GATE CS question library

Consider the following matrix.$$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$$The largest eigenvalue of the above matrix is __________.

A five-stage pipeline has stage delays of $150, 120, 150, 160$ and $140$ nanoseconds. The registers that are used between the pipeline stages have a delay of $5$ nanoseconds each. The total time to execute $100$ independent instructions on this pipeline, assuming there are no pipeline stalls, is _______ nanoseconds.

Consider the following instruction sequence where registers $\text{R1}, \text{R2}$ and $\text{R3}$ are general purpose and $\text{MEMORY[X]}$ denotes the content at the memory location $\text{X}.$ …

A sender $(\textsf{S})$ transmits a signal, which can be one of the two kinds: $H$ and $L$ with probabilities $0.1$ and $0.9$ respectively, to a receiver $(\textsf{R})$. In the graph below, the weight of edge $(u,v)$ is the probability of receiving $v$ when $u$ is transmitted, where $u,v \in \{H,L\}$. For example, the probability that the …

Gauri said that she can play the keyboard __________ her sister.

A transparent square sheet shown above is folded along the dotted line. The folded sheet will look like ___________.

If $\theta$ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, $\cos \theta =$

$\text{Pen : Write :: Knife :}$ _______ Which one of the following options maintains a similar logical relation in the above?

Listening to music during exercise improves performance and reduces discomfort. Scientists researched whether listening to music while studying can help students learn better and the results were inconclusive. Students who needed external stimulation for studying fared worse while students who did not need any external stimulation benefited from music. Which …

​​​​​​If $\left( x – \dfrac{1}{2} \right)^2 – \left( x- \dfrac{3}{2} \right) ^2 = x+2$, then the value of $x$ is:

A jigsaw puzzle has $2$ pieces. One of the pieces is shown above. Which one of the given options for the missing piece when assembled will form a rectangle? The piece can be moved, rotated or flipped to assemble with the above piece.

The number of students in three classes is in the ratio $3:13:6$. If $18$ students are added to each class, the ratio changes to $15:35:21$. The total number of students in all the three classes in the beginning was:

The number of units of a product sold in three different years and the respective net profits are presented in the figure above. The cost/unit in Year $3$ was ₹$\;1$, which was half the cost/unit in Year $2$. The cost/unit in Year $3$ was one-third of the cost/unit in Year $1$. Taxes were paid on the selling price at $10 \%$, $13 \%$, and …

Six students $\text{P, Q, R, S, T and U}$, with distinct heights, compare their heights and make the following observations. $\text{Observation I}$: $\text{S}$ is taller than $\text{R}$. $\text{Observation II}$: $\text{Q}$ is the shortest of all. $\text{Observation III}$: $\text{U}$ is taller than only one student. $\text{Observation IV}$: …

Let $G$ be a connected undirected weighted graph. Consider the following two statements. $S_1$: There exists a minimum weight edge in $G$ which is present in every minimum spanning tree of $G$. $S_2$: If every edge in $G$ has distinct weight, then $G$ has a unique minimum spanning tree. Which one of the following options is correct?

​​​​​Let $H$ be a binary min-heap consisting of $n$ elements implemented as an array. What is the worst case time complexity of an optimal algorithm to find the maximum element in $H$?

The format of the single-precision floating point representation of a real number as per the $\text{IEEE 754}$ standard is as follows: $$\begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}$$ Which one of the following choices is correct with respect to the smallest normalized positive number …

Consider the following $\text{ANSI C}$ program: int main () { Integer x; return 0; } Which one of the following phases in a seven-phase $C$ compiler will throw an error?

Consider the following statements $S1$ and $S2$ about the relational data model: $S1$: A relation scheme can have at most one foreign key. $S2$: A foreign key in a relation scheme $R$ cannot be used to refer to tuples of $R.$ Which one of the following choices is correct?

Which one of the following circuits implements the Boolean function given below? $f(x,y,z) = m_0+m_1+m_3+m_4+m_5+m_6$, where $m_i$ is the $i^{\text{th}}$ minterm.