \dots , A_n$ is

Consider the following grammar and the semantic actions to support the inherited type declaration attributes. Let $X_1, X_2, X_3, X_4, X_5$, and $X_6$ be the placeholders for the non-terminals $D, T, L$ or $L_1$ in the following table: …

Let $G$ be any connected, weighted, undirected graph. $G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight. $G$ has a unique minimum spanning tree, if, for every cut of $G$, there is a unique minimum-weight edge crossing the cut. Which of the following statements is/are TRUE?

Consider the following snapshot of a system running $n$ concurrent processes. Process $i$ is holding $X_i$ instances of a resource $R$, $1 \leq i \leq n$. Assume that all instances of $R$ are currently in use. Further, for all $i$, process $i$ can place a request for at most $Y_i$ additional instances of $R$ while holding the $X_i$ …

Consider the following statements: The smallest element in a max-heap is always at a leaf node The second largest element in a max-heap is always a child of a root node A max-heap can be constructed from a binary search tree in $\Theta(n)$ time A binary search tree can be constructed from a max-heap in $\Theta(n)$ time Which of the above statements are …

Consider the following four processes with arrival times (in milliseconds) and their length of CPU bursts (in milliseconds) as shown below: …

Consider the augmented grammar given below: $S’ \rightarrow S$ $S \rightarrow \langle L \rangle \mid id$ $L \rightarrow L, S \mid S$ Let $I_0 = \text{CLOSURE} (\{[S’ \rightarrow \cdot S ]\}).$ The number of items in the set $\text{GOTO} (I_0, \langle \: )$ is______

The index node (inode) of a Unix -like file system has $12$ direct, one single-indirect and one double-indirect pointers. The disk block size is $4$ kB, and the disk block address is $32$-bits long. The maximum possible file size is (rounded off to $1$ decimal place) ____ GB

Consider the following matrix: $R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}$ The absolute value of the product of Eigen values of $R$ is _______

A certain processor deploys a single-level cache. The cache block size is $8$ words and the word size is $4$ bytes. The memory system uses a $60$-MHz clock. To service a cache miss, the memory controller first takes $1$ cycle to accept the starting address of the block, it then takes $3$ cycles to fetch all the eight words of the block, and finally …

Let $T$ be a full binary tree with $8$ leaves. (A full binary tree has every level full.) Suppose two leaves $a$ and $b$ of $T$ are chosen uniformly and independently at random. The expected value of the distance between $a$ and $b$ in $T$ (ie., the number of edges in the unique path between $a$ and $b$) is (rounded off to $2$ decimal …

Suppose $Y$ is distributed uniformly in the open interval $(1,6)$. The probability that the polynomial $3x^2 +6xY+3Y+6$ has only real roots is (rounded off to $1$ decimal place) _______

Consider that $15$ machines need to be connected in a LAN using $8$-port Ethernet switches. Assume that these switches do not have any separate uplink ports. The minimum number of switches needed is ______

Let $\Sigma$ be the set of all bijections from $\{1, \dots , 5\}$ to $\{1, \dots , 5 \}$, where $id$ denotes the identity function, i.e. $id(j)=j, \forall j$. Let $\circ$ denote composition on functions. For a string $x=x_1 x_2 \dots x_n \in \Sigma^n, n \geq 0$, let $\pi(x)=x_1 \circ x_2 \circ \dots \circ x_n$. Consider the language …

What is the minimum number of $2$-input NOR gates required to implement a $4$ -variable function expressed in sum-of-minterms form as $f=\Sigma(0,2,5,7, 8, 10, 13, 15)?$ Assume that all the inputs and their complements are available. Answer: _______

A relational database contains two tables Student and Performance as shown below: …

Consider the following C program: #include <stdio.h> int main() { float sum = 0.0, j=1.0, i=2.0; while (i/j > 0.0625) { j=j+j; sum=sum+i/j; printf("%f\n", sum); } return 0; } The number of times the variable sum will be printed, when the above program is executed, is _________

Consider the following C program: #include <stdio.h> int main() { int a[] = {2, 4, 6, 8, 10}; int i, sum=0, *b=a+4; for (i=0; i<5; i++) sum=sum+(*b-i)-*(b-i); printf("%d\n", sum); return 0; } The output of the above C program is _______

In an RSA cryptosystem, the value of the public modulus parameter $n$ is $3007$. If it is also known as that $\phi(n)=2880$ where $\phi()$ denotes Euler’s Totient Function, then the prime factor of $n$ which is greater than $50$ is _________