GATE CS Engineering Mathematics questions

If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?

Suppose the predicate $F(x, y, t)$ is used to represent the statement that person $x$ can fool person $y$ at time $t$. Which one of the statements below expresses best the meaning of the formula, $\qquad∀x∃y∃t(¬F(x,y,t))$

Consider the following matrix $$A = \left[\begin{array}{cc}2 & 3\\x & y \end{array}\right]$$ If the eigenvalues of A are $4$ and $8$, then

The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph? $7, 6, 5, 4, 4, 3, 2, 1$ $6, 6, 6, 6, 3, 3, 2, 2$ $7, 6, 6, 4, 4, 3, 2, 2$ $8, 7, 7, 6, 4, 2, 1, 1$

What is the probability that divisor of $10^{99}$ is a multiple of $10^{96}$?

Consider a company that assembles computers. The probability of a faulty assembly of any computer is $p$. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of $q$. What is the probability of a computer being declared faulty?

What is the value of $\displaystyle\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}$ ?

Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ forms

What is the possible number of reflexive relations on a set of $5$ elements?

Let $G=(V, E)$ be a graph. Define $\xi(G) = \sum\limits_d i_d*d$, where $i_d$ is the number of vertices of degree $d$ in $G.$ If $S$ and $T$ are two different trees with $\xi(S) = \xi(T)$, then

Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices?

Consider the following well-formed formulae: $\neg \forall x(P(x))$ $\neg \exists x(P(x))$ $\neg \exists x(\neg P(x))$ $\exists x(\neg P(x))$ Which of the above are equivalent?

The binary operation $\Box$ is defined as follows $$\begin{array}{|c|c|c|} \hline \textbf{P} & \textbf{Q} & \textbf{P} \Box \textbf{Q}\\ \hline \text{T} & \text{T}& \text{T}\\ \hline \text{T} & \text{F}& \text{T} \\ \hline \text{F} & \text{T}& \text{F}\\ \hline \text{F} & \text{F}& \text{T} \\ \hline \end{array}$$ …

Which one of the following is the most appropriate logical formula to represent the statement? "Gold and silver ornaments are precious". The following notations are used: $G(x): x$ is a gold ornament $S(x): x$ is a silver ornament $P(x): x$ is precious

For the composition table of a cyclic group shown below: …

An unbalanced dice (with $6$ faces, numbered from $1$ to $6$) is thrown. The probability that the face value is odd is $90\%$ of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than $3$ is $0.75$, which one of the …

Consider the binary relation $R = \left\{(x,y), (x,z), (z,x), (z,y)\right\}$ on the set $\{x,y,z\}$. Which one of the following is TRUE?

What is the chromatic number of an $n$ vertex simple connected graph which does not contain any odd length cycle? Assume $n > 2$.

Which one of the following is NOT necessarily a property of a Group?