GATE CS question library

Let $\text{fsa}$ and $\text{pda}$ be two predicates such that $\text{fsa}(x)$ means $x$ is a finite state automaton and $\text{pda}(y)$ means that $y$ is a pushdown automaton. Let $\text{equivalent}$ be another predicate such that $\text{equivalent} (a,b)$ means $a$ and $b$ are equivalent. Which of the following first order logic …

Let $X$ be a random variable following normal distribution with mean $+1$ and variance $4$. Let $Y$ be another normal variable with mean $-1$ and variance unknown. If $P (X \leq -1) = P (Y \geq 2)$ , the standard deviation of $Y$ is

How many of the following matrices have an eigenvalue 1? $\left[\begin{array}{cc}1 & 0 \\0 & 0 \end{array} \right]\left[\begin{array}{cc}0 & 1 \\0 & 0 \end{array} \right] \left[\begin{array}{cc}1 & -1 \\1 & 1 \end{array} \right]$ and $\left[\begin{array}{cc}-1 & 0 \\1 & -1 \end{array} \right]$

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that she studies mathematics the next day is $0.6$. If she studies mathematics on a day, then the probability that she studies computer science the next day is $0.4$. Given that Aishwarya studies computer science on Monday, …

If $P, Q, R$ are Boolean variables, then $(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$ simplifies to

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve $3x^4-16x^3+24x^2+37$ is

Which of the following statements is true for every planar graph on $n$ vertices?

Let $P =\sum \limits_ {i\;\text{odd}}^{1\le i \le 2k} i$ and $Q = \sum\limits_{i\;\text{even}}^{1 \le i \le 2k} i$, where $k$ is a positive integer. Then

The Newton-Raphson iteration $x_{n+1} = \frac{1}{2}\left(x_n+\frac{R}{x_n}\right)$ can be used to compute the

The minimum number of equal length subintervals needed to approximate $\int_1^2 xe^x\,dx$ to an accuracy of at least $\frac{1}{3}\times10^{-6}$ using the trapezoidal rule is

The data blocks of a very large file in the Unix file system are allocated using

The Breadth First Search algorithm has been implemented using the queue data structure. One possible order of visiting the nodes of the following graph is:

Which combination of the integer variables $x, y,$ and $z$ makes the variable $a$ get the value $4$ in the following expression? $$a=(x > y)?((x > z) ?x:z): ((y > z) ?y:z)$$

Which of the following system calls results in the sending of SYN packets?

A clustering index is defined on the fields which are of type

Which of the following tuple relational calculus expression(s) is/are equivalent to $\forall t \in r \left(P\left(t\right)\right)$? $\neg \exists t \in r \left(P\left(t\right)\right)$ $\exists t \notin r \left(P\left(t\right)\right)$ $\neg \exists t \in r \left(\neg P\left(t\right)\right)$ $\exists t \notin r \left(\neg P\left(t\right)\right)$

What is the maximum size of data that the application layer can pass on to the TCP layer below?

If $L$ and $\overline{L}$ are recursively enumerable then $L$ is

Some code optimizations are carried out on the intermediate code because

Which of the following are decidable? Whether the intersection of two regular languages is infinite Whether a given context-free language is regular Whether two push-down automata accept the same language Whether a given grammar is context-free